Intermediate mathematics/All

Science is a wonderful thing if one does not have to earn one's living at it. One should earn one's living by work of which one is sure one is capable. Only when we do not have to be accountable to anybody can we find joy in scientific endeavor. -Albert Einstein

This article is a continuation of Introductory mathematics

It has been known since the time of Euclid^w that all of geometry can be derived from a handful of objects (points, lines...), a few actions on those objects, and a small number of axioms^w. Every field of science likewise can be reduced to a small set of objects, actions, and rules. Math itself is not a single field but rather a constellation of related fields. One way in which new fields are created is by the process of generalization.

A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole.

The purpose of this article is threefold:

• To give a broad general overview of the various fields and subfields of mathematics.
• To show how each field can be derived from first principles.
• To provide links to articles and webpages with more in depth information.

Foreword:

Mathematical notation^w can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible.

The following has been assembled from countless small pieces gathered from throughout the world wide web. I cant guarantee that there are no errors in it. Please report any errors or omissions on this articles talk page.

Numbers

Scalars

See also: Peano axioms^w, ^*Hyperoperation, ^*Algebraic extension

The basis of all of mathematics is the ^*"Next" function. See Graph theory^w.

Next(0)=1

Next(1)=2

Next(2)=3

Next(3)=4

Next(4)=5

We might express this by saying that One differs from nothing as two differs from one. This defines the Natural numbers^w (denoted $\mathbb{N}_0$). Natural numbers are those used for counting.

These have the convenient property of being transitive^w. That means that if a<b and b<c then it follows that a<c. In fact they are totally ordered^w. See ^*Order theory.

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Integers

Addition^w (See Tutorial:arithmetic) is defined as repeatedly calling the Next function, and its inverse is subtraction^w. But this leads to the ability to write equations like $1-3=x$ for which there is no answer among natural numbers. To provide an answer mathematicians generalize to the set of all integers^w (denoted $\mathbb{Z}$ because zahlen means count in german) which includes negative integers.

The Additive identity^w is zero because x + 0 = x.

The absolute value or modulus of x is defined as $|x| = \left\{ \begin{array}{rl} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0. \end{array}\right.$

^*Integers form a ring (denoted $\mathcal O_\mathbb{Q}$) over the field of rational numbers. Ring^w is defined below.

Z_n or $\mathbb{Z}/n\mathbb{Z}$ is used to denote the set of ^*integers modulo n .

^*Modular arithmetic is essentially arithmetic in the quotient ring^w Z/nZ (which has n elements).

Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z / 2Z has only two elements, zero for the even numbers and one for the odd numbers; applying the definition, [z] = z + 2Z := {z + 2y: 2y ∈ 2Z}, where 2Z is the ideal of even numbers. It is naturally isomorphic to the finite field with two elements, F₂. Intuitively: if you think of all the even numbers as 0, then every integer is either 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1).

An ^*ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.

A ^*principal ideal is an ideal $I$ in a ring $R$ that is generated by a single element $a$ of $R$ through multiplication by every element of $R$.

A ^*prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number.

The study of integers is called Number theory^w.

$a \mid b$ means a divides b.

$a \nmid b$ means a does not divide b.

$p^a \mid\mid n$ means p^a exactly divides n (i.e. p^a divides n but p^a+1 does not).

A prime number is a number that can only be divided by itself and one.

If a, b, c, and d are primes and x=abc and y=c²d then:

xy = lcm^w * gcd^w = abc²d * c

Two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. Any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.

(See Tutorial:least common multiples)

The ^*Ulam spiral. Black pixels = ^*prime numbers.

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Rational numbers

Multiplication^w (See Tutorial:multiplication) is defined as repeated addition, and its inverse is division^w. But this leads to equations like $3/2=x$ for which there is no answer. The solution is to generalize to the set of rational numbers^w (denoted $\mathbb{Q}$) which include fractions (See Tutorial:fractions). Any number which isnt rational is irrational^w. See also ^*p-adic number

The set of all rational numbers except zero forms a ^*multiplicative group which is a set of invertible elements.

Rational numbers form a ^*division algebra because every non-zero element has an inverse. The ability to find the inverse of every element turns out to be quite useful. A great deal of time and effort has been spent trying to find division algebras.

Rational numbers form a Field^w. A field is a set on which addition, subtraction, multiplication, and division are defined. See below.

The Multiplicative identity^w is one because x * 1 = x.

Division by zero is undefined and undefinable^w. 1/0 exists nowhere on the complex plane^w. It does, however, exist on the Riemann sphere^w (often called the extended complex plane) where it is surprisingly well behaved. See also ^*Wheel theory and L'Hôpital's rule^w.

$\frac{1}{nothing}=everything$

(Addition and multiplication are fast but division is slow ^*even for computers.)

Binary multiplication

From Wikipedia:Binary number 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 The binary numbers 101 and 110 are multiplied as follows: 1 0 1 (5 in decimal) × 1 1 0 (6 in decimal) -------- 0 0 0 + 1 0 1 + 1 0 1 ------------ = 1 1 1 1 0 (30 in binary) Binary numbers can also be multiplied with bits after a *binary point: 1 0 1 . 1 0 1 (5.625 in decimal) × 1 1 0 . 0 1 (6.25 in decimal) ------------------- 1 . 0 1 1 0 1 + 0 0 . 0 0 0 0 + 0 0 0 . 0 0 0 + 1 0 1 1 . 0 1 + 1 0 1 1 0 . 1 --------------------------- = 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal) From Wikipedia:Power of two 21 = 2 22 = 4 24 = 16 28 = 256 216 = 65,536 232 = 4,294,967,296 264 = 18,446,744,073,709,551,616 (20 digits) 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 (39 digits) Our universe is tiny. Starting with only 2 people and doubling the population every 100 years will in only 27,000 years result in enough people to completely fill the observable universe.

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Irrational and complex numbers

Exponentiation^w (See Tutorial:exponents) is defined as repeated multiplication, and its inverses are roots^w and logarithms^w. But this leads to multiple equations with no solutions:

Equations like $\sqrt{2}=x.$ The solution is to generalize to the set of algebraic numbers^w (denoted $\mathbb{A}$). (See also ^*algebraic integer and algebraically closed.) To see a proof that the square root of two is irrational see Square root of 2^w.

Equations like $2^{\sqrt{2}}=x$ The solution (because x is transcendental^w) is to generalize to the set of Real numbers^w (denoted $\R$).

Equations like $\sqrt{-1}=x$ and $e^x=-1.$ The solution is to generalize to the set of complex numbers^w (denoted $\mathbb{C}$) by defining i = sqrt(-1). A single complex number $z=a+bi$ consists of a real part a and an imaginary part bi (See Tutorial:complex numbers). Imaginary numbers^w (denoted $\mathbb{I}$) often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the Spacetime interval^w (given below), ^*time itself is an imaginary quantity.

The Complex conjugate^w of the complex number $z=a+bi$ is $\overline{z}=a-bi.$ (Not to be confused with the dual^w of a vector.)

$\frac{1}{a+bi} = \frac{1}{a+bi} \cdot \frac{a-bi}{a-bi} = \frac{a-bi}{a^2+b^2}$

Complex numbers form an ^*Algebra over a field (K-algebra) because complex multiplication is ^*Bilinear.

$\sqrt{-100} * \sqrt{-100} = 10i * 10i = -100 \neq \sqrt{-100 * -100}$

The complex numbers are not ordered^w. However the absolute value^w or ^*modulus of a complex number is:

$|z| = |a + ib| = \sqrt{a^2+b^2}$

A Gaussian integer a + bi is a Gaussian prime if and only if either:

• one of a, b is zero and absolute value of the other is a prime number of the form 4n + 3 (with n a nonnegative integer), or
• both are nonzero and a² + b² is a prime number (which will not be of the form 4n + 3).

There are n solutions of $\sqrt[n]{z}$

0^0 = 1. See Empty product^w.

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

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Hypercomplex numbers

Complex numbers can be used to represent and perform rotations^w but only in 2 dimensions. Hypercomplex numbers^w like quaternions^w (denoted $\mathbb{H}$), octonions^w (denoted $\mathbb{O}$), and ^*sedenions (denoted $\mathbb{S}$) are one way to generalize complex numbers to some (but not all) higher dimensions.

A quaternion can be thought of as a complex number whose coefficients are themselves complex numbers (hence a hypercomplex number).

$(a + b\boldsymbol{\hat{\imath}}) + (c + d\boldsymbol{\hat{\imath}})\boldsymbol{\hat{\jmath}} = a + b\boldsymbol{\hat{\imath}} + c\boldsymbol{\hat{\jmath}} + d\boldsymbol{\hat{\imath}\hat{\jmath}} = a + b\boldsymbol{\hat{\imath}} + c\boldsymbol{\hat{\jmath}} + d\boldsymbol{\hat{k}}$

Where

$\boldsymbol{\hat{\imath}}^2 = \boldsymbol{\hat{\jmath}}^2 = \boldsymbol{\hat{k}}^2 = \boldsymbol{\hat{\imath}} \boldsymbol{\hat{\jmath}} \boldsymbol{\hat{k}} = -1$

and

$\begin{alignat}{2} \boldsymbol{\hat{\imath}}\boldsymbol{\hat{\jmath}} & = \boldsymbol{\hat{k}}, & \qquad \boldsymbol{\hat{\jmath}}\boldsymbol{\hat{\imath}} & = -\boldsymbol{\hat{k}}, \\ \boldsymbol{\hat{\jmath}}\boldsymbol{\hat{k}} & = \boldsymbol{\hat{\imath}}, & \boldsymbol{\hat{k}}\boldsymbol{\hat{\jmath}} & = -\boldsymbol{\hat{\imath}}, \\ \boldsymbol{\hat{k}}\boldsymbol{\hat{\imath}} & = \boldsymbol{\hat{\jmath}}, & \boldsymbol{\hat{\imath}}\boldsymbol{\hat{k}} & = -\boldsymbol{\hat{\jmath}}. \end{alignat}$

Any real finite-dimensional ^*division algebra over the reals must be:

• isomorphic to R or C if ^*unitary and commutative (equivalently: associative and commutative)
• isomorphic to the quaternions if noncommutative but associative
• isomorphic to the octonions if non-associative but alternative.

The following is known about the dimension of a finite-dimensional division algebra A over a field K:

• dim A = 1 if K is algebraically closed,
• dim A = 1, 2, 4 or 8 if K is ^*real closed, and
• If K is neither algebraically nor real closed, then there are infinitely many dimensions in which there exist division algebras over K.

^*Split-complex numbers (hyperbolic complex numbers) are similar to complex numbers except that i² = +1.

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Tetration

Tetration^w is defined as repeated exponentiation and its inverses are called super-root and super-logarithm.

$\begin{matrix} {}^{b}a & = & \underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} & = & a\uparrow\uparrow b = & \underbrace{a\uparrow (a\uparrow(\dots\uparrow a))} & \\ & & b\mbox{ copies of }a & & & b\mbox{ copies of }a \end{matrix}$

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Hyperreal numbers

See also: ^*Non-standard calculus

When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If $q$ is a finite $( q \cdot 1 )$ amount of charge then using Leibniz's notation^w $dq$ would be an infinitesimal $( q \cdot 1/\infty )$ amount of charge. See Differential^w

Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite $( M \cdot \infty )$. But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite $( M \cdot \infty^2 )$.

Infinity and the infinitesimal are called Hyperreal numbers^w (denoted ${}^*\mathbb{R}$). Hyperreals behave, in every way, exactly like real numbers. For example, $2 \cdot \infty$ is exactly twice as big as $\infty.$ In reality, the mass of the ocean is a real number so it is hardly surprising that it behaves like one. See ^*Epsilon numbers and ^*Big O notation

In ancient times infinity was called the "all".

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Groups and rings

Main articles: Algebraic structure^w, Abstract algebra^w, and ^*group theory

Addition and multiplication can be generalized in so many ways that mathematicians have created a whole system just to categorize them.

Any straight line through the origin forms a group. Adding any 2 points on the line results in a 3rd point that is also on the line.

A ^*magma is a set with a single ^*closed binary operation (usually, ^*but not always, addition. See ^*Additive group).

a + b = c

A ^*semigroup is a magma where the addition is associative. See also ^*Semigroupoid

a + (b + c) = (a + b) + c

A ^*monoid is a semigroup with an additive identity element.

a + 0 = a

A ^*group is a monoid with additive inverse elements.

a + (-a) = 0

An ^*abelian group is a group where the addition is commutative.

a + b = b + a

A ^*pseudo-ring is an abelian group that also has a second closed, associative, binary operation (usually, but not always, multiplication).

a * (b * c) = (a * b) * c

And these two operations satisfy a distribution law.

a(b + c) = ab + ac

A ^*ring is a pseudo-ring that has a multiplicative identity

a * 1 = a

A ^*commutative ring is a ring where multiplication commutes, (e.g. ^*integers)

a * b = b * a

A ^*field is a commutative ring where every element has a multiplicative inverse (and thus there is a multiplicative identity),

a * (1/a) = 1

The existence of a multiplicative inverse for every nonzero element automatically implies that there are no ^*zero divisors in a field

if ab=0 for some a≠0, then we must have b=0 (we call this having no zero-divisors).

$\mathbb{Z}/n\mathbb{Z}$ is the ^*quotient ring of $\mathbb{Z}$ by the ideal $n\mathbb{Z}$ containing all integers divisible by n.

Thus $\mathbb{Z}/n\mathbb{Z}$ is a field when $n\mathbb{Z}$ is a ^*maximal ideal, that is, when n is prime.

The ^*center of a ^*group is the commutative subgroup of elements c such that c+x = x+c for every x. See also: ^*Centralizer and normalizer.

The ^*center of a ^*noncommutative ring is the commutative subring of elements c such that cx = xc for every x.

The ^*characteristic of ring R, denoted char(R), is the number of times one must add the ^*multiplicative identity to get the ^*additive identity.

The circle of center 0 and radius 1 in the complex plane is a Lie group with complex multiplication.

A ^*Lie group is a group that is also a smooth differentiable manifold, in which the group operation is multiplication rather than addition. (Differentiation requires the ability to multiply and divide which is usually impossible with most groups.)

All non-zero ^*nilpotent elements are ^*zero divisors.

The square matrix^w $A = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}$ is nilpotent

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Numbers dont lie. (But they sure help)

From Wikipedia:Mathematical fallacy:

$\begin{align} a &= b \\ a^2 &= ab \\ a^2 - b^2 &= ab - b^2 \\ (a - b)(a + b) &= b(a - b) \\ a + b &= b \\ b + b &= b \\ 2b &= b \\ 2 &= 1 \end{align}$

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a = b. Since division by zero is undefined, the argument is invalid.

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Intervals

[-2,5[ or [-2,5) denotes the interval^w from -2 to 5, including -2 but excluding 5.

[3..7] denotes all integers from 3 to 7.

The set of all reals is unbounded at both ends.

An open interval does not include its endpoints.

^*Compactness is a property that generalizes the notion of a subset being closed and bounded.

The ^*unit interval is the closed interval [0,1]. It is often denoted I.

The ^*unit square is a square whose sides have length 1.

Often, "the" unit square refers specifically to the square in the Cartesian plane^w with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).

The ^*unit disk in the complex plane is the set of all complex numbers of absolute value less than one and is often denoted $\mathbb {D}$

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Vectors

See also: ^*Algebraic geometry, ^*Algebraic variety, ^*Scheme, ^*Algebraic manifold, and Linear algebra^w

The one dimensional number line can be generalized to a multidimensional Cartesian coordinate system^w thereby creating multidimensional math (i.e. geometry^w). See also ^*Curvilinear coordinates

For sets A and B, the Cartesian product^w A × B is the set of all ordered pairs^w (a, b) where a ∈ A and b ∈ B.

$\mathbb R^{3}$ is the Cartesian product $\mathbb{R} \times \mathbb{R} \times \mathbb{R}.$

$\mathbb{R}^\infty = \mathbb{R}^\mathbb{N}$

$\mathbb{C}^3$ is the Cartesian product^w $\mathbb{C} \times \mathbb{C} \times \mathbb{C}$ (See ^*Complexification)

The *direct product generalizes the Cartesian product.

(See also *Direct sum) If we think of $\R$ as the *set of real numbers, then the direct product $\R \times \R$ is precisely just the Cartesian product. If we think of $\R$ as the *group of real numbers under addition, then the direct product $\R \times \R$ still consists of the Cartesian product. The difference is that $\R \times \R$ is now a group. We have to also say how to add their elements. To add ordered pairs, we define the sum $(a, b) + (c, d)$ to be $(a + c, b + d)$. However, if we think of $\R$ as the *field of real numbers, then the direct product $\R \times \R$ does not exist – naively defining $\{ (x,y) | x,y \in \mathbb{R} \}$ in a similar manner to the above examples would not result in a field since the element $(1,0)$ does not have a multiplicative inverse. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the *direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression $xy$) we use direct product. In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product.

A vector space^w is a coordinate space^w with vector addition^w and scalar multiplication^w (multiplication of a vector and a scalar^w belonging to a field^w).

i, j, and k are basis vectors
a = a_xi + a_yj + a_zk

If ${\mathbf e_1} , {\mathbf e_2} , {\mathbf e_3}$ are orthogonal^w unit^{w *}basis vectors

and ${\mathbf u} , {\mathbf v} , {\mathbf x}$ are arbitrary vectors

and are $u_n , v_n , x_n$ scalars belonging to a field then we can (and usually do) write:

$\mathbf{u} = u_1 \mathbf{e_1} + u_2 \mathbf{e_2} + u_3 \mathbf{e_3} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix}$

$\mathbf{v} = v_1 \mathbf{e_1} + v_2 \mathbf{e_2} + v_3 \mathbf{e_3} = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix}$

$\mathbf{x} = x_1 \mathbf{e_1} + x_2 \mathbf{e_2} + x_3 \mathbf{e_3} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}$

See also: Linear independence^w

A ^*module generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a ring^w.

Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the "norm^w" which works for all vectors. The norm of vector $\mathbf{v}$ is denoted $\|\mathbf{v}\|.$ The double bars are used to avoid confusion with the absolute value of the function.

Taxicab metric^w (called L¹ norm. See ^*L^p space. Sometimes called Lebesgue spaces. See also Lebesgue measure^w.) A circle in L¹ space is shaped like a diamond.

$\|\mathbf{v}\| = v_1 + v_2 + v_3$

c² = (a+b)² - 4ab/2
c² = a² + b²

In Euclidean space^w the norm (called L² norm) doesnt depend on the choice of coordinate system. As a result, rigid objects can rotate in Euclidean space. See proof of the Pythagorean theorem^w to the right. L² is the only ^*Hilbert space among L^p spaces.

$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$

In Minkowski space^w (See ^*Pseudo-Euclidean space) the Spacetime interval^w is

$\|s\| = \sqrt{x^2 + y^2 + z^2 + (cti)^2}$

In ^*complex space the most common norm of an n dimensional vector is obtained by treating it as though it were a regular real valued 2n dimensional vector in Euclidean space

$\left\| \boldsymbol{z} \right\| = \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}$

Infinity norm. (In this space a circle is shaped like a square.)

$\left\| \mathbf{x} \right\| _\infty := \max \left( \left| x_1 \right| , \ldots , \left| x_n \right| \right) .$

A ^*Banach space is a ^*normed vector space that is also a complete metric space^w (there are no points missing from it).

Manifolds

Tangent bundle of a circle A manifoldw $\mathbf{M}$ is a type of topological spacew in which each point has an infinitely small neighbourhoodw that is homeomorphicw to Euclidean spacew. A manifold is locally, but not globally, Euclidean. A *Riemannian metric on a manifold allows distances and angles to be measured. A *Tangent space $\mathbf{T}_p \mathbf{M}$ is the set of all vectors tangent to $\mathbf{M}$ at point p. Informally, a *tangent bundle $\mathbf{TM}$ (red cylinder in image to the right) on a differentiable manifold $\mathbf{M}$ (blue circle) is obtained by joining all the *tangent spaces (red lines) together in a smooth and non-overlapping manner. The tangent bundle always has twice as many dimensions as the original manifold. A *vector bundle is the same thing minus the requirement that it be tangent. A *fiber bundle is the same thing minus the requirement that the fibers be vector spaces. The *cotangent bundle (*Dual bundle) of a differentiable manifold is obtained by joining all the *cotangent spaces (pseudovector spaces). The cotangent bundle always has twice as many dimensions as the original manifold. Sections of that bundle are known as differential one-formsw. The circle of center 0 and radius 1 in the *complex plane is a Lie group with complex multiplication. A *Lie group is a group that is also a finite-dimensional smooth manifold, in which the group operation is multiplication rather than addition. *n×n invertible matrices (See below) are a Lie group. A *Lie algebra (See *Infinitesimal transformation) is a local or linearized version of a Lie group. The Lie derivativew generalizes the Lie bracketw which generalizes the wedge productw which is a generalization of the cross productw which only works in 3 dimensions.

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Spaces

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An arrow from A to B means that space A is also a kind of space B.

Around 1735, Euler discovered the formula $V - E + F = 2$ relating the number of vertices, edges and faces of a convex polyhedron, and hence of a ^*planar graph. No metric is required to prove this formula. The study and generalization of this formula is the origin of topology^w.

A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

Hierarchy of mathematical spaces.

The metric is a function that defines a concept of distance between any two points. The distance from a point to itself is zero. The distance between two distinct points is positive.

1.	$d(x,y) \ge 0$
2.	$d(x,y) = 0 \quad$ iff $\quad x = y$
3.	$d(x,z) \le d(x,y) + d(y, z)$
4.	$d(x,y) = d(y,x)$

A norm is the generalization to real vector spaces of the intuitive notion of distance in the real world. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same).

A norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero.

1. Failed to parse (syntax error): {\displaystyle \|\mathbf{v}\| ≥ 0}
2. $\|\mathbf{v}\| = 0 \quad$ iff   $\quad \mathbf{v} = \mathbf{0}$ (the zero vector)
3. Failed to parse (syntax error): {\displaystyle \|\mathbf{u} + \mathbf{v}\| ≤ \|\mathbf{u}\| + \|\mathbf{v}\| \quad} (The ^*Triangle inequality)
4. $\|\mathbf{v}\| = \|\mathbf{-v}\|$
5. $\|a \cdot \mathbf{v}\| = |a| \cdot \|\mathbf{v}\|$

A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

From Wikipedia:List of vector spaces in mathematics:

• ^*Topological space
  • ^*Manifold
  • ^*Metric space

• ^*Vector space
  • Dual space^w
  • ^*Riesz space
  • ^*Topological vector space
    • ^*Montel space
    • ^*Locally convex topological vector space
      • ^*Normed vector space
        • ^*Inner product space
        • ^*Banach space
          • ^*Tsirelson space
          • ^*Orlicz space
          • ^*Morrey–Campanato space
          • ^*Hilbert space
            • ^*Real coordinate space
              • ^*L^p space
                • Euclidean space^w
                • ^*Pseudo-Euclidean space
                  • ^*Minkowski space
            • ^*Sobolev space

• ^*Besov space
• ^*Bochner space
• ^*Fock space
• ^*Fréchet space
  • ^*Schwartz space
• ^*Hardy space
• ^*Hölder space
• ^*LF-space

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Multiplication of vectors

Multiplication can be generalized to allow for multiplication of vectors in 3 different ways:

Dot product

Dot product^w (a Scalar^w): $\mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \|\ \| \mathbf{v}\| \cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3$

$\mathbf{u}\cdot\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 + u_2 v_2 + u_3 v_3 \end{bmatrix}$

Strangely, only parallel components multiply.

The dot product can be generalized to the bilinear form^w $\beta(\mathbf{u,v}) = u^T Av = scalar$ where A is an (0,2) tensor. (For the dot product in Euclidean space A is the identity tensor. But in Minkowski space A is the ^*Minkowski metric).

Two vectors are orthogonal if $\beta(\mathbf{u,v}) = 0.$

A bilinear form is symmetric if $\beta(\mathbf{u,v}) = \beta(\mathbf{v,u})$

Its associated ^*quadratic form is $Q(\mathbf{x}) = \beta(\mathbf{x,x}).$

In Euclidean space $\|\mathbf{v}\|^2 = \mathbf{v}\cdot\mathbf{v}= Q(\mathbf{x}).$

A nondegenerate bilinear form is one for which the associated matrix is invertible (its determinate is not zero)

$\beta(\mathbf{u,v})=0 \,$ for all v implies that u = 0.

The inner product^w is a generalization of the dot product to complex vector space.

$\langle u,v\rangle=\overline{\langle v,u\rangle}=u\cdot \bar{v}=\langle v \mid u\rangle$ (See ^*Bra–ket notation.)

The inner product can be generalized to a sesquilinear form^w

A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V × V → C such that $h(w,z) = \overline{h(z, w)}.$

A is a ^*Hermitian operator iff^w $\langle v \mid A u\rangle = \langle A v \mid u\rangle.$ Often written as $\langle v \mid A \mid u\rangle.$

The curl operator, $\nabla\times$ is Hermitian.

A ^*Hilbert space is an inner product space^w that is also a Complete metric space^w.

The inner product of 2 functions $f$ and $g$ between $a$ and $b$ is

$\langle f,g\rangle=\int\limits_a^b f\cdot\overline{g}\,dx$

If this is equal to 0, the functions are said to be orthogonal on the interval. Unlike with vectors, this has no geometric significance but this definition is useful in ^*Fourier analysis. See below.

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Outer product

Outer product^w (a tensor^w called a dyadic^w):$\mathbf{u} \otimes \mathbf{v}.$

As one would expect, every component of one vector multipies with every component of the other vector.

$\begin{align} \mathbf{u} \otimes \mathbf{v} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} = \begin{bmatrix} {\color{red} u_1 v_1 } & {\color{blue} u_1 v_2 } & {\color{blue} u_1 v_3 } \\ {\color{blue} u_2 v_1 } & {\color{red} u_2 v_2 } & {\color{blue} u_2 v_3 } \\ {\color{blue} u_3 v_1 } & {\color{blue} u_3 v_2 } & {\color{red} u_3 v_3 } \end{bmatrix} \end{align}$

For complex vectors, it is customary to use the conjugate transpose of v (denoted v^H or v*):

$\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{H} = \mathbf{u} \mathbf{v}^*\$

Taking the dot product of u⊗v and any vector x (See Visualization of Tensor multiplication^w) causes the components of x not pointing in the direction of v to become zero. What remains is then rotated from v to u. Therefore an outer product rotates one component of a vector and causes all other components to become zero.

$\mathbf{e}_1 \otimes \mathbf{e}_2 \cdot \mathbf{e}_2 = \mathbf{e}_1$

To rotate a vector with 2 components you need the sum of at least 2 outer products (a bivector). But this is still not perfect. Any 3rd component not in the plane of rotation will become zero.

A true 3 dimensional rotation matrix can be constructed by summing three outer products. The first two sum to form a bivector. The third one rotates the axis of rotation zero degrees but is necessary to prevent that dimension from being squashed to nothing. $\mathbf{e}_1 \otimes \mathbf{e}_2 - \mathbf{e}_2 \otimes \mathbf{e}_1 + \mathbf{e}_3 \otimes \mathbf{e}_3$

The Tensor product^w generalizes the outer product^w.

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Geometric product

The geometric product^w will be explained in detail below.

Wedge product

A unit vector and a unit bivector are shown in red

Wedge product^w (a simple bivector^w): $\mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = [\overline{\mathbf{u}}, \overline{\mathbf{v}}]$

The wedge product of 2 vectors is equal to the ^*geometric product minus the inner product as will be explained in detail below.

The wedge product is also called the exterior product^w (sometimes mistakenly called the outer product).

The term "exterior" comes from the exterior product of two vectors not being a vector.

Just as a vector has length and direction so a bivector has an area and an orientation.

In three dimensions $\mathbf{u} \wedge \mathbf{v}$ is the dual^w of the cross product^w which is a pseudovector^w. $\overline{\mathbf{u} \wedge \mathbf{v}} = \mathbf{u} \times \mathbf{v}$

$\mathbf{a \wedge b \wedge c = a \otimes b \otimes c - a \otimes c \otimes b + c \otimes a \otimes b - c \otimes b \otimes a + b \otimes c \otimes a - b \otimes a \otimes c}$

The magnitude of a∧b∧c equals the volume of the parallelepiped.

The triple product^w a∧b∧c is a trivector which is a 3rd degree tensor.

In 3 dimensions a trivector is a pseudoscalar so in 3 dimensions every trivector can be represented as a scalar times the unit trivector. See Levi-Civita symbol^w

$\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) \cdot \mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3$

The dual^w of vector a is bivector ā:

$\overline{\mathbf{a}} \quad\stackrel{\rm def}{=} \quad\begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}$

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Covectors

The Mississippi flows at about 3 km per hour. Km per hour has both direction and magnitude and is a vector.

The Mississippi flows downhill about one foot per km. Feet per km has direction and magnitude but is not a vector. Its a covector.

The difference between a vector and a covector becomes apparent when doing a changing units. If we measured in meters instead of km then 3 km per hour become 3000 meters per hour. The numerical value increases. Vectors are therefore contravariant.

But 1 Foot per km becomes 0.001 foot per meter. The numerical value decreases. Covectors are therefore covariant.

Tensors are more complicated. They can be part contravariant and part covariant.

A (1,1) Tensor is one part contravariant and one part covariant. It is totally unaffected by a change of units. It is these that we will study in the next section.

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Tensors

See also: ^*Matrix norm and ^*Tensor contraction

External links: Review of Linear Algebra and High-Order Tensors

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Just as a vector is a sum of unit vectors multiplied by constants so a tensor is a sum of unit dyadics ($e_1 \otimes e_2$) multiplied by constants. Each dyadic is associated with a certain plane segment having a certain orientation and magnitude. (But a dyadic is not the same thing as a bivector^w.)

A simple tensor is a tensor that can be written as a product of tensors of the form $T=a\otimes b\otimes\cdots\otimes d.$ (See Outer Product above.) The rank of a tensor T is the minimum number of simple tensors that sum to T. A bivector^w is a tensor of rank 2.

The order or degree of the tensor is the dimension of the tensor which is the total number of indices required to identify each component uniquely. A vector is a 1st-order tensor.

Complex numbers can be used to represent and perform rotations^w but only in 2 dimensions.

Tensors^w, on the other hand, can be used in any number of dimensions to represent and perform rotations and other linear transformations^w. See the image to the right.

Any affine transformation^w is equivalent to a linear transformation followed by a translation^w of the origin. (The origin^w is always a fixed point for any linear transformation.) "Translation" is just a fancy word for "move".

Multiplying a tensor and a vector results in a new vector that can not only have a different magnitude but can even point in a completely different direction:

$\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a_1x+a_2y+a_3z\\ b_1x+b_2y+b_3z\\ c_1x+c_2y+c_3z\\ \end{bmatrix}$

Some special cases:

$\begin{bmatrix} {\color{green}a_1} & a_2 & a_3 \\ {\color{green}b_1} & b_2 & b_3 \\ {\color{green}c_1} & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix} = \begin{bmatrix} a_1\\ b_1\\ c_1\\ \end{bmatrix}$

$\begin{bmatrix} a_1 & {\color{green}a_2} & a_3 \\ b_1 & {\color{green}b_2} & b_3 \\ c_1 & {\color{green}c_2} & c_3 \\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix} = \begin{bmatrix} a_2\\ b_2\\ c_2\\ \end{bmatrix}$

$\begin{bmatrix} a_1 & a_2 & {\color{green}a_3} \\ b_1 & b_2 & {\color{green}b_3} \\ c_1 & c_2 & {\color{green}c_3} \\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix} = \begin{bmatrix} a_3\\ b_3\\ c_3\\ \end{bmatrix}$

One can also multiply a tensor with another tensor. Each column of the second tensor is transformed exactly as a vector would be.

$\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{bmatrix}$

And we can also switch things around using a ^*Permutation matrix. (See also ^*Permutation group):

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} x\\ z\\ y\\ \end{bmatrix}$

$\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} z\\ y\\ x\\ \end{bmatrix}$

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} y\\ x\\ z\\ \end{bmatrix}$

Matrices do not in general commute:

$\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & a \\ b & 0 \end{pmatrix}$

but

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} = \begin{pmatrix} 0 & b \\ a & 0 \end{pmatrix}$

The Determinant^w of a matrix is the area or volume of the n-dimensional parallelepiped spanned by its column (or row) vectors and is frequently useful.

$|A|=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$

Matrices do have zero divisors:

$A= \begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}, \quad A^2= \begin{pmatrix}0&0&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{pmatrix}, \quad A^3= \begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}, \quad A^4=0$

Decomposition of tensors

Every tensor of degree 2 can be decomposed into a symmetric and an anti-symmetric tensor $\text{Symmetric tensor} = \begin{bmatrix} x & a & b \\ a & y & c \\ b & c & z \\ \end{bmatrix}$ $\text{Anti-symmetric tensor} = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \\ \end{bmatrix}$ The Outer product (tensor product) of a vector with itself is a symmetric tensor: $\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} \otimes \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} xx & xy & xz \\ yx & yy & yz \\ zx & zy & zz \\ \end{bmatrix}$ The wedge product of 2 vectors is anti-symmetric: $\mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = \begin{bmatrix} 0 & u_1 v_2 - v_1 u_2 & u_1 v_3 - v_1 u_3 \\ u_2 v_1 - v_2 u_1 & 0 & u_2 v_3 - v_2 u_3 \\ u_3 v_1 - v_3 u_1 & u_3 v_2 - v_3 u_2 & 0 \end{bmatrix}$ Any $n \times n$ matrix X with complex entries can be expressed as $X = A + N \,$ where A is diagonalizable N is nilpotent A commutes with N (i.e. AN = NA) This is the *Jordan–Chevalley decomposition.

Block matrix

From Wikipedia:Block matrix The matrix $\mathbf{P} = \begin{bmatrix} 1 & 1 & 2 & 2\\ 1 & 1 & 2 & 2\\ 3 & 3 & 4 & 4\\ 3 & 3 & 4 & 4\end{bmatrix}$ can be partitioned into 4 2×2 blocks $\mathbf{P}_{11} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \mathbf{P}_{12} = \begin{bmatrix} 2 & 2\\ 2 & 2\end{bmatrix}, \mathbf{P}_{21} = \begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}, \mathbf{P}_{22} = \begin{bmatrix} 4 & 4\\ 4 & 4\end{bmatrix}.$ The partitioned matrix can then be written as $\mathbf{P} = \begin{bmatrix} \mathbf{P}_{11} & \mathbf{P}_{12}\\ \mathbf{P}_{21} & \mathbf{P}_{22}\end{bmatrix}.$ the matrix product $\mathbf{C}=\mathbf{A}\mathbf{B}$ can be formed blockwise, yielding $\mathbf{C}$ as an $(m\times n)$ matrix with $q$ row partitions and $r$ column partitions. The matrices in the resulting matrix $\mathbf{C}$ are calculated by multiplying: $\mathbf{C}_{\alpha \beta} = \sum^s_{\gamma=1}\mathbf{A}_{\alpha \gamma}\mathbf{B}_{\gamma \beta}.$ Or, using the *Einstein notation that implicitly sums over repeated indices: $\mathbf{C}_{\alpha \beta} = \mathbf{A}_{\alpha \gamma}\mathbf{B}_{\gamma \beta}.$

Normal matrices

A diagonal matrix^w:

$\begin{bmatrix} 6 & 0 & 0 \\ 0 & -7i & 0 \\ 0 & 0 & 1+9i \end{bmatrix}$

The determinate of a diagonal matrix:

$\begin{vmatrix} a&0&0\\ 0&b&0\\ 0&0&c \end{vmatrix} =abc$

A superdiagonal entry is one that is directly above and to the right of the main diagonal. A subdiagonal entry is one that is directly below and to the left of the main diagonal. The eigenvalues of diag(λ₁, ..., λ_n) are λ₁, ..., λ_n with associated eigenvectors of e₁, ..., e_n.

A ^*spectral theorem is a result about when a matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations.

A matrix is normal if and only if it is ^*diagonalizable.

All unitaryw, Hermitianw, and *skew-Hermitian matrices are normal.	All orthogonalw, symmetricw, and skew-symmetricw matrices are normal.
A Unitary matrixw is a complex square matrix whose rows (or columns) form an *orthonormal basis of $\C^n$ with respect to the usual inner product. $\begin{bmatrix} a & b \\ -e^{i\varphi} b^* & e^{i\varphi} a^* \\ \end{bmatrix},\qquad \left| a \right| ^2 + \left| b \right| ^2 = 1 ,$	An orthogonal matrix is a real unitary matrix. Its columns and rows are orthogonal unit vectors (i.e., *orthonormal vectors). A permutation matrix is an orthogonal matrix. $\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{bmatrix}.$
A Hermitian matrixw is a complex square matrix that is equal to its own conjugate transposew. $A = \overline {A^\text{T}} = A^\dagger = A^\text{H}$ The diagonal elements must be real. $\begin{bmatrix} 2 & 2+i & -4 \\ 2-i & 0 & i \\ -4 & -i & -1 \\ \end{bmatrix}$	A symmetric matrixw is a real Hermitian matrix. It is equal to its transposew. $A = A^\mathrm{T}.$ $\begin{bmatrix} 2 & 2 & -4\\ 2 & 0 & 0\\ -4 & 0 & -1\end{bmatrix}$
A *Skew-Hermitian matrix is a complex square matrix whose conjugate transpose is its negative. $A^\dagger = -A,\;$ The diagonal elements must be imaginary. $\begin{bmatrix} i & 2+i & -4 \\ -2+i & 0 & i \\ 4 & i & -i \\ \end{bmatrix}$	A Skew-symmetric matrixw is a real Skew-Hermitian matrix. Its transpose equals its negative. AT = −A The diagonal elements must be zero. $\begin{bmatrix} 0 & 2 & -4\\ -2 & 0 & 0\\ 4 & 0 & 0\end{bmatrix}$

Change of basis

An n-by-n square matrix A is invertible if there exists an n-by-n square matrix A^-1 such that

$\mathbf{AA^{-1}} = \mathbf{A^{-1}A} = \mathbf{I}_n \$

A matrix is invertible if and only if its determinant is non-zero.

$\begin{align} |\mathbf{AA^{-1}}| & = |\mathbf{I}_n| \ \\ |\mathbf{A}| |\mathbf{A^{-1}}| & = 1 \\ \frac{|\mathbf{A}|}{|\mathbf{A}|} & = 1 \\ |\mathbf{A}| & \neq 0 \end{align}$

The standard basis for $R^{2}$ would be:

$E_2 = \left\{ \begin{pmatrix} 1\\ 0\end{pmatrix}, \begin{pmatrix} 0\\ 1\end{pmatrix} \right\}$

Given a matrix $M$ whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector $v$ are given by the matrix product $M^{-1}v$.

From Wikipedia:Matrix similarity:

Given a linear transformation:

$y = Tx$,

it can be the case that a change of basis can result in a simpler form of the same transformation.

$y' = Sx'$,

where x' and y' are in the new basis.

$x' = P x$

$y' = P y$

and P is the change-of-basis matrix.

To derive T in terms of the simpler matrix, we use:

$\begin{align} y' &= Sx' \\ Py &= SPx \\ P^{-1}Py &= P^{-1}SPx \\ y &= \left(P^{-1}SP\right)x = Tx \end{align}$

Thus, the matrix in the original basis is given by

$T = P^{-1}SP$

Therefore

$S = PTP^{-1}$

From Wikipedia:Matrix similarity

Two n-by-n matrices A and B are called similar if

$B = P^{-1} A P$

for some invertible n-by-n matrix P.

A transformation A ↦ P⁻¹AP is called a similarity transformation or conjugation of the matrix A. In the ^*general linear group, similarity is therefore the same as ^*conjugacy, and similar matrices are also called conjugate.

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Linear groups

A square matrix^w of order n is an n-by-n matrix. Any two square matrices of the same order can be added and multiplied. A matrix is invertible if and only if its determinant is nonzero.

GL_n(F) or GL(n, F), or simply GL(n) is the ^*Lie group of n×n invertible matrices with entries from the field F. The group operation is matrix multiplication^w. The group GL(n, F) and its subgroups are often called linear groups or matrix groups.

SL(n, F) or SL_n(F), is the ^*subgroup of GL(n, F) consisting of matrices with a determinant^w of 1.

U(n), the Unitary group of degree n is the group^w of n × n unitary matrices^w. The group operation is matrix multiplication^w. The determinant of a unitary matrix is a complex number with norm 1.

SU(n), the special unitary group of degree n, is the ^*Lie group of n×n unitary matrices^w with determinant^w 1.

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Symmetry groups

^*Affine group

^*Poincaré group: boosts, rotations, translations

^*Lorentz group: boosts, rotations

The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.)

Aff(n,K): the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

E(n): rotations, reflections, and translations.

O(n): rotations, reflections

SO(n): rotations

so(3) is the Lie algebra of SO(3) and consists of all skew-symmetric^w 3 × 3 matrices.

Clifford group: The set of invertible elements x such that for all v in V $x v \alpha(x)^{-1}\in V .$ The ^*spinor norm Q is defined on the Clifford group by $Q(x) = x^\mathrm{t}x.$

Pin_V(K): The subgroup of elements of spinor norm 1. Maps 2-to-1 to the orthogonal group

Spin_V(K): The subgroup of elements of Dickson invariant 0 in Pin_V(K). When the characteristic is not 2, these are the elements of determinant 1. Maps 2-to-1 to the special orthogonal group. Elements of the spin group act as linear transformations on the space of spinors

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Rotations

In 4 spatial dimensions a rigid object can ^*rotate in 2 different ways simultaneously.

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Stereographic projection of four-dimensional Tesseract^w in double rotation

See also: ^*Hypersphere of rotations, ^*Rotation group SO(3), ^*Special unitary group, ^*Plate trick, ^*Spin representation, ^*Spin group, ^*Pin group, ^*Spinor, Clifford algebra^w, ^*Indefinite orthogonal group, ^*Root system, Bivectors^w, Curl^w

From Wikipedia:Rotation group SO(3):

Consider the solid ball in R³ of radius π. For every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The two rotations through π and through −π are the same. So we ^*identify (or "glue together") ^*antipodal points on the surface of the ball.

The ball with antipodal surface points identified is a ^*smooth manifold, and this manifold is ^*diffeomorphic to the rotation group. It is also diffeomorphic to the ^*real 3-dimensional projective space RP³, so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is ^*connected but not ^*simply connected. As to the latter, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". (In other words one full rotation is not equivalent to doing nothing.)

A set of belts can be continuously rotated without becoming twisted or tangled. The cube must go through two full rotations for the system to return to its initial state. See ^*Tangloids.

Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second half of the path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The ^*Balinese plate trick and similar tricks demonstrate this practically.

The same argument can be performed in general, and it shows that the ^*fundamental group of SO(3) is cyclic group^w of order 2. In physics applications, the non-triviality of the fundamental group allows for the existence of objects known as ^*spinors, and is an important tool in the development of the ^*spin-statistics theorem.

Spin group

The *universal cover of SO(3) is a *Lie group called *Spin(3). The group Spin(3) is isomorphic to the *special unitary group SU(2); it is also diffeomorphic to the unit *3-sphere S3 and can be understood as the group of *versors (quaternionsw with absolute valuew 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in *quaternions and spatial rotation. The map from S3 onto SO(3) that identifies antipodal points of S3 is a *surjective *homomorphism of Lie groups, with *kernel {±1}. Topologically, this map is a two-to-one *covering map. (See the *plate trick.) From Wikipedia:Spin group: The spin group Spin(n) is the *double cover of the *special orthogonal group SO(n) = SO(n, R), such that there exists a *short exact sequence of *Lie groups (with n ≠ 2) $1 \to \mathrm{Z}_2 \to \operatorname{Spin}(n) \to \operatorname{SO}(n) \to 1.$ As a Lie group, Spin(n) therefore shares its *dimension, n(n − 1)/2, and its *Lie algebra with the special orthogonal group. For n > 2, Spin(n) is *simply connected and so coincides with the *universal cover of *SO(n). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of *reflection through the origin, generally denoted −I . Spin(n) can be constructed as a *subgroup of the invertible elements in the Clifford algebraw Cl(n). A distinct article discusses the *spin representations.

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Matrix representations

See also: ^*Group representation, ^*Presentation of a group, ^*Abstract algebra

Real numbers

If a vector is multiplied with the the ^*identity matrix $I$ then the vector is completely unchanged:

$I \cdot v = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = 1 \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}$

And if $A=a \cdot I$ then

$A \cdot v = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = a \begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} a\cdot x\\ a \cdot y\\ a\cdot z\\ \end{bmatrix}$

Therefore $A=a \cdot I$ can be thought of as the matrix form of the scalar a. The scalar matrices are the center of the algebra of matrices.

$A + B = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} + \begin{bmatrix} b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \\ \end{bmatrix} = \begin{bmatrix} a + b & 0 & 0 \\ 0 & a + b & 0 \\ 0 & 0 & a + b \\ \end{bmatrix}$

$A \cdot B = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{bmatrix} \begin{bmatrix} b & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & b \\ \end{bmatrix} = \begin{bmatrix} ab & 0 & 0 \\ 0 & ab & 0 \\ 0 & 0 & ab \\ \end{bmatrix}$

$1/A = \begin{bmatrix} 1/a & 0 & 0 \\ 0 & 1/a & 0 \\ 0 & 0 & 1/a \\ \end{bmatrix}$

$|A| = \begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{vmatrix} = a^3$

$A^B = \begin{bmatrix} a^b & 0 & 0 \\ 0 & a^b & 0 \\ 0 & 0 & a^b \\ \end{bmatrix}$

$e^A= \begin{bmatrix} e^a & 0 & 0 \\ 0 & e^a & 0 \\ 0 & 0 & e^a \\ \end{bmatrix}$.

$\ln A= \begin{bmatrix} \ln a & 0 & 0 \\ 0 & \ln a & 0 \\ 0 & 0 & \ln a \\ \end{bmatrix}$.

(Note: Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.)

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Complex numbers

Complex numbers can also be written in matrix form^w in such a way that complex multiplication corresponds perfectly to matrix multiplication:

$\begin{align} (a+ib)(c+id) &= \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} c & d \\ -d & c \end{bmatrix} \\ &= \begin{bmatrix} ac-bd & ad+bc \\ -(ad+bc) & ac-bd \end{bmatrix} \end{align}$

$\begin{align} (i)(i) &= \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \\ &= \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \\ &= -I \end{align}$

The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin computed using the Pythagorean theorem.

$|z|^2 = \begin{vmatrix} a & -b \\ b & a \end{vmatrix} = a^2 + b^2.$

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Quaternions

There are at least two ways of representing quaternions as matrices^w in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication^w.

Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as

$\begin{bmatrix} z_1 & z_2 \\ -\overline{z_2} & \overline{z_1} \end{bmatrix} = \begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}= a \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} + b \begin{bmatrix} i & 0 \\ 0 & -i \\ \end{bmatrix} + c \begin{bmatrix} 0 & 1 \\ -1 & 0 \\ \end{bmatrix} + d \begin{bmatrix} 0 & i \\ i & 0 \\ \end{bmatrix}.$

Multiplying any two Pauli matrices always yields a quaternion unit matrix. See Isomorphism to quaternions below.

By replacing each 0, 1, and i with its 2 × 2 matrix representation that same quaternion can be written as a 4 × 4 real (^*block) matrix:

$\begin{bmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\ -d & -c & b & a \end{bmatrix}= a \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} + b \begin{bmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{bmatrix} + c \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{bmatrix}.$

Therefore:

$\begin{align} b \cdot i &= b \cdot (e_1 \wedge e_2 + e_4 \wedge e_3)\\ c \cdot j &= c \cdot (e_1 \wedge e_3 + e_2 \wedge e_4)\\ d \cdot k &= d \cdot (e_1 \wedge e_4 + e_3 \wedge e_2)\\ \end{align}$

However, the representation of quaternions in M(4,ℝ) is not unique. In fact, there exist 48 distinct representations of this form. Each 4x4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. See Wikipedia:Quaternion#Matrix_representations.

The obvious way of representing quaternions with 3 × 3 real matrices does not work because:

$b \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \cdot c \begin{bmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \neq d \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & {\color{red}1} & 0 \end{bmatrix}.$

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Vectors

Euclidean

See also: ^*Split-complex numbers

Unfortunately the matrix representation of a vector is not so obvious. First we must decide what properties the matrix should have. To see consider the square (^*quadratic form) of a single vector:

$Q(c) = c^2 = \langle c , c \rangle = (a\mathbf{e_1} + b\mathbf{e_2})^2$

$Q(c) = aa\mathbf{e_1}\mathbf{e_1} + bb\mathbf{e_2}\mathbf{e_2} + ab\mathbf{e_1}\mathbf{e_2} + ba\mathbf{e_2}\mathbf{e_1}$

$Q(c) = a^2\mathbf{e_1}\mathbf{e_1} + b^2\mathbf{e_2}\mathbf{e_2} + ab(\mathbf{e_1}\mathbf{e_2} + \mathbf{e_2}\mathbf{e_1})$

From the Pythagorean theorem we know that:

$c^2 = a^2 + b^2 + ab(0) = Scalar$

So we know that

$e_1^2 = e_2^2 = 1$

$e_1 e_2 = -e_2 e_1$

This particular Clifford algebra is known as Cl_2,0. The subscript 2 indicates that the 2 basis vectors are square roots of +1. See ^*Metric signature. If we had used $c^2 = -a^2 -b^2$ then the result would have been Cl_0,2.

The set of 3 matrices in 3 dimensions that have these properties are called ^*Pauli matrices. The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of ℝ³.

From Wikipedia:Pauli matrices

The Pauli matrices are a set of three 2 × 2 complex^w matrices^w which are Hermitian^w and unitary^w. They are

$\begin{align} \sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \,. \end{align}$

Squaring a Pauli matrix results in a "scalar":

$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} = \sigma_0 = I$

Do NOT confuse this scalar with the vectors above. It may look similar to the Pauli matrices but it is not the matrix representation of a vector. It is the matrix representation of a scalar. Scalars are totally different from vectors and the matrix representations of scalars are totally different from the matrix representations of vectors. They are NOT the same.

Multiplication is ^*anticommutative:

$\sigma_1 \sigma_2 = - \sigma_2 \sigma_1$

$\sigma_2 \sigma_3 = - \sigma_3 \sigma_2$

$\sigma_3 \sigma_1 = - \sigma_1 \sigma_3$

And

$\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} i&0\\0&i\end{pmatrix} = i$

Exponential of a Pauli vector which is analogous to Euler's formula, extended to quaternions:

$e^{i a(\hat{n} \cdot \vec{\sigma})} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a}$

commutation^w relations:

$\begin{align} \left[\sigma_1, \sigma_2\right] &= 2i\sigma_3 \, \\ \left[\sigma_2, \sigma_3\right] &= 2i\sigma_1 \, \\ \left[\sigma_3, \sigma_1\right] &= 2i\sigma_2 \, \\ \left[\sigma_1, \sigma_1\right] &= 0\, \\ \end{align}$

^*anticommutation relations:

$\begin{align} \left\{\sigma_1, \sigma_1\right\} &= 2I\, \\ \left\{\sigma_1, \sigma_2\right\} &= 0\,.\\ \end{align}$

Adding the commutator ($ab - ba$) to the anticommutator ($ab + ba$) gives the general formula for multiplying any 2 arbitrary "vectors" (or rather their matrix representations):

$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) \, I + i ( \vec{a} \times \vec{b} )\cdot \vec{\sigma}$

If $i$ is identified with the pseudoscalar $\sigma_x \sigma_y \sigma_z$ then the right hand side becomes $a \cdot b + a \wedge b$ which is also the definition for the geometric product^w of two vectors in geometric algebra^w (Clifford algebra^w). The geometric product of two vectors is a multivector^w.

For any 2 arbitrary vectors:

$fd = force*distance$

$fd = (f_1\mathbf{e_1} + f_2\mathbf{e_2})(d_1\mathbf{e_1} + d_2\mathbf{e_2})$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} + f_2d_1\mathbf{e_2}\mathbf{e_1}$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + f_1d_2\mathbf{e_1}\mathbf{e_2} - f_2d_1\mathbf{e_1}\mathbf{e_2}$

$fd = f_1d_1\mathbf{e_1}\mathbf{e_1} + f_2d_2\mathbf{e_2}\mathbf{e_2} + (f_1d_2 - f_2d_1)\mathbf{e_1}\mathbf{e_2}$

Applying the rules of Clifford algebra we get:

$fd = f_1d_1 + f_2d_2 + (f_1d_2 - f_2d_1)\mathbf{e_1} \wedge \mathbf{e_2}$

$fd = Energy + Torque$

$fd = {\color{red} f \cdot d} + {\color{blue} f \wedge d}$

$fd = {\color{red} Scalar} + {\color{blue} Bivector} = Multivector$

Isomorphism to quaternions

Multiplying any 2 Pauli matrices results in a quaternion. Hence the geometric interpretation of the quaternion units $\boldsymbol{\hat{\imath}}, \boldsymbol{\hat{\jmath}}, \boldsymbol{\hat{k}} \quad$ as bivectors in 3 dimensional (not 4 dimensional) space. Quaternions form a *division algebra—every non-zero element has an inverse—whereas Pauli matrices do not. $\begin{array}{c|ccc|ccc|c} {\color{red} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}} & {\color{green} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{blue} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} \\\hline {\color{green} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{black} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} \\ {\color{green} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}} & {\color{red} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{black} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} \\ {\color{green} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}} & {\color{blue} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} \\\hline {\color{blue} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{red} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}} & {\color{green} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}} \\ {\color{blue} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{black} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}} \\ {\color{blue} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{black} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} & {\color{black} \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}} & {\color{red} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}} \\\hline {\color{red} \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}} & {\color{blue} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}} & {\color{blue} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}} & {\color{green} \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}} & {\color{green} \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}} & {\color{red} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}} \end{array}$ $\begin{array}{r|rrr|rrr|r} {\color{red} I_2} & {\color{green} \sigma_1} & {\color{green} \sigma_2} & {\color{green} \sigma_3} & {\color{blue} \boldsymbol{\hat{\imath}}} & {\color{blue} \boldsymbol{\hat{\jmath}}} & {\color{blue} \boldsymbol{\hat{k}}} & {\color{red} i_2} \\\hline {\color{green} \sigma_1} & {\color{red} I_2} & {\color{blue} \boldsymbol{\hat{\imath}}} & {\color{blue} \boldsymbol{-\hat{\jmath}}} & {\color{green} \sigma_2} & {\color{green} -\sigma_3} & {\color{red} i_2} & {\color{blue} \boldsymbol{\hat{k}}} \\ {\color{green} \sigma_2} & {\color{blue} \boldsymbol{-\hat{\imath}}} & {\color{red} I_2} & {\color{blue} \boldsymbol{\hat{k}}} & {\color{green} -\sigma_1} & {\color{red} i_2} & {\color{green} \sigma_3} & {\color{blue} \boldsymbol{\hat{\jmath}}} \\ {\color{green} \sigma_3} & {\color{blue} \boldsymbol{\hat{\jmath}}} & {\color{blue} \boldsymbol{-\hat{k}}} & {\color{red} I_2} & {\color{red} i_2} & {\color{green} \sigma_1} & {\color{green} -\sigma_2} & {\color{blue} \boldsymbol{\hat{\imath}}} \\\hline {\color{blue} \boldsymbol{\hat{\imath}}} & {\color{green} -\sigma_2} & {\color{green} \sigma_1} & {\color{red} i_2} & {\color{red} -I_2} & {\color{blue} \boldsymbol{\hat{k}}} & {\color{blue} \boldsymbol{-\hat{\jmath}}} & {\color{green} -\sigma_3} \\ {\color{blue} \boldsymbol{\hat{\jmath}}} & {\color{green} \sigma_3} & {\color{red} i_2} & {\color{green} -\sigma_1} & {\color{blue} \boldsymbol{-\hat{k}}} & {\color{red} -I_2} & {\color{blue} \boldsymbol{\hat{\imath}}} & {\color{green} -\sigma_2} \\ {\color{blue} \boldsymbol{\hat{k}}} & {\color{red} i_2} & {\color{green} -\sigma_3} & {\color{green} \sigma_2} & {\color{blue} \boldsymbol{\hat{\jmath}}} & {\color{blue} \boldsymbol{-\hat{\imath}}} & {\color{red} -I_2} & {\color{green} -\sigma_1} \\\hline {\color{red} i_2} & {\color{blue} \boldsymbol{\hat{k}}} & {\color{blue} \boldsymbol{\hat{\jmath}}} & {\color{blue} \boldsymbol{\hat{\imath}}} & {\color{green} -\sigma_3} & {\color{green} -\sigma_2} & {\color{green} -\sigma_1} & {\color{red} I_2} \end{array}$ ${\color{green} \sigma_1 \sigma_2} = {\color{blue} \boldsymbol{\hat{\imath}}} = {\color{blue} \begin{bmatrix} i & 0 \\ 0 & -i \\ \end{bmatrix}}$ ${\color{green} \sigma_3 \sigma_1} = {\color{blue} \boldsymbol{\hat{\jmath}}} = {\color{blue} \begin{bmatrix} 0 & 1 \\ -1 & 0 \\ \end{bmatrix}}$ ${\color{green} \sigma_2 \sigma_3} = {\color{blue} \boldsymbol{\hat{k}}} = {\color{blue} \begin{bmatrix} 0 & i \\ i & 0 \\ \end{bmatrix}}$ And multiplying a Pauli matrix and a quaternion results in a Pauli matrix:

Further reading: ^*Generalizations of Pauli matrices, ^*Gell-Mann matrices and ^*Pauli equation

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Pseudo-Euclidean

See also: ^*Electron magnetic moment

From Wikipedia:Gamma matrices

Gamma ^*matrices, $\{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \}$, also known as the Dirac matrices, are a set of 4 × 4 conventional matrices with specific ^*anticommutation relations that ensure they ^*generate a matrix representation of the Clifford algebra^w Cℓ_1,3(R). One gamma matrix squares to 1 times the ^*identity matrix and three gamma matrices square to -1 times the identity matrix.

$(\gamma^0)^2 = I$

$(\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = -I$

The defining property for the gamma matrices to generate a Clifford algebra^w is the anticommutation relation

$\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_4$

where $\{ , \}$ is the ^*anticommutator, $\eta^{\mu \nu}$ is the ^*Minkowski metric with signature (+ − − −) and $I_4$ is the 4 × 4 identity matrix.

Minkowski metric

From Wikipedia:Minkowski_space#Minkowski_metric The simplest example of a Lorentzian manifold is *flat spacetime, which can be given as R4 with coordinates $(t, x, y, z)$ and the metric $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 = \eta_{\mu\nu} dx^{\mu} dx^{\nu}. \,$ Note that these coordinates actually cover all of R4. The flat space metric (or *Minkowski metric) is often denoted by the symbol η and is the metric used in *special relativity. A standard basis for Minkowski space is a set of four mutually orthogonal vectors { e0, e1, e2, e3 } such that $-\eta(e_0, e_0) = \eta(e_1, e_1) = \eta(e_2, e_2) = \eta(e_3, e_3) = 1 .$ These conditions can be written compactly in the form $\eta(e_\mu, e_\nu) = \eta_{\mu \nu}.$ Relative to a standard basis, the components of a vector v are written (v0, v1, v2, v3) where the *Einstein summation convention is used to write v = vμeμ. The component v0 is called the timelike component of v while the other three components are called the spatial components. The spatial components of a 4-vector v may be identified with a 3-vector v = (v1, v2, v3). In terms of components, the Minkowski inner product between two vectors v and w is given by $\eta(v, w) = \eta_{\mu \nu} v^\mu w^\nu = v^0 w_0 + v^1 w_1 + v^2 w_2 + v^3 w_3 = v^\mu w_\mu = v_\mu w^\mu,$ and $\eta(v, v) = \eta_{\mu \nu} v^\mu v^\nu = v^0v_0 + v^1 v_1 + v^2 v_2 + v^3 v_3 = v^\mu v_\mu.$ Here lowering of an index with the metric was used. The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally a constant pseudo-Riemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type (0,2) tensor. It accepts two arguments u, v. The definition $u \cdot v =\eta(u, v)$ yields an inner product-like structure on M, previously and also henceforth, called the Minkowski inner product, similar to the Euclidean inner product, but it describes a different geometry. It is also called the relativistic dot product. If the two arguments are the same, $u \cdot u =\eta(u, u) \equiv ||u||^2 \equiv u^2,$ the resulting quantity will be called the Minkowski norm squared. This bilinear form can in turn be written as $u \cdot v = u^{\mathrm T}[\eta] v,$ where [η] is a 4×4 matrix associated with η. Possibly confusingly, denote [η] with just η as is common practice. The matrix is read off from the explicit bilinear form as $\eta = \pm \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix},$ and the bilinear form $u \cdot v =\eta(u, v),$ with which this section started by assuming its existence, is now identified.

When interpreted as the matrices of the action of a set of orthogonal basis vectors for ^*contravariant vectors in Minkowski space^w, the column vectors on which the matrices act become a space of ^*spinors, on which the Clifford algebra of ^*spacetime acts. This in turn makes it possible to represent infinitesimal ^*spatial rotations and Lorentz boosts^w. Spinors facilitate spacetime computations in general, and in particular are fundamental to the ^*Dirac equation for relativistic spin-½ particles.

In Dirac representation^w, the four ^*contravariant gamma matrices are

$\begin{align} \gamma^0 &= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},\quad& \gamma^1 &= \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix} \\ \gamma^2 &= \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix},\quad& \gamma^3 &= \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}. \end{align}$

$\gamma^0$ is the time-like matrix and the other three are space-like matrices.

$(\gamma^0)^2 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

$(\gamma^1)^2 = (\gamma^2)^2 = (\gamma^3)^2 = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$

The matrices are also sometimes written using the 2×2 ^*identity matrix, $I_2$, and the ^*Pauli matrices.

The gamma matrices we have written so far are appropriate for acting on ^*Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:

$\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}.$

Another common choice is the Weyl or chiral basis, in which $\gamma^k$ remains the same but $\gamma^0$ is different, and so $\gamma^5$ is also different, and diagonal,

$\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix},\quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -I_2 & 0 \\ 0 & I_2 \end{pmatrix},$

Original Dirac matrices

Source: Weisstein, Eric W. "Dirac Matrices." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiracMatrices.html $\begin{array}{c|c|c|c} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&-i\\ 0&0&i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&0&-1 \end{pmatrix} \\\hline \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix} & {\color{red} \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix} } & {\color{red} \begin{pmatrix} 0&0&0&-i\\ 0&0&i&0\\ 0&-i&0&0\\ i&0&0&0 \end{pmatrix} } & {\color{red} \begin{pmatrix} 0&0&1&0\\ 0&0&0&-1\\ 1&0&0&0\\ 0&-1&0&0 \end{pmatrix} } \\\hline \begin{pmatrix} 0&0&-i&0\\ 0&0&0&-i\\ i&0&0&0\\ 0&i&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&0&-i\\ 0&0&-i&0\\ 0&i&0&0\\ i&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&0&-1\\ 0&0&1&0\\ 0&1&0&0\\ -1&0&0&0 \end{pmatrix} & \begin{pmatrix} 0&0&-i&0\\ 0&0&0&i\\ i&0&0&0\\ 0&-i&0&0 \end{pmatrix} \\\hline {\color{red} \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix}} & \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&-1&0 \end{pmatrix} & \begin{pmatrix} 0&-i&0&0\\ i&0&0&0\\ 0&0&0&i\\ 0&0&-i&0 \end{pmatrix} & \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&1 \end{pmatrix} \end{array}$ Surprisingly the 4 by 4 table above forms a multiplication table even though it is actually created by the following rules: $\begin{array}{c|c|c|c} \sigma_{00} & \sigma_{01} & \sigma_{02} & \sigma_{03} \\\hline \sigma_{10} & \sigma_{11} & \sigma_{12} & \sigma_{13} \\\hline \sigma_{20} & \sigma_{21} & \sigma_{22} & \sigma_{23} \\\hline \sigma_{30} & \sigma_{31} & \sigma_{32} & \sigma_{33} \end{array} \quad \text{where} \quad$ $\quad\quad\begin{align} \sigma_{ij} &= \sigma_i \otimes \sigma_j \end{align}$ where $\sigma_i$ and $\sigma_j$ are the original 2x2 Pauli matrices and $\otimes$ is the *Kronecker product (not the tensor product) $\sigma_0 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \quad \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$ The Dirac matrices are commonly referred to by the following name. Note that $\sigma_i$ do not refer to the original Pauli matrices. $\begin{array}{c|c|c|c} \sigma_0 & \sigma_1 & \sigma_2 & \sigma_3 \\\hline \rho_1 & \alpha_1 & \alpha_2 & \alpha_3 \\\hline \rho_2 & y_1 & y_2 & y_3 \\\hline \rho_3 & \delta_1 & \delta_2 & \delta_3 \end{array} \quad \text{where} \quad \begin{align} \sigma_0 &= I_4 \\ -\rho_1 &= y_5 \\ \rho_2 &= \alpha_5 \\ \rho_3 &= \alpha_4 = y_4 \end{align}$ The 16 original Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214). $\alpha_1, \alpha_2, \alpha_3, \quad \rho_3, \rho_2 \quad (\alpha_4, \alpha_5)$ $y_1, y_2, y_3, \quad \rho_3, -\rho_1 \quad (y_4, y_5)$ $\delta_1, \delta_2, \delta_3, \quad \rho_1, \rho_2$ $\alpha_1, y_1, \delta_1, \quad \sigma_2, \sigma_3$ $\alpha_2, y_2, \delta_2, \quad \sigma_1, \sigma_3$ $\alpha_3, y_3, \delta_3, \quad \sigma_1, \sigma_2$ Any of the 15 original Dirac matrices (excluding the identity matrix $\sigma_0$) anticommute with eight other original Dirac matrices and commute with the remaining eight, including itself and the identity matrix. Any of the 16 original Dirac matrices multiplied times itself equals $I_4$

Higher-dimensional gamma matrices

*Analogous sets of gamma matrices can be defined in any dimension and for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma matrix to be presented below generate the Clifford algebra. It is useful to define the product of the four gamma matrices as follows: $\gamma^5 := i\gamma^0\gamma^1\gamma^2\gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$ (in the Dirac basis). Although $\gamma^5$ uses the letter gamma, it is not one of the gamma matrices of Cℓ1,3(R). The number 5 is a relic of old notation in which $\gamma^0$ was called "$\gamma^4$". From Wikipedia:Higher-dimensional gamma matrices Consider a space-time of dimension d with the flat *Minkowski metric, $\eta = \parallel \eta_{a b} \parallel = \text{diag}(+1,-1, \dots, -1) ~,$ where a,b = 0,1, ..., d−1. Set N= 2⌊d/2⌋. The standard Dirac matrices correspond to taking d = N = 4. The higher gamma matrices are a d-long sequence of complex N×N matrices $\Gamma_i,\ i=0,\ldots,d-1$ which satisfy the *anticommutator relation from the *Clifford algebra Cℓ1,d−1(R) (generating a representation for it), $\{ \Gamma_a ~,~ \Gamma_b \} = \Gamma_a\Gamma_b + \Gamma_b\Gamma_a = 2 \eta_{a b} I_N ~,$ where IN is the *identity matrix in N dimensions. (The spinors acted on by these matrices have N components in d dimensions.) Such a sequence exists for all values of d and can be constructed explicitly, as provided below. The gamma matrices have the following property under hermitian conjugation, $\Gamma_0^\dagger= +\Gamma_0 ~,~ \Gamma_i^\dagger= -\Gamma_i ~(i=1,\dots,d-1) ~.$

Further reading: Quan­tum Me­chan­ics for En­gi­neers and How (not) to teach Lorentz covariance of the Dirac equation

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Multivectors

See also: ^*Dirac algebra

External links:

A brief introduction to geometric algebra

A brief introduction to Clifford algebra

The Construction of Spinors in Geometric Algebra

Functions of Multivector Variables

Clifford Algebra Representations

Clifford algebra is a type of algebra characterized by the geometric product of scalars, vectors, bivectors, trivectors...etc.

Just as a vector has length so a bivector has area and a trivector has volume.

Just as a vector has direction so a bivector has orientation. In three dimensions a trivector has only one possible orientation and is therefore a pseudoscalar. But in four dimensions a trivector becomes a pseudovector and the quadvector becomes the pseudoscalar.

Multiplication of arbitrary vectors

The dot product of two vectors is:

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} \mathbf{u} \cdot \mathbf{v} &= {\color{blue}\text{vector}} \cdot {\color{blue}\text{vector}} \\ &= (u_{x} + u_{y})(v_{x} + v_{y}) \\ &= {\color{red}u_{x} v_{x} + u_{y} v_{y}} \end{split} }

But this is actually quite mysterious. When we multiply $(a_1 + a_2)$ and $(b_1 + b_2)$ we dont get $(a_1 b_1 + a_2 b_2)$ so why is it that when we multiply vectors we only multiply parallel components? Clifford algebra has a surprisingly simple answer. The answer is: We dont! Instead of the dot product or the wedge product Clifford algebra uses the geometric product.

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} \mathbf{u} \mathbf{v} &= (u_{x} {\color{blue}e_{x}} + u_{y} {\color{blue} e_{y}} ) (v_{x} {\color{blue}e_{x}} + v_{y} {\color{blue} e_{y}} ) \\ &= u_{x} v_{x} {\color{red}e_{x} e_{x}} + u_{x} v_{y} {\color{green}e_{x} e_{y}} + u_{y} v_{x} {\color{green}e_{y} e_{x}} + u_{y} v_{y} {\color{red}e_{y} e_{y}} \\ &= u_{x} v_{x} {\color{red}(1)} + u_{y} v_{y} {\color{red}(1)} + u_{x} v_{y} {\color{green}e_{x} e_{y}} - u_{y} v_{x} {\color{green}e_{x} e_{y}} \\ &= (u_{x} v_{x} + u_{y} v_{y}){\color{red}(1)} + (u_{x} v_{y} - u_{y} v_{x}) {\color{green}e_{xy}} \\ &= {\color{red}\text{scalar}} + {\color{green}\text{bivector}} \end{split} }

A scalar plus a bivector (or any number of blades of different grade) is called a multivector. The idea of adding a scalar and a bivector might seem wrong but in the real world it just means that what appears to be a single equation is in fact a set of ^*simultaneous equations.

For example:

$\mathbf{u} \mathbf{v} = 5$

would just mean that:

Failed to parse (syntax error): {\displaystyle (u_{x} v_{x} + u_{y} v_{y}){\color{red}(1)} = 5 \\ \quad \quad \quad \text{and} \\ (u_{x} v_{y} - u_{y} v_{x}) {\color{green}e_{xy}} = 0 }

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Rules

All the properties of Clifford algebra derive from a few simple rules.

Let ${\color{blue}e_{x}},$ ${\color{blue}e_{y}},$ and ${\color{blue}e_{z}}$ be perpendicular unit vectors.

Multiplying two perpendicular vectors results in a bivector:

$e_x e_y = {\color{green}e_{xy}}$

Multiplying three perpendicular vectors results in a trivector:

$e_x e_y e_z = {\color{orange}e_{xyz}}$

Multiplying parallel vectors results in a scalar:

$e_x e_x = {\color{red}e_{xx}} = {\color{red}1}$

Clifford algebra is associative therefore the fact that multiplying parallel vectors results in a scalar means that:

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} {\color{green}(e_{x} e_{y})} {\color{blue} (e_{y}) } &= {\color{blue} e_{x}} {\color{red}(e_{y} e_{y})} \\ &= {\color{blue} e_{x}} {\color{red}(1)} \\ &= {\color{blue} e_{x}} \end{split} }

and:

$\begin{align} {\color{green} (e_{x} e_{y})} {\color{blue} (e_{y} + e_{z}) } &= {\color{blue} e_{xyy}} + {\color{orange}e_{xyz}} \\ &= {\color{blue} e_{x}} + {\color{orange}e_{xyz}} \end{align}$

and:

$\begin{align} {\color{green} (e_{x} e_{y}) } {\color{blue} (e_{z}) } &= {\color{orange} e_{xyz}} \end{align}$

Rotation from x to y is the negative of rotation from y to x:

$e_{xy} = -e_{yx}$

Therefore:

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} {\color{green}(e_{x} e_{y})} {\color{blue} (e_{x}) } &= \phantom{-}{\color{blue} e_{x}} {\color{green}(e_{y} e_{x})} \\ &= {\color{blue} -e_{x}} {\color{green}(e_{x} e_{y})} \\ &= -{\color{red} (e_{x} e_{x})} {\color{blue} e_{y}} \\ &= -{\color{red} (1)} {\color{blue} e_{y}} \\ &= -{\color{blue} e_{y}} \end{split} }

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Multiplication tables

In one dimension:

$\begin{array}{r|r} {\color{red} 1 } & {\color{blue} e_{x}} \\\hline {\color{blue} e_{x}} & {\color{red} 1 } \end{array}$

In two dimensions:

$\begin{array}{r|rr|r} {\color{red} 1 } & {\color{blue} e_{x}} & {\color{blue} e_{y}} & {\color{green} e_{xy}} \\\hline {\color{blue} e_{x}} & {\color{red} 1 } & {\color{green} e_{xy}} & {\color{blue} {e_{y}}} \\ {\color{blue} e_{y}} & {\color{green} -e_{xy}} & {\color{red} 1 } & {\color{blue} -e_{x}} \\\hline {\color{green} e_{xy}} & {\color{blue} -e_{y}} & {\color{blue} {e_{x}}} & {\color{red} {-1}} \end{array}$

In three dimensions:

$\begin{array}{r|rrr|rrr|r} {\color{red} 1 } & {\color{blue} e_{x}} & {\color{blue} e_{y}} & {\color{blue} e_{z}} & {\color{green} e_{xy}} & {\color{green} e_{xz}} & {\color{green} e_{yz}} & {\color{orange} e_{xyz}} \\\hline {\color{blue} e_{x}} & {\color{red} 1 } & {\color{green} e_{xy}} & {\color{green} e_{xz}} & {\color{blue} e_{y}} & {\color{blue} e_{z}} & {\color{orange} e_{xyz}} & {\color{green} e_{yz}} \\ {\color{blue} e_{y}} & {\color{green} -e_{xy}} & {\color{red} 1 } & {\color{green} e_{yz}} & {\color{blue} -e_{x}} & {\color{orange} -e_{xyz}} & {\color{blue} e_{z}} & {\color{green} -e_{xz}} \\ {\color{blue} e_{z}} & {\color{green} -e_{xz}} & {\color{green} -e_{yz}} & {\color{red} 1 } & {\color{orange} e_{xyz}}& {\color{blue} -e_{x}} & {\color{blue} -e_{y}} & {\color{green} e_{xy}} \\\hline {\color{green} e_{xy}} & {\color{blue} -e_{y}} & {\color{blue} e_{x}} & {\color{orange} e_{xyz}} & {\color{red} -1} & {\color{green} -e_{yz}} & {\color{green} e_{xz}} & {\color{blue} -e_{z}} \\ {\color{green} e_{xz}} & {\color{blue} -e_{z}} & {\color{orange} -e_{xyz}} & {\color{blue} e_{x}} & {\color{green} e_{yz}} & {\color{red} -1 } & {\color{green} -e_{xy}} & {\color{blue} e_{y}} \\ {\color{green} e_{yz}} & {\color{orange} e_{xyz}} & {\color{blue} -e_{z}} & {\color{blue} e_{y}} & {\color{green} -e_{xz}} & {\color{green} e_{xy}} & {\color{red} -1 } & {\color{blue} -e_{x}} \\\hline {\color{orange} e_{xyz}} & {\color{green} e_{yz}} & {\color{green} -e_{xz}} & {\color{green} e_{xy}} & {\color{blue} -e_{z}} & {\color{blue} e_{y}} & {\color{blue} -e_{x}} & {\color{red} -1 } \end{array}$

In four dimensions:

$\begin{array}{r|rrrr|rrrrrr|rrrr|r} {\color{red} 1 } & {\color{black} e_{1}} & {\color{black} e_{2}} & {\color{black} e_{3}} & {\color{black} e_{4}} & {\color{black} e_{12}} & {\color{black} e_{13}} & {\color{black} e_{14}} & {\color{black} e_{23}} & {\color{black} e_{24}} & {\color{black} e_{34}} & {\color{black} e_{123}} & {\color{black} e_{124}} & {\color{black} e_{134}} & {\color{black} e_{234}} & {\color{red} e_{1234}} \\\hline {\color{black} e_{1}} & {\color{red} 1 } & {\color{black} e_{12}} & {\color{black} e_{13}} & {\color{black} e_{14}} & {\color{black} e_{2}} & {\color{black} e_{3}} & {\color{black} e_{4}} & {\color{black} e_{123}} & {\color{black} e_{124}} & {\color{black} e_{134}} & {\color{black} e_{23}} & {\color{black} e_{24}} & {\color{black} e_{34}} & {\color{red} e_{1234}} & {\color{black} e_{234}} \\ {\color{black} e_{2}} & {\color{black} -e_{12}} & {\color{red} 1 } & {\color{black} e_{23}} & {\color{black} e_{24}} & {\color{black} -e_{1}} & {\color{black} -e_{123}} & {\color{black} -e_{124}} & {\color{black} e_{3}} & {\color{black} e_{4}} & {\color{black} e_{234}} & {\color{black} -e_{13}} & {\color{black} -e_{14}} & {\color{red} -e_{1234}} & {\color{black} e_{34}} & {\color{black} -e_{134}} \\ {\color{black} e_{3}} & {\color{black} -e_{13}} & {\color{black} -e_{23}} & {\color{red} 1 } & {\color{black} e_{34}} & {\color{black} e_{123}} & {\color{black} -e_{1}} & {\color{black} -e_{134}} & {\color{black} -e_{2}} & {\color{black} -e_{234}} & {\color{black} e_{4}} & {\color{black} e_{12}} & {\color{red} e_{1234}} & {\color{black} -e_{14}} & {\color{black} -e_{24}} & {\color{black} e_{124}} \\ {\color{black} e_{4}} & {\color{black} -e_{14}} & {\color{black} -e_{24}} & {\color{black} -e_{34}} & {\color{red} 1 } & {\color{black} e_{124}} & {\color{black} e_{123}} & {\color{black} -e_{1}} & {\color{black} e_{234}} & {\color{black} -e_{2}} & {\color{black} -e_{3}} & {\color{red} -e_{1234}} & {\color{black} e_{12}} & {\color{black} e_{13}} & {\color{black} e_{23}} & {\color{black} -e_{123}} \\\hline {\color{black} e_{12}} & {\color{black} -e_{2}} & {\color{black} e_{1}} & {\color{black} e_{123}} & {\color{black} e_{124}} & {\color{red} -1 } & {\color{black} -e_{23}} & {\color{black} -e_{24}} & {\color{black} e_{13}} & {\color{black} e_{14}} & {\color{red} e_{1234}} & {\color{black} -e_{3}} & {\color{black} -e_{4}} & {\color{black} -e_{234}} & {\color{black} e_{134}} & {\color{black} -e_{34}} \\ {\color{black} e_{13}} & {\color{black} -e_{3}} & {\color{black} -e_{123}} & {\color{black} e_{1}} & {\color{black} e_{134}} & {\color{black} e_{23}} & {\color{red} -1 } & {\color{black} -e_{34}} & {\color{black} -e_{12}} & {\color{red} -e_{1234}} & {\color{black} e_{14}} & {\color{black} e_{2}} & {\color{black} e_{234}} & {\color{black} -e_{4}} & {\color{black} -e_{124}} & {\color{black} e_{24}} \\ {\color{black} e_{14}} & {\color{black} -e_{4}} & {\color{black} -e_{124}} & {\color{black} -e_{123}} & {\color{black} e_{1}} & {\color{black} e_{24}} & {\color{black} e_{34}} & {\color{red} -1 } & {\color{red} e_{1234}} & {\color{black} -e_{12}} & {\color{black} -e_{13}} & {\color{black} -e_{234}} & {\color{black} e_{2}} & {\color{black} e_{3}} & {\color{black} e_{123}} & {\color{black} -e_{23}} \\ {\color{black} e_{23}} & {\color{black} e_{123}} & {\color{black} -e_{3}} & {\color{black} e_{2}} & {\color{black} e_{234}} & {\color{black} -e_{13}} & {\color{black} e_{12}} & {\color{red} e_{1234}} & {\color{red} -1 } & {\color{black} -e_{34}} & {\color{black} e_{24}} & {\color{black} -e_{1}} & {\color{black} -e_{134}} & {\color{black} e_{124}} & {\color{black} -e_{4}} & {\color{black} -e_{14}} \\ {\color{black} e_{24}} & {\color{black} e_{124}} & {\color{black} -e_{4}} & {\color{black} -e_{234}} & {\color{black} e_{2}} & {\color{black} -e_{14}} & {\color{red} -e_{1234}} & {\color{black} e_{12}} & {\color{black} e_{34}} & {\color{red} -1 } & {\color{black} -e_{23}} & {\color{black} e_{123}} & {\color{black} -e_{144}} & {\color{black} -e_{123}} & {\color{black} e_{3}} & {\color{black} e_{13}} \\ {\color{black} e_{34}} & {\color{black} e_{134}} & {\color{black} e_{234}} & {\color{black} -e_{4}} & {\color{black} e_{3}} & {\color{red} e_{3412}} & {\color{black} -e_{14}} & {\color{black} e_{13}} & {\color{black} -e_{24}} & {\color{black} e_{23}} & {\color{red} -1 } & {\color{black} -e_{124}} & {\color{black} e_{123}} & {\color{black} -e_{1}} & {\color{black} -e_{2}} & {\color{black} -e_{12}} \\\hline {\color{black} e_{123}} & {\color{black} e_{23}} & {\color{black} -e_{13}} & {\color{black} e_{12}} & {\color{red} e_{1234}} & {\color{black} -e_{3}} & {\color{black} e_{2}} & {\color{black} e_{234}} & {\color{black} -e_{1}} & {\color{black} -e_{134}} & {\color{black} e_{124}} & {\color{red} -1 } & {\color{black} -e_{34}} & {\color{black} e_{24}} & {\color{black} -e_{14}} & {\color{black} -e_{4}} \\ {\color{black} e_{124}} & {\color{black} e_{24}} & {\color{black} -e_{14}} & {\color{red} -e_{1234}} & {\color{black} e_{12}} & {\color{black} -e_{4}} & {\color{black} -e_{234}} & {\color{black} e_{2}} & {\color{black} e_{123}} & {\color{black} -e_{1}} & {\color{black} -e_{123}} & {\color{black} e_{34}} & {\color{red} -1 } & {\color{black} -e_{23}} & {\color{black} e_{13}} & {\color{black} e_{3}} \\ {\color{black} e_{134}} & {\color{black} e_{34}} & {\color{red} e_{1234}} & {\color{black} -e_{14}} & {\color{black} e_{13}} & {\color{black} e_{234}} & {\color{black} -e_{4}} & {\color{black} e_{3}} & {\color{black} -e_{124}} & {\color{black} e_{123}} & {\color{black} -e_{1}} & {\color{black} -e_{24}} & {\color{black} e_{23}} & {\color{red} -1 } & {\color{black} -e_{12}} & {\color{black} -e_{2}} \\ {\color{black} e_{234}} & {\color{red} -e_{1234}} & {\color{black} e_{34}} & {\color{black} -e_{24}} & {\color{black} e_{23}} & {\color{black} -e_{134}} & {\color{black} e_{124}} & {\color{black} -e_{123}} & {\color{black} -e_{4}} & {\color{black} e_{3}} & {\color{black} -e_{2}} & {\color{black} e_{14}} & {\color{black} -e_{13}} & {\color{black} e_{12}} & {\color{red} -1 } & {\color{black} e_{1}} \\\hline {\color{red} e_{1234}} & {\color{black} -e_{234}} & {\color{black} e_{134}} & {\color{black} -e_{124}} & {\color{black} e_{123}} & {\color{black} -e_{34}} & {\color{black} e_{24}} & {\color{black} -e_{23}} & {\color{black} -e_{14}} & {\color{black} e_{13}} & {\color{black} -e_{12}} & {\color{black} e_{4}} & {\color{black} -e_{3}} & {\color{black} e_{2}} & {\color{black} -e_{1}} & {\color{red} 1 } \end{array}$

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Basis

Every multivector of the Clifford algebra can be expressed as a linear combination of the canonical basis elements. The basis elements of the Clifford algebra Cℓ₃ are $\{{\color{red}1}, \, {\color{blue}e_x}, \, {\color{blue}e_y}, \, {\color{blue}e_z}, \, {\color{green}e_{xy}}, \, {\color{green}e_{xz}}, \, {\color{green}e_{yz}}, \, {\color{orange}e_{xyz}} \}$ and the general element of Cℓ₃ is given by

$A = a_0{\color{red}(1)} + a_1 {\color{blue}e_x} + a_2 {\color{blue}e_y} + a_3 {\color{blue}e_z} + a_4 {\color{green}e_{xy}} + a_5 {\color{green}e_{xz}} + a_6 {\color{green}e_{yz}} + a_7 {\color{orange}e_{xyz}}.$

If $a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7$ are all real then the Clifford algebra is Cℓ₃(R). If the coefficients are allowed to be complex then the Clifford algebra is Cℓ₃(C).

A multivector can be separated into components of different grades:

Failed to parse (syntax error): {\displaystyle \langle \mathbf{A} \rangle_0 = a_0{\color{red}(1)}\\ \langle \mathbf{A} \rangle_1 = a_1 {\color{blue}e_x} + a_2 {\color{blue}e_y} + a_3 {\color{blue}e_z}\\ \langle \mathbf{A} \rangle_2 = a_4 {\color{green}e_{xy}} + a_5 {\color{green}e_{xz}} + a_6 {\color{green}e_{yz}}\\ \langle \mathbf{A} \rangle_3 = a_7 {\color{orange}e_{xyz}}\\ }

The elements of even grade form a subalgebra because the sum or product of even grade elements always results in an element of even grade. The elements of odd grade do not form a subalgebra.

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Relation to other algebras

Failed to parse (syntax error): {\displaystyle Cℓ_0 (\mathbf{R})} : Real numbers (scalars). A scalar can (and should) be thought of as zero vectors multiplied together. See Empty product.

Failed to parse (syntax error): {\displaystyle Cℓ_0 (\mathbf{C})} : Complex numbers

Failed to parse (syntax error): {\displaystyle Cℓ_1 (\mathbf{R})} : Split-complex numbers

$\begin{array}{r|r} {\color{red} 1 } & {\color{blue} e_{x}} \\\hline {\color{blue} e_{x}} & {\color{red} 1 } \end{array}$

Failed to parse (syntax error): {\displaystyle Cℓ_1 (\mathbf{C})} : Bicomplex numbers

Failed to parse (syntax error): {\displaystyle Cℓ_2^0 (\mathbf{R})} : Complex numbers (The superscript 0 indicates the even subalgebra)

$\begin{array}{r|r} {\color{red} 1} & {\color{green} e_{xy}} \\\hline {\color{green} e_{xy}} & {\color{red} {-1}} \end{array}$

Failed to parse (syntax error): {\displaystyle Cℓ_3^0 (\mathbf{R})} : Quaternions

$\begin{array}{r|rrr} {\color{red} 1} & {\color{green} e_{xy}} & {\color{green} e_{xz}} & {\color{green} e_{yz}} \\\hline {\color{green} e_{xy}} & {\color{red} -1 } & {\color{green} -e_{yz}} & {\color{green} e_{xz}} \\ {\color{green} e_{xz}} & {\color{green} e_{yz}} & {\color{red} -1 } & {\color{green} -e_{xy}} \\ {\color{green} e_{yz}} & {\color{green} -e_{xz}} & {\color{green} e_{xy}} & {\color{red} -1 } \end{array}$

Failed to parse (syntax error): {\displaystyle Cℓ_3^0 (\mathbf{C})} : Biquaternions

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Multivector multiplication using tensors

To find the product

$AB = (a_0 {\color{red} 1 } + a_1 {\color{blue} e_{x}} + a_2 {\color{blue} e_{y}} + a_3 {\color{green} e_{xy}}) (b_0 {\color{red} 1 } + b_1 {\color{blue} e_{x}} + b_2 {\color{blue} e_{y}} + b_3 {\color{green} e_{xy}})$

we have to multiply every component of the first multivector with every component of the second multivector.

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} AB = & \phantom{+} (a_0 b_0 {\color{red}1}{\color{red}1} + a_0 b_1 {\color{red}1}{\color{blue} e_{x}} + a_0 b_2 {\color{red}1}{\color{blue} e_{y}} + a_0 b_3 {\color{red}1}{\color{green} e_{xy}}) \\ &+ (a_1 b_0 {\color{blue} e_{x}}{\color{red}1} + a_1 b_1 {\color{blue} e_{x}}{\color{blue} e_{x}} + a_1 b_2 {\color{blue} e_{x}}{\color{blue} e_{y}} + a_1 b_3 {\color{blue} e_{x}}{\color{green} e_{xy}}) \\ &+ (a_2 b_0 {\color{blue} e_{y}}{\color{red}1} + a_2 b_1 {\color{blue} e_{y}}{\color{blue} e_{x}} + a_2 b_2 {\color{blue} e_{y}}{\color{blue} e_{y}} + a_2 b_3 {\color{blue} e_{y}}{\color{green} e_{xy}}) \\ &+ (a_3 b_0 {\color{green} e_{xy}}{\color{red}1} + a_3 b_1 {\color{green} e_{xy}}{\color{blue} e_{x}} + a_3 b_2 {\color{green} e_{xy}}{\color{blue} e_{y}} + a_3 b_3 {\color{green} e_{xy}}{\color{green} e_{xy}}) \end{split} }

Then we reduce each of the 16 resulting terms to its standard form.

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} AB = & \phantom{+} (a_0 b_0 {\color{red}1} + a_0 b_1 {\color{blue} e_{x}} + a_0 b_2 {\color{blue} e_{y}} + a_0 b_3 {\color{green} e_{xy}}) \\ &+ (a_1 b_0 {\color{blue} e_{x}} + a_1 b_1 {\color{red}1} + a_1 b_2 {\color{green} e_{xy}} + a_1 b_3 {\color{blue} e_{y}}) \\ &+ (a_2 b_0 {\color{blue} e_{y}} - a_2 b_1 {\color{green} e_{xy}} + a_2 b_2 {\color{red}1} - a_2 b_3 {\color{blue} e_{x}}) \\ &+ (a_3 b_0 {\color{green} e_{xy}} - a_3 b_1 {\color{blue} e_{y}} + a_3 b_2 {\color{blue} e_{x}} - a_3 b_3 {\color{red}1}) \end{split} }

Finally we collect like products into the four components of the final multivector.

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} AB = & \phantom{+} ( a_0 b_0 + a_1 b_1 + a_2 b_2 - a_3 b_3 ) {\color{red}1} \\ &+ ( a_1 b_0 + a_0 b_1 + a_3 b_2 - a_2 b_3 ) {\color{blue} e_{x} } \\ &+ ( a_2 b_0 - a_3 b_1 + a_0 b_2 + a_1 b_3 ) {\color{blue} e_{y} } \\ &+ ( a_3 b_0 - a_2 b_1 + a_1 b_2 + a_0 b_3 ) {\color{green} e_{xy} } \end{split} }

This is all very tedious and error-prone. It would be nice if there was some way to cut straight to the end. Tensor notation allows us to do just that.

To find the tensor that we need we first need to know which terms end up as scalars, which terms end up as vectors...etc. There is an easy way to do this and it involves the multiplication table.

First lets start with an easy one.

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Complex numbers

The multiplication table for Failed to parse (syntax error): {\displaystyle Cℓ_2^0 (\mathbf{R})} (Which is isomorphic to complex numbers)

$\begin{array}{r|r} {\color{red} 1} & {\color{green} e_{xy}} \\\hline {\color{green} e_{xy}} & {\color{red} -1 } \end{array}$

We can see then that:

$\begin{align} \left( \begin{array}{rrrr} {\color{red} 1} & {\color{green} e_{xy}} \\ {\color{green} e_{xy}} & {\color{red} 1 } \end{array} \right) \left( \begin{array}{rrrr} {\color{red} 1} \\ {\color{green} e_{xy}} \end{array} \right) &= \left( \begin{array}{lcl} {\color{red} 1} \cdot {\color{red} 1} & + & {\color{green} e_{xy}} {\color{green} e_{xy}} \\ {\color{green} e_{xy}} {\color{red} 1} & + & {\color{red} 1 } {\color{green} e_{xy}} \end{array} \right) \\ &= \left( \begin{array}{lcl} {\color{red} 1} & - & {\color{red} 1 } \\ {\color{green} e_{xy}} & + & {\color{green} e_{xy}} \end{array} \right) \end{align}$

It worked! All the terms in the first row are scalars and all the terms in the second row are bivectors. This is exactly what we are looking for.

$\begin{align} \left( \begin{array}{rrrr} b_0 & \cdot & {\color{red} 1} \quad & b_1 & \cdot & {\color{green} e_{xy}} \\ b_1 & \cdot & {\color{green} e_{xy}} \quad & b_0 & \cdot & {\color{red} 1 } \end{array} \right) \left( \begin{array}{rrrr} a_0 & \cdot & {\color{red} 1} \\ a_1 & \cdot & {\color{green} e_{xy}} \end{array} \right) &= \left( \begin{array}{lcl} b_0 a_0 & \cdot & {\color{red} 1} \cdot {\color{red} 1} & + & b_1 a_1 & \cdot & {\color{green} e_{xy}} {\color{green} e_{xy}} \\ b_1 a_0 & \cdot & {\color{green} e_{xy}} {\color{red} 1} & + & b_0 a_1 & \cdot & {\color{red} 1 } {\color{green} e_{xy}} \end{array} \right) \\ &= \left( \begin{array}{lcl} b_0 a_0 & \cdot & {\color{red} 1} & - & b_1 a_1 & \cdot & {\color{red} 1 } \\ b_1 a_0 & \cdot & {\color{green} e_{xy}} & + & b_0 a_1 & \cdot & {\color{green} e_{xy}} \end{array} \right) \end{align}$

Pay special attention to the signs in the final matrix above.

Therefore to find the product

$(a_0 {\color{red} 1 } + a_1 {\color{green} e_{xy}} ) (b_0 {\color{red} 1 } + b_1 {\color{green} e_{xy}} )$

We would multiply:

$\left( \begin{array}{rr} {\color{red} b_0 } & {\color{green} -b_1 } \\ {\color{green} b_1 } & {\color{red} b_0 } \end{array} \right) \left( \begin{array}{r} {\color{red} a_0 } \\ {\color{green} a_1 } \end{array} \right) = \left( \begin{array}{rcr} {\color{red} b_0 } {\color{red} a_0 }& - & {\color{red} b_1 } {\color{red} a_1 } \\ {\color{green} b_1 } {\color{green} a_0 }& + & {\color{green} b_0 } {\color{green} a_1 } \end{array} \right)$

Each row of the final matrix has exactly the right terms with exactly the right signs.

The vector above represents a complex number. You should think of the first column of the matrix above as representing another complex number. All the other terms in the matrix are just there to make our lives a little bit easier.

It works. It works so well that complex numbers can be represented as matrices as:

$\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$

Which corresponds perfectly to a multiplication table for complex numbers:

$\begin{array}{r|r} {\color{red} 1} & {\color{green} -i} \\\hline {\color{green} i} & {\color{red} 1 } \end{array}$

$\quad \quad \begin{array}{r|r} {\color{red} 1} & {\color{green} -e_{xy}} \\\hline {\color{green} e_{xy}} & {\color{red} 1 } \end{array}$

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Quaternions

The multiplication table for Failed to parse (syntax error): {\displaystyle Cℓ_3^0 (\mathbf{R})} (Which is isomorphic to quaternions) is:

$\begin{array}{r|rrr} {\color{red} 1} & {\color{green} e_{xy}} & {\color{green} e_{xz}} & {\color{green} e_{yz}} \\\hline {\color{green} e_{xy}} & {\color{red} -1 } & {\color{green} -e_{yz}} & {\color{green} e_{xz}} \\ {\color{green} e_{xz}} & {\color{green} e_{yz}} & {\color{red} -1 } & {\color{green} -e_{xy}} \\ {\color{green} e_{yz}} & {\color{green} -e_{xz}} & {\color{green} e_{xy}} & {\color{red} -1 } \end{array} \quad \quad \quad \begin{array}{l} i = e_{yz} \\ j = e_{xz} \\ k = e_{xy} \end{array}$

The entire 2nd row of the multiplication table is just ${\color{green} e_{xy}}$ multiplied by the entire first row.
The entire 3rd row of the multiplication table is just ${\color{green} e_{xz}}$ multiplied by the entire first row.
The entire 4th row of the multiplication table is just ${\color{green} e_{yz}}$ multiplied by the entire first row.

We can see then that if we multiply each row by the first row again then we get:

$\begin{align} \left( \begin{array}{rrrr} {\color{red} 1} & {\color{green} e_{xy}} & {\color{green} e_{xz}} & {\color{green} e_{yz}} \\ {\color{green} e_{xy}} & {\color{red} 1 } & {\color{green} e_{yz}} & {\color{green} e_{xz}} \\ {\color{green} e_{xz}} & {\color{green} e_{yz}} & {\color{red} 1 } & {\color{green} e_{xy}} \\ {\color{green} e_{yz}} & {\color{green} e_{xz}} & {\color{green} e_{xy}} & {\color{red} 1 } \end{array} \right) \left( \begin{array}{rrrr} {\color{red} 1} \\ {\color{green} e_{xy}} \\ {\color{green} e_{xz}} \\ {\color{green} e_{yz}} \end{array} \right) &= \left( \begin{array}{rrrr} {\color{red} 1} \cdot {\color{red} 1} & + & {\color{green} e_{xy}} {\color{green} e_{xy}} & + & {\color{green} e_{xz}} {\color{green} e_{xz}} & + & {\color{green} e_{yz}} {\color{green} e_{yz}} \\ {\color{green} e_{xy}} {\color{red} 1} & + & {\color{red} 1 } {\color{green} e_{xy}} & + & {\color{green} e_{yz}} {\color{green} e_{xz}} & + & {\color{green} e_{xz}} {\color{green} e_{yz}} \\ {\color{green} e_{xz}} {\color{red} 1} & + & {\color{green} e_{yz}} {\color{green} e_{xy}} & + & {\color{red} 1 } {\color{green} e_{xz}} & + & {\color{green} e_{xy}} {\color{green} e_{yz}} \\ {\color{green} e_{yz}} {\color{red} 1} & + & {\color{green} e_{xz}} {\color{green} e_{xy}} & + & {\color{green} e_{xy}} {\color{green} e_{xz}} & + & {\color{red} 1 } {\color{green} e_{yz}} \end{array} \right) \\ &= \left( \begin{array}{llll} {\color{red} 1} & - & {\color{red} 1 } & - & {\color{red} 1 } & - & {\color{red} 1 } \\ {\color{green} e_{xy}} & + & {\color{green} e_{xy}}& + & {\color{green} e_{xy}} & - & {\color{green} e_{xy}} \\ {\color{green} e_{xz}} & - & {\color{green} e_{xz}} & + & {\color{green} e_{xz}} & + & {\color{green} e_{xz}} \\ {\color{green} e_{yz}} & + & {\color{green} e_{yz}} & - & {\color{green} e_{yz}} & + & {\color{green} e_{yz}} \end{array} \right) \end{align}$

This works because we have in effect multiplied each term by a second term twice. In other words we have multiplied every term by the square of another term and the square of every term is either 1 or -1.

Therefore to find the product

$(a_0 {\color{red} 1 } + a_1 {\color{green} e_{xy}} + a_2 {\color{green} e_{xz}} + a_3 {\color{green} e_{yz}}) (b_0 {\color{red} 1 } + b_1 {\color{green} e_{xy}} + b_2 {\color{green} e_{xz}} + b_3 {\color{green} e_{yz}})$

We would multiply:

$\left( \begin{array}{rrrr} {\color{red} b_0 } & {\color{green} -b_1 } & {\color{green} -b_2 } & {\color{green} -b_3 } \\ {\color{green} b_1 } & {\color{red} b_0 } & {\color{green} b_3 } & {\color{green} {-b_2 }} \\ {\color{green} b_2 } & {\color{green} -b_3 } & {\color{red} b_0 } & {\color{green} b_1 } \\ {\color{green} b_3 } & {\color{green} b_2 } & {\color{green} {-b_1 }} & {\color{red} b_0 } \end{array} \right) \left( \begin{array}{rrrr} {\color{red} a_0 } \\ {\color{green} a_1 } \\ {\color{green} a_2 } \\ {\color{green} a_3 } \end{array} \right) = \left( \begin{array}{rrrr} {\color{red} b_0 } {\color{red} a_0 } & - & {\color{red} b_1 } {\color{red} a_1 } & - & {\color{red} b_2 } {\color{red} a_2 } & - & {\color{red} b_3 } {\color{red} a_3 } \\ {\color{green} b_1 } {\color{green} a_0 } & + & {\color{green} b_0 } {\color{green} a_1 } & + & {\color{green} b_3 } {\color{green} a_2 } & - & {\color{green} {b_2 }} {\color{green} a_3 } \\ {\color{green} b_2 } {\color{green} a_0 } & - & {\color{green} b_3 } {\color{green} a_1 } & + & {\color{green} b_0 } {\color{green} a_2 } & + & {\color{green} b_1 } {\color{green} a_3 } \\ {\color{green} b_3 } {\color{green} a_0 } & + & {\color{green} b_2 } {\color{green} a_1 } & - & {\color{green} {b_1 }} {\color{green} a_2 } & + & {\color{green} b_0 } {\color{green} a_3 } \end{array} \right)$

Just as complex numbers can be represented as matrices, so a quaternion can be represented as:

$\left( \begin{array}{rrrr} a & -b & -c & -d \\ b & a & d &-c \\ c & -d & a & b \\ d & c & -b & a \end{array} \right)= \left( \begin{array}{rr} \left( \begin{array}{rr} a & -b \\ b & a \end{array} \right) & - \left( \begin{array}{rr} c & d \\ -d &c \end{array} \right) \\ \left( \begin{array}{rr} c & -d \\ d & c \end{array} \right) & \left( \begin{array}{rr} a & b \\ -b & a \end{array} \right) \end{array} \right)$

Which corresponds to a multiplication table for quaternions:

1 −k −j −i k 1 i −j j −i 1 k i j −k 1

$\begin{array}{r|rrr} {\color{red} 1} & {\color{green} -e_{xy}} & {\color{green} -e_{xz}} & {\color{green} -e_{yz}} \\\hline {\color{green} e_{xy}} & {\color{red} 1 } & {\color{green} e_{yz}} & {\color{green} -e_{xz}} \\ {\color{green} e_{xz}} & {\color{green} -e_{yz}} & {\color{red} 1 } & {\color{green} e_{xy}} \\ {\color{green} e_{yz}} & {\color{green} e_{xz}} & {\color{green} -e_{xy}} & {\color{red} 1 } \end{array}$

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CL₂

The multiplication table for Failed to parse (syntax error): {\displaystyle Cℓ_{2} (\mathbf{R})} is:

$\begin{array}{r|rr|r} {\color{red} 1 } & {\color{blue} e_{x}} & {\color{blue} e_{y}} & {\color{green} e_{xy}} \\\hline {\color{blue} e_{x}} & {\color{red} 1 } & {\color{green} e_{xy}} & {\color{blue} {e_{y}}} \\ {\color{blue} e_{y}} & {\color{green} -e_{xy}} & {\color{red} 1 } & {\color{blue} -e_{x}} \\\hline {\color{green} e_{xy}} & {\color{blue} -e_{y}} & {\color{blue} {e_{x}}} & {\color{red} {-1}} \end{array}$

We can see then that:

$\begin{align} \left( \begin{array}{rrrr} {\color{red} 1 } & {\color{blue} e_{x}} & {\color{blue} e_{y}} & {\color{green} e_{xy}} \\ {\color{blue} e_{x}} & {\color{red} 1 } & {\color{green} e_{xy}} & {\color{blue} {e_{y}}} \\ {\color{blue} e_{y}} & {\color{green} e_{xy}} & {\color{red} 1 } & {\color{blue} e_{x}} \\ {\color{green} e_{xy}} & {\color{blue} e_{y}} & {\color{blue} {e_{x}}} & {\color{red} {1}} \end{array} \right) \left( \begin{array}{rrrr} {\color{red} 1 } \\ {\color{blue} e_{x}} \\ {\color{blue} e_{y}} \\ {\color{green} e_{xy}} \end{array} \right) &= \left( \begin{array}{rrrr} {\color{red} 1 } \cdot {\color{red} 1 } & + & {\color{blue} e_{x}} {\color{blue} e_{x}} & + & {\color{blue} e_{y}} {\color{blue} e_{y}} & + & {\color{green} e_{xy}} {\color{green} e_{xy}} \\ {\color{blue} e_{x}} {\color{red} 1 } & + & {\color{red} 1 } {\color{blue} e_{x}} & + & {\color{green} e_{xy}} {\color{blue} e_{y}} & + & {\color{blue} {e_{y}}} {\color{green} e_{xy}} \\ {\color{blue} e_{y}} {\color{red} 1 } & + & {\color{green} e_{xy}} {\color{blue} e_{x}} & + & {\color{red} 1 } {\color{blue} e_{y}} & + & {\color{blue} e_{x}} {\color{green} e_{xy}} \\ {\color{green} e_{xy}} {\color{red} 1 } & + & {\color{blue} e_{y}} {\color{blue} e_{x}} & + & {\color{blue} {e_{x}}} {\color{blue} e_{y}} & + & {\color{red} {1}} {\color{green} e_{xy}} \end{array} \right) \\ &= \left( \begin{array}{llll} {\color{red}1} & + & {\color{red}1} & + & {\color{red}1} & - & {\color{red}1} \\ {\color{blue}e_{x}} & + & {\color{blue}e_{x}} & + & {\color{blue}e_{x}} & - & {\color{blue}e_{x}} \\ {\color{blue}e_{y}} & - & {\color{blue}e_{y}} & + & {\color{blue}e_{y}} & + & {\color{blue}e_{y}} \\ {\color{green}e_{xy}} & - & {\color{green}e_{xy}} & + & {\color{green}e_{xy}} & + & {\color{green}e_{xy}} \end{array} \right) \end{align}$

Therefore to find the product

$(a_0 {\color{red} 1 } + a_1 {\color{blue} e_{x}} + a_2 {\color{blue} e_{y}} + a_3 {\color{green} e_{xy}}) (b_0 {\color{red} 1 } + b_1 {\color{blue} e_{x}} + b_2 {\color{blue} e_{y}} + b_3 {\color{green} e_{xy}})$

We would multiply:

$\left( \begin{array}{rrrr} {\color{red} b_0 } & {\color{blue} b_1 } & \phantom{-} {\color{blue} b_2 } & {\color{green} -b_3 } \\ {\color{blue} b_1 } & {\color{red} b_0 } & {\color{green} b_3 } & {\color{blue} {-b_2 }} \\ {\color{blue} b_2 } & {\color{green} -b_3 } & {\color{red} b_0 } & {\color{blue} b_1 } \\ {\color{green} b_3 } & {\color{blue} -b_2 } & {\color{blue} {b_1 }} & {\color{red} b_0 } \end{array} \right) \left( \begin{array}{rrrr} {\color{red} a_0 } \\ {\color{blue} a_1 } \\ {\color{blue} a_2 } \\ {\color{green} a_3 } \end{array} \right) = \left( \begin{array}{rrrr} {\color{red} b_0 } {\color{red} a_0 } & + & {\color{red} b_1 } {\color{red} a_1 } & + & {\color{red} b_2 } {\color{red} a_2 } & - & {\color{red} b_3 } {\color{red} a_3 } \\ {\color{blue} b_1 } {\color{blue} a_0 } & + & {\color{blue} b_0 } {\color{blue} a_1 } & + & {\color{blue} b_3 } {\color{blue} a_2 } & - & {\color{blue} {b_2 }} {\color{blue} a_3 } \\ {\color{blue} b_2 } {\color{blue} a_0 } & - & {\color{blue} b_3 } {\color{blue} a_1 } & + & {\color{blue} b_0 } {\color{blue} a_2 } & + & {\color{blue} b_1 } {\color{blue} a_3 } \\ {\color{green} b_3 } {\color{green} a_0 } & - & {\color{green} b_2 } {\color{green} a_1 } & + & {\color{green} {b_1 }} {\color{green} a_2 } & + & {\color{green} b_0 } {\color{green} a_3 } \end{array} \right)$

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Squares of pseudoscalars are either 1 or -1

In 0 dimensions:

$(1)^2 = 1$

In 1 dimension:

$(e_{1})^2 = e_{11} = 1$

In 2 dimensions:

$(e_{12})^2 = e_{1212} = -e_{11 \color{red}{22}} = -1$

In 3 dimensions:

$(e_{123})^2 = e_{123123} = e_{1212 \color{red}{33}} = -1$

In 4 dimensions:

$(e_{1234})^2 = e_{12341234} = -e_{123123 \color{red}{44}} = 1$

In 5 dimensions:

$(e_{12345})^2 = e_{1234512345} = e_{12341234 \color{red}{55}} = 1$

In 6 dimensions:

$(e_{123456})^2 = e_{123456123456} = -e_{1234512345 \color{red}{66}} = -1$

In 7 dimensions:

$(e_{1234567})^2 = e_{12345671234567} = e_{123456123456 \color{red}{77}} = -1$

In 8 dimensions:

$(e_{12345678})^2 = e_{1234567812345678} = -e_{12345671234567 \color{red}{88}} = 1$

In 9 dimensions:

$(e_{123456789})^2 = e_{123456789123456789} = e_{1234567812345678 \color{red}{99}} = 1$

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Bivectors in higher dimensions

A simple bivector can be used to represent a single rotation.

In four dimensions a rigid object can rotate in two different ways simultaneously. Such a rotation can only be represented as the sum of two simple bivectors.

In six dimensions a rigid object can rotate in three different ways simultaneously. Such a rotation can only be represented as the sum of three simple bivectors.

From Wikipedia:Bivector

The wedge product of two vectors is a bivector, but not all bivectors are wedge products of two vectors. For example, in four dimensions the bivector

$\mathbf{B} = \mathbf{e}_1 \wedge \mathbf{e}_2 + \mathbf{e}_3 \wedge \mathbf{e}_4 = \mathbf{e}_1 \mathbf{e}_2 + \mathbf{e}_3\mathbf{e}_4 = \mathbf{e}_{12} + \mathbf{e}_{34}$

cannot be written as the wedge product of two vectors. A bivector that can be written as the wedge product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions;

A bivector has a real square if and only if it is simple.

$\begin{align} (\mathbf{e}_{12})^2 &= e_{1212} \\ &= 1 \end{align}$

But:

$\begin{align} (\mathbf{e}_{12} + \mathbf{e}_{34})^2 &= e_{1212} + e_{1234} + e_{3412} + e_{3434} \\ &= -e_{1122} + e_{1234} + e_{1234} - e_{3344} \\ &= -1 + 2 e_{1234} -1 \\ &= 2 e_{1234} - 2 \end{align}$

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Other quadratic forms

The square of a vector is:

Failed to parse (unknown function "\begin{split}"): {\displaystyle \begin{split} \mathbf{v} \mathbf{v} &= (v_{x} {\color{blue}e_{x}} + v_{y} {\color{blue}e_{y}}) (v_{x} {\color{blue}e_{x}} + v_{y} {\color{blue}e_{y}}) \\ &= v_{x} v_{x} {\color{red}e_{x} e_{x}} + v_{x} v_{y} {\color{green}e_{x} e_{y}} + v_{y} v_{x} {\color{green}e_{y} e_{x}} + v_{y} v_{y} {\color{red}e_{y} e_{y}} \\ &= v_{x} v_{x} {\color{red}(1)} + v_{y} v_{y} {\color{red}(1)} + v_{x} v_{y} {\color{green}e_{x} e_{y}} - v_{y} v_{x} {\color{green}e_{x} e_{y}} \\ &= (v_{x} v_{x} + v_{y} v_{y}){\color{red}(1)} + (v_{x} v_{y} - v_{y} v_{x}) {\color{green}e_{xy}} \\ &= (v_{x}^2 + v_{y}^2){\color{red}(1)} + (0) {\color{green}e_{xy}} \\ &= (v_{x}^2 + v_{y}^2){\color{red}(1)} \\ &= {\color{red}\text{scalar}} \end{split} }

($v_{x}^2 + v_{y}^2$) is called the quadratic form. In this case both terms are positive but some Clifford algebras have quadratic forms with negative terms. Some have both positive and negative terms.

From Wikipedia:Clifford algebra:

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:

$Q(v) = v^2 = v_1^2 + \cdots + v_p^2 - v_{p+1}^2 - \cdots - v_{p+q}^2 ,$

where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the ^*signature of the quadratic form. The real vector space with this quadratic form is often denoted R^p,q. The Clifford algebra on R^p,q is denoted Cℓ_p,q(R). The symbol Cℓ_n(R) means either Cℓ_n,0(R) or Cℓ_0,n(R) depending on whether the author prefers positive-definite or negative-definite spaces.

A standard basis^w {e_i} for R^p,q consists of n = p + q mutually orthogonal vectors, p of which square to +1 and q of which square to −1. The algebra Cℓ_p,q(R) will therefore have p vectors that square to +1 and q vectors that square to −1.

From Wikipedia:Spacetime algebra:

^*Spacetime algebra (STA) is a name for the Clifford algebra^w Cl_3,1(R), or equivalently the geometric algebra^w G(M⁴), which can be particularly closely associated with the geometry of special relativity^w and relativistic spacetime^w. See also ^*Algebra of physical space.

The spacetime algebra may be built up from an orthogonal basis of one time-like vector $\gamma_0$ and three space-like vectors, $\{\gamma_1, \gamma_2, \gamma_3\}$, with the multiplication rule

$\gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu}$

where $\eta_{\mu \nu}$ is the Minkowski metric^w with signature (− + + +).

Thus:

$\gamma_0^2 = {-1}$

$\gamma_1^2 = \gamma_2^2 = \gamma_3^2 = {+1}$

$\gamma_\mu \gamma_\nu = - \gamma_\nu \gamma_\mu$

The basis vectors $\gamma_k$ share these properties with the ^*Gamma matrices, but no explicit matrix representation need be used in STA.

Failed to parse (syntax error): {\displaystyle Cℓ_{3,0} (\mathbf{R})} :Algebra of physical space (Time = scalar)

Failed to parse (syntax error): {\displaystyle Cℓ_{3,1} (\mathbf{R})} :Spacetime algebra (Time = vector)

Failed to parse (syntax error): {\displaystyle Cℓ_{0,2} (\mathbf{R})} :Quaternions (Three quaternions = two vectors that square to -1 and one bivector that squares to -1)

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Rotors

See also: ^*Rotor (mathematics)

From Wikipedia:Geometric algebra

The inverse of a vector is:

$v^{-1} = \frac{1}{v} = \frac{v}{vv} = \frac{v}{v \cdot v + v \wedge v} = \frac{v}{v \cdot v}$

The projection of $v$ onto $a$ (or the parallel part) is

$v_{\| a} = (v \cdot a)a^{-1}$

and the rejection of $v$ from $a$ (or the orthogonal part) is

$v_{\perp a} = v - v_{\| a} = (v\wedge a)a^{-1} .$

The reflection $v'$ of a vector $v$ along a vector $a$, or equivalently across the hyperplane orthogonal to $a$, is the same as negating the component of a vector parallel to $a$. The result of the reflection will be

$v' = {-v_{\| a} + v_{\perp a}} = {-(v \cdot a)a^{-1} + (v \wedge a)a^{-1}} = {(-a \cdot v - a \wedge v)a^{-1}} = -ava^{-1}$

If a is a unit vector then $a^{-1}=\frac{a}{1} = a$ and therefore $v' = -ava$

$-ava$ is called the sandwich product which is called a double-sided product.

If we have a product of vectors $R = a_1a_2 \cdots a_r$ then we denote the reverse as

$R^\dagger = (a_1a_2\cdots a_r)^\dagger = a_r\cdots a_2 a_1.$

Any rotation is equivalent to 2 reflections.

$v'' = bv'b = bavab = RvR^\dagger$

R is called a Rotor

$R = ba = b \cdot a + b \wedge a = Scalar + Bivector = Multivector$

If a and b are unit vectors then the rotor is automatically normalised:

$RR^\dagger = R^\dagger R=1 .$

2 rotations becomes:

$R_2R_1MR_1^\dagger R_2^\dagger$

R₂R₁ represents Rotor R₁ rotated by Rotor R₂. This would be called a single-sided transformation. (R₂R₁R₂ would be double-sided.) Therefore rotors do not transform double-sided the same way that other objects do. They transform single-sided.

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Quaternions

The square root of the product of a quaternion with its conjugate is called its ^*norm:

$\lVert q \rVert = \sqrt{qq^*} = \sqrt{q^*q} = \sqrt{a^2 + b^2 + c^2 + d^2}$

A unit quaternion is a quaternion of norm one. Unit quaternions, also known as ^*versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions.

From Wikipedia:Quaternions and spatial rotation

Every nonzero quaternion has a multiplicative inverse

$(a+bi+cj+dk)^{-1} = \frac{1}{a^2+b^2+c^2+d^2}\,(a-bi-cj-dk).$

Thus quaternions form a ^*division algebra.

The inverse of a unit quaternion is obtained simply by changing the sign of its imaginary components.

A ^*3-D Euclidean vector such as (2, 3, 4) or (a_x, a_y, a_z) can be rewritten as 0 + 2 i + 3 j + 4 k or 0 + a_x i + a_y j + a_z k, where i, j, k are unit vectors representing the three ^*Cartesian axes. A rotation through an angle of θ around the axis defined by a unit vector

$\vec{u} = (u_x, u_y, u_z) = 0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}$

can be represented by a quaternion. This can be done using an ^*extension of Euler's formula^w:

$\mathbf{q} = e^{\frac{\theta}{2}{(0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})}} = \cos \frac{\theta}{2} + (0 + u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) \sin \frac{\theta}{2}$

It can be shown that the desired rotation can be applied to an ordinary vector $\mathbf{p} = (p_x, p_y, p_z) = 0 + p_x\mathbf{i} + p_y\mathbf{j} + p_z\mathbf{k}$ in 3-dimensional space, considered as a quaternion with a real coordinate equal to zero, by evaluating the conjugation of p by q:

$\mathbf{p'} = \mathbf{q} \mathbf{p} \mathbf{q}^{-1}$

using the ^*Hamilton product

The conjugate of a product of two quaternions is the product of the conjugates in the reverse order.

Conjugation by the product of two quaternions is the composition of conjugations by these quaternions: If p and q are unit quaternions, then rotation (conjugation) by pq is

$\mathbf{p q} \vec{v} (\mathbf{p q})^{-1} = \mathbf{p q} \vec{v} \mathbf{q}^{-1} \mathbf{p}^{-1} = \mathbf{p} (\mathbf{q} \vec{v} \mathbf{q}^{-1}) \mathbf{p}^{-1}$,

which is the same as rotating (conjugating) by q and then by p. The scalar component of the result is necessarily zero.

The imaginary part $b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ of a quaternion behaves like a vector $\vec{v} = (b,c,d)$ in three dimension vector space, and the real part a behaves like a ^*scalar in R. When quaternions are used in geometry, it is more convenient to define them as ^*a scalar plus a vector:

$a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = a + \vec{v}.$

When multiplying the vector/imaginary parts, in place of the rules i² = j² = k² = ijk = −1 we have the quaternion multiplication rule:

$\vec{v} \vec{w} = \vec{v} \times \vec{w} - \vec{v} \cdot \vec{w},$

From these rules it follows immediately that (^*see details):

$(s + \vec{v}) (t + \vec{w}) = (s t - \vec{v} \cdot \vec{w}) + (s \vec{w} + t \vec{v} + \vec{v} \times \vec{w}).$

It is important to note, however, that the vector part of a quaternion is, in truth, an "axial" vector or "pseudovector", not an ordinary or "polar" vector.

From Wikipedia:Quaternion#Quaternions_as_the_even_part_of_Cℓ3,0(R):

the reflection of a vector r in a plane perpendicular to a unit vector w can be written:

$r^{\prime} = - w\, r\, w.$

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

$v^{\prime\prime} = \sigma_2 \sigma_1 \, v \, \sigma_1 \sigma_2$

corresponds to a rotation of 180° in the plane containing σ₁ and σ₂.

This is very similar to the corresponding quaternion formula,

$v^{\prime\prime} = -\mathbf{k}\, v\, \mathbf{k}.$

In fact, the two are identical, if we make the identification

$\mathbf{k} = \sigma_2 \sigma_1, \mathbf{i} = \sigma_3 \sigma_2, \mathbf{j} = \sigma_1 \sigma_3$

and it is straightforward to confirm that this preserves the Hamilton relations

$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i} \mathbf{j} \mathbf{k} = -1.$

In this picture, quaternions correspond not to vectors but to bivectors^w – quantities with magnitude and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers^w becomes clearer, too: in 2D, with two vector directions σ₁ and σ₂, there is only one bivector basis element σ₁σ₂, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ₁σ₂, σ₂σ₃, σ₃σ₁, so three imaginaries.

The usefulness of quaternions for geometrical computations can be generalised to other dimensions, by identifying the quaternions as the even part Cℓ⁺_3,0(R) of the Clifford algebra^w Cℓ_3,0(R).

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Spinors

See also: ^*Bispinor

External link:An introduction to spinors

Spinors may be regarded as non-normalised rotors which transform single-sided.

Note: The (real) ^*spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°. This property characterizes spinors.

From Wikipedia:Orientation entanglement

In three dimensions...the ^*Lie group ^*SO(3) is not ^*simply connected. Mathematically, one can tackle this problem by exhibiting the ^*special unitary group SU(2), which is also the ^*spin group in three ^*Euclidean dimensions, as a ^*double cover of SO(3).

SU(2) is the following group,

$\mathrm{SU}(2) = \left \{ \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta\in\mathbf{C}, |\alpha|^2 + |\beta|^2 = 1\right \} ~,$

where the overline denotes ^*complex conjugation.

For comparison: Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as

$\begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}.$

If X = (x₁,x₂,x₃) is a vector in R³, then we identify X with the 2 × 2 matrix with complex entries

$X=\left(\begin{matrix}x_1&x_2-ix_3\\x_2+ix_3&-x_1\end{matrix}\right)$

Note that −det(X) gives the square of the Euclidean length of X regarded as a vector, and that X is a ^*trace-free, or better, trace-zero ^*Hermitian matrix.

The unitary group acts on X via

$X\mapsto MXM^+$

where M ∈ SU(2). Note that, since M is unitary,

$\det(MXM^+) = \det(X)$, and

$MXM^+$ is trace-zero Hermitian.

Hence SU(2) acts via rotation on the vectors X. Conversely, since any ^*change of basis which sends trace-zero Hermitian matrices to trace-zero Hermitian matrices must be unitary, it follows that every rotation also lifts to SU(2). However, each rotation is obtained from a pair of elements M and −M of SU(2). Hence SU(2) is a double-cover of SO(3). Furthermore, SU(2) is easily seen to be itself simply connected by realizing it as the group of unit ^*quaternions, a space ^*homeomorphic to the ^*3-sphere.

A unit quaternion has the cosine of half the rotation angle as its scalar part and the sine of half the rotation angle multiplying a unit vector along some rotation axis (here assumed fixed) as its pseudovector (or axial vector) part. If the initial orientation of a rigid body (with unentangled connections to its fixed surroundings) is identified with a unit quaternion having a zero pseudovector part and +1 for the scalar part, then after one complete rotation (2pi rad) the pseudovector part returns to zero and the scalar part has become -1 (entangled). After two complete rotations (4pi rad) the pseudovector part again returns to zero and the scalar part returns to +1 (unentangled), completing the cycle.

From Wikipedia:Spinors in three dimensions

The association of a spinor with a 2×2 complex ^*Hermitian matrix was formulated by Élie Cartan.

In detail, given a vector x = (x₁, x₂, x₃) of real (or complex) numbers, one can associate the complex matrix

$\vec{x} \rightarrow X \ =\left(\begin{matrix}x_3&x_1-ix_2\\x_1+ix_2&-x_3\end{matrix}\right).$

Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:

• det X = – (length x)².
• X ² = (length x)²I, where I is the identity matrix.
• $\frac{1}{2}(XY+YX)=({\bold x}\cdot{\bold y})I$
• $\frac{1}{2}(XY-YX)=iZ$ where Z is the matrix associated to the cross product z = x × y.
• If u is a unit vector, then −UXU is the matrix associated to the vector obtained from x by reflection in the plane orthogonal to u.
• It is an elementary fact from ^*linear algebra that any rotation in 3-space factors as a composition of two reflections. (Similarly, any orientation reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation, decomposing as the reflection in the plane perpendicular to a unit vector u₁ followed by the plane perpendicular to u₂, then the matrix U₂U₁XU₁U₂ represents the rotation of the vector x through R.

Having effectively encoded all of the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the ^*column vectors) play. Provisionally, a spinor is a column vector

$\xi=\left[\begin{matrix}\xi_1\\\xi_2\end{matrix}\right],$ with complex entries ξ₁ and ξ₂.

The space of spinors is evidently acted upon by complex 2×2 matrices. Furthermore, the product of two reflections in a given pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation, so there is an action of rotations on spinors.

Often, the first example of spinors that a student of physics encounters are the 2×1 spinors used in Pauli's theory of electron spin. The ^*Pauli matrices are a vector of three 2×2 ^*matrices that are used as ^*spin ^*operators.

Given a ^*unit vector in 3 dimensions, for example (a, b, c), one takes a ^*dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector.

The ^*eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector.

Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli spin matrices gives the matrix:

$S_u = (0.8,-0.6,0.0)\cdot \vec{\sigma}=0.8 \sigma_{1}-0.6\sigma_{2}+0.0\sigma_{3} = \begin{bmatrix} 0.0 & 0.8+0.6i \\ 0.8-0.6i & 0.0 \end{bmatrix}$

The eigenvectors may be found by the usual methods of ^*linear algebra, but a convenient trick is to note that a Pauli spin matrix is an ^*involutory matrix, that is, the squareof the above matrix is the identity matrix.

Thus a (matrix) solution to the eigenvector problem with eigenvalues of ±1 is simply 1 ± S_u. That is,

$S_u (1\pm S_u) = \pm 1 (1 \pm S_u)$

One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen is not zero. Taking the first column of the above, eigenvector solutions for the two eigenvalues are:

$\begin{bmatrix} 1.0+ (0.0)\\ 0.0 +(0.8-0.6i) \end{bmatrix}, \begin{bmatrix} 1.0- (0.0)\\ 0.0-(0.8-0.6i) \end{bmatrix}$

The trick used to find the eigenvectors is related to the concept of ^*ideals, that is, the matrix eigenvectors (1 ± S_u)/2 are ^*projection operators or ^*idempotents and therefore each generates an ideal in the Pauli algebra. The same trick works in any ^*Clifford algebra, in particular the ^*Dirac algebra that are discussed below. These projection operators are also seen in ^*density matrix theory where they are examples of pure density matrices.

More generally, the projection operator for spin in the (a, b, c) direction is given by

$\frac{1}{2}\begin{bmatrix}1+c&a-ib\\a+ib&1-c\end{bmatrix}$

and any non zero column can be taken as the projection operator. While the two columns appear different, one can use a² + b² + c² = 1 to show that they are multiples (possibly zero) of the same spinor.

From Wikipedia:Tensor#Spinors:

When changing from one ^*orthonormal basis (called a frame) to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not ^*simply connected (see ^*orientation entanglement and ^*plate trick): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1. A ^*spinor is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.

Succinctly, spinors are elements of the ^*spin representation of the rotation group, while tensors are elements of its ^*tensor representations. Other ^*classical groups have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.

From Wikipedia:Spinor:

Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966: "Spinors...provide a linear representation of the group of rotations in a space with any number $n$ of dimensions, each spinor having $2^\nu$ components where $n = 2\nu+1$ or $2\nu$." The star (*) refers to Cartan 1913.

(Note: $\nu$ is the number of ^*simultaneous independent rotations an object can have in n dimensions.)

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, realizing it as a group of rotations among them, but it also acts on the column vectors (that is, the spinors).

From Wikipedia:Spinor:

In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. More precisely, it is the fermions of spin-1/2 that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic ^*Dirac equation for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.

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Functions

A function f is like a "black box" that takes an input x, and returns a single corresponding output f(x).

The red curve is the graph of a function f in the Cartesian plane, consisting of all points with coordinates of the form (x, f(x)). The property of having one output for each input is represented geometrically by the fact that each vertical line (such as the yellow line through the origin) has exactly one crossing point with the curve.

From Wikipedia:Function (mathematics)

In mathematics, a function is a ^*relation between a ^*set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function $f(x)=x^2$ that relates each ^*real number x to its square x². The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. See Tutorial:Evaluate by Substitution. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input ^*variable(s) are sometimes referred to as the argument(s) of the function.

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Euclids "common notions"

From Wikipedia:Euclidean geometry:

Things that do not differ from one another are equal to one another

a=a

Things that are equal to the same thing are also equal to one another

If

a=b b=c

then a=c

If equals are added to equals, then the wholes are equal

If

a=b c=d

then a+c=b+d

If equals are subtracted from equals, then the remainders are equal

If

a=b c=d

then a-c=b-d

The whole is greater than the part.

If

b≠0

then a+b>a

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Elementary algebra

ƒ(x) = x² is an example of an even function.

From Wikipedia:Elementary algebra:

Elementary algebra builds on and extends arithmetic by introducing letters called ^*variables to represent general (non-specified) numbers.

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition^w, subtraction^w, multiplication^w, division^w and exponentiation^w). For example,

• Added terms are simplified using coefficients. For example, $x + x + x$ can be simplified as $3x$ (where 3 is a numerical coefficient).

• Multiplied terms are simplified using exponents. For example, $x \times x \times x$ is represented as $x^3$

• Like terms are added together, for example, $2x^2 + 3ab - x^2 + ab$ is written as $x^2 + 4ab$, because the terms containing $x^2$ are added together, and, the terms containing $ab$ are added together.

• Brackets can be "multiplied out", using the distributive property^w. For example, $x (2x + 3)$ can be written as $(x \times 2x) + (x \times 3)$ which can be written as $2x^2 + 3x$

• Expressions can be factored. For example, $6x^5 + 3x^2$, by dividing both terms by $3x^2$ can be written as $3x^2 (2x^3 + 1)$

ƒ(x) = x³ is an example of an odd function.

For any function $f$, if $a = b$ then:

• $f(a) = f(b)$
• $a + c = b + c$
• $ac = bc$
• $a^c = b^c$

One must be careful though when squaring both sides of an equation since this can result is solutions that dont satisfy the original equation.

$1 \neq -1$

yet

$1^2 = -1^2$

A function is an even function^w if f(x) = f(-x)

A function is an odd function^w if f(x) = -f(-x)

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Trigonometry

 

The law of cosines^w reduces to the Pythagorean theorem^w when gamma=90 degrees

$c^2 = a^2 + b^2 - 2ab\cos\gamma,$

The law of sines^w (also known as the "sine rule") for an arbitrary triangle states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{abc}{2\Delta},$

where $\Delta$ is the area of the triangle

$\mbox{Area} = \Delta = \frac{1}{2}a b\sin C.$

The law of tangents^w:

$\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}$

The parallelogram law reduces to the Pythagorean theorem when the parallelogram is a rectangle

$(AB)^2+(BC)^2+(CD)^2+(DA)^2=(AC)^2+(BD)^2\,$

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Right triangles

A right triangle is a triangle with gamma=90 degrees.

For small values of x, sin x ≈ x. (If x is in radians).

SOH → sin = "opposite" / "hypotenuse" CAH → cos = "adjacent" / "hypotenuse" TOA → tan = "opposite" / "adjacent"

= sin A = a/c = cos A = b/c = tan A = a/b

$\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.$

(Note: the expression of tan(x) has i in the numerator, not in the denominator, because the order of the terms (and thus the sign) of the numerator is changed w.r.t. the expression of sin(x).)

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Hyperbolic functions

See also: ^*Hyperbolic angle

From Wikipedia:Hyperbolic function:

A ray through the ^*unit hyperbola $x^2 - y^2 = 1$ in the point $(\cosh a, \sinh a),$ where $a$ is twice the area between the ray, the hyperbola, and the $x$-axis. For points on the hyperbola below the $x$-axis, the area is considered negative (see animated version^w with comparison with the trigonometric (circular) functions).

Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of ^*circular sector area u and hyperbolic functions depending on ^*hyperbolic sector area u.

Hyperbolic functions^w are analogs of the ordinary trigonometric, or circular, functions.

• Hyperbolic sine:

$\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.$

• Hyperbolic cosine:

$\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.$

• Hyperbolic tangent:

$\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} =$

$= \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}.$

• Hyperbolic cotangent:

$\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} =$

$= \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}, \qquad x \neq 0.$

• Hyperbolic secant:

$\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} =$

$= \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}.$

• Hyperbolic cosecant:

$\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} =$

$= \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}, \qquad x \neq 0.$

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Areas and volumes

The length of the circumference C of a circle is related to the radius r and diameter d by:

$\mathrm{Circumference} = \tau r = 2 \pi r = \pi d$

where

$\pi$ = 3.141592654

$\tau$ = 2 * π

The area of a circle is:

$\mathrm{Area} = \pi r^2$

The surface area of a sphere is

$\mathrm{\text{Surface area}} = 4 \cdot \pi r^2$

The surface area of a sphere 1 unit in radius is:

$4 \pi (1 \text{ unit})^2 = 12.56637 \text{ unit}^2$

The surface area of a sphere 128 units in radius is:

$4 \pi (128 \text{ unit})^2 = 205,887 \text{ unit}^2$

The volume inside a sphere is

$\mathrm{Volume} = \frac{4}{3} \cdot \pi r^3$

The volume of a sphere 1 unit in radius is:

$V = \frac{4}{3} \cdot \pi (1 \text{ unit})^3 = 4.1888 \text{ unit}^3$

The moment of inertia of a hollow sphere is:

$I = \frac{2}{3} m r^2\,\!$

Moment of inertia of a sphere is:

$I = \frac{2}{5} m r^2\,\!$

The area of a hexagon is:

$Area = \frac{3 \sqrt{3}}{2} a^2 = 2.59807621135 \cdot a^2$

where a is the length of any side.

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Fractals

A square that is twice as big is four times as massive because it is 2 dimensional (2² = 4). A cube that is twice as big is eight times as massive because it is 3 dimensional (2³ = 8).

A triangle that is twice as big is four times as massive.

But a Sierpiński triangle that is twice as big is exactly three times as massive. It therefore has a Hausdorff dimension of 1.5849. (2^1.5849 = 3)

A pyramid that is twice as big is eight times as massive.

But a Sierpiński pyramid that is twice as big is exactly five times as massive. It therefore has a Hausdorff dimension of 2.3219 (2^2.3219 = 5)

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Polynomials

See also: ^*Runge's phenomenon, ^*Polynomial ring, ^*System of polynomial equations, ^*Rational root theorem, ^*Descartes' rule of signs, and ^*Complex conjugate root theorem

From Wikipedia:Polynomial:

A polynomial^w can always be written in the form

$Z(x) = a_0 + a_1 x + a_2 x^2 + \dotsb + a_{n-1}x^{n-1} + a_n x^n$

where $a_0, \ldots, a_n$ are constants called coefficients and n is the degree^w of the polynomial.

A ^*linear polynomial is a polynomial of degree one.

Each individual ^*term is the product of the ^*coefficient and a variable raised to a nonnegative integer power.

A ^*monomial has only one term.

A ^*binomial has 2 terms.

^*Fundamental theorem of algebra:

Every single-variable, degree n polynomial with complex coefficients has exactly n complex roots^w.

However, some or even all of the roots might be the same number.

A root (or zero) of a function is a value of x for which Z(x)=0.

$Z(x) = a_n(x - z_1)(x - z_2)\dotsb(x - z_n)$

If $Z(x) = (x - z_1)(x - z_2)^k$ then z₂ is a root of ^*multiplicity k. z₂ is a root of multiplicity k-1 of the derivative (Derivative is defined below) of Z(x).

If k=1 then z₂ is a simple root.

The graph is tangent to the x axis at the multiple roots of f and not tangent at the simple roots.

The graph crosses the x-axis at roots of odd multiplicity and bounces off (not goes through) the x-axis at roots of even multiplicity.

Near x=z₂ the graph has the same general shape as $A(x - z_2)^k$

The ^*complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib).

The roots of the formula $ax^{2} + bx + c = 0$ are given by the Quadratic formula^w:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$ See Completing the squarew

$b^2 - 4ac$ is called the discriminant.

$ax^2+bx+c \quad = \quad a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a} \quad = \quad a(x-h)^2+k$

This is a parabola shifted to the right h units, stretched by a factor of a, and moved upward k units.

k is the value at x=h and is either the maximum or the minimum value.

The roots of

$a_0 + a_1 x + \cdots + a_n x^n,$

are the multiplicative inverses of

$a_n + a_{n-1} x + \cdots + a_0 x^n,$

There is no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc). See ^*Galois theory.

^*Binomial therorem:

$(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n,$

Where $\binom{n}{k} = \frac{n!}{k! (n-k)!}.$ See Binomial coefficient^w

Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series.

$x^2 - y^2 = (x + y)(x - y)$

$x^2 + y^2 = (x + yi)(x - yi)$

The polynomial remainder theorem^w states that the remainder of the division of a polynomial Z(x) by the linear polynomial x-a is equal to Z(a). See ^*Ruffini's rule.

Determining the value at Z(a) is sometimes easier if we use ^*Horner's method (^*synthetic division) by writing the polynomial in the form

$Z(x) = a_0 + x(a_1 + x(a_2 + \cdots + x(a_{n-1} + x(a_n)))).$

A ^*monic polynomial is a one variable polynomial in which the leading coefficient is equal to 1.

$a_0 + a_1x + a_2x^2 + \cdots + a_{n-1}x^{n-1} + 1x^n$

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Rational functions

A ^*rational function is a function of the form

$f(x) = k{(x - z_1)(x - z_2)\dotsb(x - z_n) \over (x - p_1)(x - p_2)\dotsb(x - p_m)} = {Z(x) \over P(x)}$

It has n zeros^w and m poles^w. A pole is a value of x for which |f(x)| = infinity.

The vertical asymptotes^w are the poles of the rational function.

If n<m then f(x) has a horizontal asymptote at the x axis

If n=m then f(x) has a horizontal asymptote at k.

If n>m then f(x) has no horizontal asymptote.

See also ^*Wikipedia:Asymptote#Oblique_asymptotes

Given two polynomials $Z(x)$ and $P(x) = (x-p_1)(x-p_2) \cdots (x-p_m)$, where the p_i are distinct constants and deg Z < m, partial fractions^w are generally obtained by supposing that

$\frac{Z(x)}{P(x)} = \frac{c_1}{x-p_1} + \frac{c_2}{x-p_2} + \cdots + \frac{c_m}{x-p_m}$

and solving for the c_i constants, by substitution, by ^*equating the coefficients of terms involving the powers of x, or otherwise.

(This is a variant of the ^*method of undetermined coefficients.)

If the degree of Z is not less than m then use long division to divide P into Z. The remainder then replaces Z in the equation above and one proceeds as before.

If $P(x) = (x-p)^m$ then $\frac{Z(x)}{P(x)} = \frac{c_1}{(x-p)} + \frac{c_2}{(x-p)^2} + \cdots + \frac{c_m}{(x-p)^m}$

A ^*Generalized hypergeometric series is given by

$\sum_{x=0} c_x$ where c₀=1 and ${c_{x+1} \over c_x} = {Z(x) \over P(x)} = f(x)$

The function f(x) has n zeros and m poles.

^*Basic hypergeometric series, or hypergeometric q-series, are ^*q-analogue generalizations of generalized hypergeometric series.

Roughly speaking a ^*q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1

We define the q-analog of n, also known as the q-bracket or q-number of n, to be

$[n]_q=\frac{1-q^n}{1-q} = q^0 + q^1 + q^2 + \ldots + q^{n - 1}$

one may define the q-analog of the factorial^w, known as the ^*q-factorial, by

$[n]_q! = [1]_q \cdot [2]_q \cdots [n-1]_q \cdot [n]_q$

^*Elliptic hypergeometric series are generalizations of basic hypergeometric series.

An elliptic function is a meromorphic function that is periodic in two directions.

A ^*generalized hypergeometric function is given by

$F(x) = {}_nF_m(z_1,...z_n;p_1,...p_m;x) = \sum_{y=0} c_y x^y$

So for e^x (see below) we have:

$c_y = \frac{1}{y!}, \qquad \frac{c_{y+1}}{c_y} = \frac{1}{y+1}.$

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Integration and differentiation

Force • distance = energy

See also: Hyperreal number^w and Implicit differentiation^w

The integral^w is a generalization of multiplication.

For example: a unit mass dropped from point x₂ to point x₁ will release energy.

The usual equation is is a simple multiplication:

$gravity \cdot (x_2 - x_1) = energy$

But that equation cant be used if the strength of gravity is itself a function of x.

The strength of gravity at x₁ would be different than it is at x₂.

And in reality gravity really does depend on x (x is the distance from the center of the earth):

$gravity(x) = 1/x^2$ (See inverse-square law^w.)

However, the corresponding Definite integral^w is easily solved:

$\int_{x_1}^{x_2} gravity(x) \cdot dx$

The surprisingly simple rules for solving definite integrals

$\int_{x_1}^{x_2} f(x) \cdot dx \quad = \quad F(x_2)-F(x_1)$ F(x) is called the indefinite integralw. (antiderivativew) $F(x) = \int f(x) \cdot dx$ k and y are arbitrary constants: $\int k \cdot x^y \cdot dx \quad = \quad k \cdot \int x^y \cdot dx \quad = \quad k \cdot \frac{x^{y+1}}{y+1}$ (Units (feet, mm...) behave exactly like constants.) And most conveniently : $\int \bigg (f(x) + g(x) \bigg) \cdot dx = \int f(x) \cdot dx + \int g(x) \cdot dx$

The integral of a function is equal to the area under the curve.

When the "curve" is a constant (in other words, k•x⁰) then the integral reduces to ordinary multiplication.

The derivative^w is a generalization of division.

The derivative of the integral of f(x) is just f(x).

The derivative of a function at any point is equal to the slope of the function at that point.

$f'(x)=\frac{f(x+dx)-f(x)}{dx}.$

The equation of the line tangent to a function at point a is

$y(x) = f(a) + f'(a)(x-a)$

The Lipschitz constant^w of a function is a real number for which the absolute value of the slope of the function at every point is not greater than this real number.

The derivative of f(x) where f(x) = k•x^y is

$f'(x) = {df \over dx} = {d(k \cdot x^y) \over dx} \quad = \quad k \cdot {d(x^y) \over dx} \quad = \quad k \cdot y \cdot x^{y-1}$

The derivative of a $k \cdot x^0$ is $k \cdot 0 \cdot x^{-1}$

The integral of $x^{-1}$ is ln(x). See natural log^w

Chain rule^w for the derivative of a function of a function:

$f(g(x))' = \frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$

The Chain rule for a function of 2 functions:

$f(g(x), h(x))' = \frac{\operatorname df}{\operatorname dx} = { \partial f \over \partial g}{\operatorname dg \over \operatorname dx} + {\partial f \over \partial h}{\operatorname dh \over \operatorname dx }$ (See "partial derivatives" below)

The Product rule^w can be considered a special case of the chain rule^w for several variables

$\frac{df}{dx} = {d (g(x) \cdot h(x)) \over dx} = \frac{\partial(g \cdot h)}{\partial g}\frac{dg}{dx}+\frac{\partial (g \cdot h)}{\partial h}\frac{dh}{dx} = \frac{dg}{dx} h + g \frac{dh}{dx}$

Product rule^w:

$(g \cdot h)' = \frac{(g+dg) \cdot (h+dh) - g \cdot h}{dx} = g' \cdot h + g \cdot h'$ (because $dh \cdot dg$ is negligible)

$(g \cdot h \cdot j)' = g' \cdot h \cdot j + g \cdot h' \cdot j + g \cdot h \cdot j'$

^*General Leibniz rule:

$(gh)^{(n)}=\sum_{k=0}^n {n \choose k} g^{(n-k)} h^{(k)}$

By the chain rule:

$\bigg(\frac{1}{h}\bigg)' = \frac{-1}{h^2} \cdot h'$

Therefore the Quotient rule^w:

$\bigg( \frac{g(x)}{h(x)} \bigg)' = \bigg( g \cdot \frac{1}{h} \bigg)' = g' \cdot \frac{1}{h} + g \cdot \frac{-h'}{h^2} = \frac{g' \cdot h - g \cdot h'}{h^2}$

There is a chain rule for integration but the inner function must have the form $g=ax+c$ so that its derivative $\frac{dg}{dx} = a$ and therefore $dx=\frac{dg}{a}$

$\int f(g(x)) \cdot dx = \int f(g) \cdot \frac{dg}{a} = \frac{1}{a} \int f(g) \cdot dg$

Actually the inner function can have the form $g=ax^y+c$ so that its derivative $\frac{dg}{dx} = a \cdot y \cdot x^{y-1}$ and therefore $dx=\frac{dg}{a \cdot y \cdot x^{y-1}}$ provided that all factors involving x cancel out.

$\int x^{y-1} \cdot f(g(x)) \cdot dx = \int {\color{red} x^{y-1}} \cdot f(g) \cdot \frac{dg}{a \cdot y \cdot {\color{red} x^{y-1}}} = \frac{1}{a \cdot y} \int f(g) \cdot dg$

The product rule for integration is called Integration by parts^w

$g \cdot h' = (g \cdot h)' - g' \cdot h$

$\int g \cdot h' \cdot dx = g \cdot h - \int g' \cdot h \cdot dx$

One can use partial fractions^w or even the Taylor series^w to convert difficult integrals into a more manageable form.

$\frac{f(x)}{(x-1)^2} = \frac{a_0(x-1)^0 + a_1(x-1)^1 + \dots + a_n(x-1)^n}{(x-1)^2}$

The fundamental theorem of Calculus is:

$F(x) - F(a) = \int_a^x\!f(t)\, dt \quad \text{and} \quad F'(x) = f(x)$

The fundamental theorem of calculus is just the particular case of the ^*Leibniz integral rule:

$\frac{d}{dx} \left (\int_{a(x)}^{b(x)}f(x,t)\,dt \right) = f\big(x,b(x)\big)\cdot \frac{d}{dx} b(x) - f\big(x,a(x)\big)\cdot \frac{d}{dx} a(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x} f(x,t) \,dt.$

In calculus, a function f defined on a subset of the real numbers with real values is called ^*monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing.

A differential form^w is a generalisation of the notion of a differential^w that is independent of the choice of ^*coordinate system. f(x,y) dx ∧ dy is a 2-form in 2 dimensions (an area element). The derivative^w operation on an n-form is an n+1-form; this operation is known as the exterior derivative^w. By the generalized Stokes' theorem^w, the integral of a function over the boundary of a manifold^w is equal to the integral of its exterior derivative on the manifold itself.

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Taylor & Maclaurin series

If we know the value of a smooth function^w at x=0 (smooth means all its derivatives are continuous^w) and we also know the value of all of its derivatives at x=0 then we can determine the value at any other point x by using the Maclaurin series^w. ("!" means factorial^w)

$f(x) = a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 \cdots \quad \text{where} \quad a_n = {f^{(n)}(0) \over n!}$

The proof of this is actually quite simple. Plugging in a value of x=0 causes all terms but the first to become zero. So, assuming that such a function exists, a₀ must be the value of the function at x=0. Simply differentiate both sides of the equation and repeat for the next term. And so on.

Because the functions $x^n$ can be multiplied by scalars and added they therefore form an infinite dimensional vector space. (An infinite dimensional space is not a ^*Compact space.) The function f(x) occupies a single point in that infinite dimensional space corresponding to a vector whose components are $a_n$

$\mathbf{v} = \begin{bmatrix} a_0 & a_1 & a_2 & \ldots & a_{\infty} \end{bmatrix}$

The Taylor series^w generalizes the Maclaurin series.

$f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k$

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Riemann surface for the function ƒ(z) = √z. For the imaginary part rotate 180°.

An analytic function^w is a function whose Taylor series converges for every z₀ in its domain^w; analytic functions are infinitely differentiable^w.

Any vector g = (z₀, α₀, α₁, ...) is a ^*germ if it represents a power series of an analytic function^w around z₀ with some radius of convergence r > 0.

The set of germs $\mathcal G$ is a Riemann surface^w.

Riemann surfaces are the objects on which multi-valued functions become single-valued.

A ^*connected component of $\mathcal G$ (i.e., an equivalence class) is called a ^*sheaf.

We can easily determine the Maclaurin series expansion of the exponential function^w $e^x$ because it is equal to its own derivative.

$e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = {x^0 \over 0!} + {x^1 \over 1!} + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots$

The above holds true even if x is a matrix. See ^*Matrix exponential

And likewise for cos(x)^w and sin(x)^w because cosine is the derivative of sine which is the derivative of -cosine

$\cos x = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

$\sin x = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$

It then follows that $e^{ix}=\cos x+i\sin x=\operatorname{cis} x$ and therefore $e^{i \pi}=-1 + i\cdot0$ See Euler's formula^w

x is the angle in ^*radians.

This makes the equation for a circle in the complex plane, and by extension sine and cosine, extremely simple and easy to work with especially with regard to differentiation and integration.

$\frac{d(e^{i \cdot k \cdot t})}{dt} = i \cdot k \cdot e^{i \cdot k \cdot t}$

For sine waves differentiation and integration are replaced with multiplication and division. Calculus is replaced with algebra. Therefore any expression that can be represented as a sum of sine waves can be easily differentiated or integrated.

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Fourier Series

The Maclaurin series cant be used for a discontinuous function like a square wave because it is not differentiable. (^*Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. See ^*Generalized function.)

But remarkably we can use the Fourier series^w to expand it or any other periodic function^w into an infinite sum of sine waves each of which is fully differentiable^w!

$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n\cos\left(nt\right)+b_n\sin\left(nt\right)\right]$

$a_n = \frac{2}{p}\int_{t_0}^{t_p} f(t)\cdot \cos\left(\tfrac{2\pi nt}{p}\right)\ dt$

$b_n = \frac{2}{p}\int_{t_0}^{t_p} f(t)\cdot \sin\left(\tfrac{2\pi nt}{p}\right)\ dt$

A square wave consists of all odd frequencies with the amplitude of each frequency being $\frac{1}{f}$.

A dirac comb consists of all integer frequencies with the amplitude of each frequency being 1.

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sin²(x) = 0.5*cos(0x) - 0.5*cos(2x)

The reason this works is because sine and cosine are ^*orthogonal functions.

Two vectors are said to be orthogonal when:

$\mathbf{u} \cdot \mathbf{v} = 0$

or more generally when:

$\beta(\mathbf{u,v}) = 0$

Two functions are said to be orthogonal when:

$\langle f,g \rangle = \int \overline{f(x)}g(x)\,dx = 0.$

That means that multiplying any 2 sine waves of frequency n and frequency m and integrating over one period will always equal zero unless n=m.

See the graph of sin²(x) to the right.

$\sin mx \cdot \sin nx = \frac{\cos (m - n)x - \cos (m+n) x}{2}$

See ^*Amplitude_modulation

And of course ∫ f_n*(f₁+f₂+f₃+...) = ∫ (f_n*f₁) + ∫ (f_n*f₂) + ∫ (f_n*f₃) +...

The complex form of the Fourier series uses complex exponentials instead of sine and cosine and uses both positive and negative frequencies (clockwise and counter clockwise) whose imaginary parts cancel.

The complex coefficients encode both amplitude and phase and are complex conjugates of each other.

$F(\nu) = \mathcal{F}\{f\} = \int_{\mathbb{R}^n} f(x) e^{-2 \pi i x\cdot\nu} \, \mathrm{d}x$

where the dot between x and ν indicates the inner product^w of Rⁿ.

A 2 dimensional Fourier series is used in video compression.

A ^*discrete Fourier transform can be computed very efficiently by a ^*fast Fourier transform (FFT).

The FFT has been described as "the most important numerical algorithm of our lifetime"

In mathematical analysis, many generalizations of Fourier series have proven to be useful.

They are all special cases of decompositions over an orthonormal basis of an inner product space.

^*Spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.

Spherical harmonics are ^*basis functions for SO(3). See Laplace series^w.

Every continuous function in the function space can be represented as a ^*linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

Every quadratic polynomial can be written as a1+bt+ct², that is, as a linear combination of the basis functions 1, t, and t².

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Transforms

Fourier transforms^w generalize Fourier series to nonperiodic functions like a single pulse of a square wave.

The more localized in the time domain (the shorter the pulse) the more the Fourier transform is spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle^w.

Using the Fourier transform we can determine that the Dirac delta function^w consists of all frequencies with the amplitude of each frequency being 1.

$G(\omega)=\mathcal{F}\{f(t)\}=\int_{-\infty}^\infty f(t) e^{-i\omega t}dt$

Laplace transforms^w generalize Fourier transforms to complex frequency $s=\sigma+i\omega$.

Complex frequency includes a term corresponding to the amount of damping.

See http://www.dspguide.com/CH32.PDF.

$F(s)=\mathcal{L}\{f(t)\}=\int_0^\infty f(t) e^{-\sigma t}e^{-i \omega t}dt$

$\mathcal{L}\{ \delta(t-a) \} = e^{-as}$, (assuming a > 0)

$\mathcal{L}\{e^{at} \}= \frac{1}{s - a}$

The inverse Laplace transform^w is given by

$f(t) = \mathcal{L}^{-1} \{F\} = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}F(s)e^{st}\,ds,$

where the integration is done along the vertical line Re(s) = γ in the complex plane^w such that γ is greater than the real part of all ^*singularities of F(s) and F(s) is bounded on the line, for example if contour path is in the ^*region of convergence.

If all singularities are in the left half-plane, or F(s) is an ^*entire function , then γ can be set to zero and the above inverse integral formula becomes identical to the ^*inverse Fourier transform.

The ^*Z-transform can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of ^*time-scale calculus.

Integral transforms^w generalize Fourier transforms to other ^*kernals (besides sine^w and cosine^w)

Cauchy kernel =$\frac{1}{\zeta-x} \quad \text{or} \quad \frac{1}{2\pi i} \cdot \frac{1}{\zeta-x}$

Hilbert kernel = $cot\frac{\theta-t}{2}$

Poisson Kernel:

For the ball of radius r, $B_{r}$, in Rⁿ, the Poisson kernel takes the form:

$P(x,\zeta) = \frac{r^2-|x|^2}{r} \cdot \frac{1}{|\zeta-x|^n} \cdot \frac{1}{\omega_{n}}$

where $x\in B_{r}$, $\zeta\in S$ (the surface of $B_{r}$), and $\omega_n$ is the ^*surface area of the unit n-sphere.

unit disk (r=1) in the complex plane:

$K(x,\phi) = \frac{1^2-|x|^2}{1} \cdot \frac{1}{|e^{i\phi}-x|^2}\cdot \frac{1}{2\pi}$

Dirichlet kernel

$D_n(x)=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n\cos(kx)=\frac{\sin\left(\left(n + \frac{1}{2}\right) x \right)}{\sin(\frac{x}{2})} \approx 2\pi\delta(x)$

The ^*convolution theorem states that

$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$

where $\cdot$ denotes point-wise multiplication. It also works the other way around:

$\mathcal{F}\{f \cdot g\}= \mathcal{F}\{f\}*\mathcal{F}\{g\}$

By applying the inverse Fourier transform $\mathcal{F}^{-1}$, we can write:

$f*g= \mathcal{F}^{-1}\big\{\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}\big\}$

and:

$f \cdot g= \mathcal{F}^{-1}\big\{\mathcal{F}\{f\}*\mathcal{F}\{g\}\big\}$

This theorem also holds for the Laplace transform^w.

The ^*Hilbert transform is a ^*multiplier operator. The multiplier of H is σ_H(ω) = −i sgn(ω) where sgn is the ^*signum function. Therefore:

$\mathcal{F}(H(u))(\omega) = (-i\,\operatorname{sgn}(\omega)) \cdot \mathcal{F}(u)(\omega)$

where $\mathcal{F}$ denotes the Fourier transform^w.

Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of $\mathcal{F}$.

By Euler's formula^w,

$\sigma_H(\omega) = \begin{cases} i = e^{+\frac{i\pi}{2}}, & \text{for } \omega < 0\\ 0, & \text{for } \omega = 0\\ -i = e^{-\frac{i\pi}{2}}, & \text{for } \omega > 0 \end{cases}$

Therefore, H(u)(t) has the effect of shifting the phase of the ^*negative frequency components of u(t) by +90° (π/2 radians) and the phase of the positive frequency components by −90°.

And i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.

In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI).

At any given moment, the output is an accumulated effect of all the prior values of the input function

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Differential equations

See also: ^*Variation of parameters

Simple harmonic motion shown both in real space and ^*phase space.

^*Simple harmonic motion of a mass on a spring is a second-order linear ordinary differential equation^w.

$Force = mass*acc = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,$

where m is the inertial mass, x is its displacement from the equilibrium, and k is the spring constant.

Solving for x produces

$x(t) = A\cos\left(\omega t - \varphi\right),$

A is the amplitude (maximum displacement from the equilibrium position), $\omega = 2\pi f = \sqrt{k/m}$ is the angular frequency^w, and φ is the phase.

Energy passes back and forth between the potential energy in the spring and the kinetic energy of the mass.

The important thing to note here is that the frequency of the oscillation depends only on the mass and the stiffness of the spring and is totally independent of the amplitude.

That is the defining characteristic of resonance.

File:RLC series circuit v1.svg

RLC series circuit

^*Kirchhoff's voltage law states that the sum of the emfs in any closed loop of any electronic circuit is equal to the sum of the ^*voltage drops in that loop.

$V(t) = V_R + V_L + V_C$

V is the voltage, R is the resistance, L is the inductance, C is the capacitance.

$V(t) = RI(t) + L \frac{dI(t)}{dt} + \frac{1}{C} \int_{0}^t I(\tau)\, d\tau$

I = dQ/dt is the current.

It makes no difference whether the current is a small number of charges moving very fast or a large number of charges moving slowly.

In reality ^*the latter is the case.

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^*Damping oscillation is a typical ^*transient response

If V(t)=0 then the only solution to the equation is the transient response which is a rapidly decaying sine wave with the same frequency as the resonant frequency of the circuit.

Like a mass (inductance) on a spring (capacitance) the circuit will resonate at one frequency.

Energy passes back and forth between the capacitor and the inductor with some loss as it passes through the resistor.

If V(t)=sin(t) from -∞ to +∞ then the only solution is a sine wave with the same frequency as V(t) but with a different amplitude and phase.

If V(t) is zero until t=0 and then equals sin(t) then I(t) will be zero until t=0 after which it will consist of the steady state response plus a transient response.

From Wikipedia:Characteristic equation (calculus):

Starting with a linear homogeneous differential equation with constant coefficients $a_{n}, a_{n-1}, \ldots , a_{1}, a_{0}$,

$a_{n}y^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_{1}y^\prime + a_{0}y = 0$

it can be seen that if $y(x) = e^{rx} \,$, each term would be a constant multiple of $e^{rx} \,$. This results from the fact that the derivative of the exponential function^w $e^{rx} \,$ is a multiple of itself. Therefore, $y' = re^{rx} \,$, $y'' = r^{2}e^{rx} \,$, and $y^{(n)} = r^{n}e^{rx} \,$ are all multiples. This suggests that certain values of $r \,$ will allow multiples of $e^{rx} \,$ to sum to zero, thus solving the homogeneous differential equation. In order to solve for $r \,$, one can substitute $y = e^{rx} \,$ and its derivatives into the differential equation to get

$a_{n}r^{n}e^{rx} + a_{n-1}r^{n-1}e^{rx} + \cdots + a_{1}re^{rx} + a_{0}e^{rx} = 0$

Since $e^{rx} \,$ can never equate to zero, it can be divided out, giving the characteristic equation

$a_{n}r^{n} + a_{n-1}r^{n-1} + \cdots + a_{1}r + a_{0} = 0$

By solving for the roots, $r \,$, in this characteristic equation, one can find the general solution to the differential equation. For example, if $r \,$ is found to equal to 3, then the general solution will be $y(x) = ce^{3x} \,$, where $c \,$ is an arbitrary constant^w.

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Partial derivatives

See also: ^*Currying

^*Partial derivatives and ^*multiple integrals generalize derivatives and integrals to multiple dimensions.

The partial derivative with respect to one variable $\frac{\part f(x,y)}{\part x}$ is found by simply treating all other variables as though they were constants.

Multiple integrals are found the same way.

Let f(x, y, z) be a scalar function^w (for example electric potential energy or temperature).

A 2 dimensional example of a scalar function would be an elevation map.

(Contour lines of an elevation map are an example of a ^*level set.)

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The total derivative^w of f(x(t), y(t)) with respect to t is

$\frac{\operatorname df}{\operatorname dt} = { \partial f \over \partial x}{\operatorname dx \over \operatorname dt} + {\partial f \over \partial y}{\operatorname dy \over \operatorname dt }$

And the differential^w is

$\operatorname df = { \partial f \over \partial x}\operatorname dx + {\partial f \over \partial y} \operatorname dy .$

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Gradient of scalar field

The Gradient^w of f(x, y, z) is a vector field whose value at each point is a vector (technically its a covector^w because it has units of distance⁻¹) that points "downhill" with a magnitude equal to the slope^w of the function at that point.

You can think of it as how much the function changes per unit distance.

The gradient of temperature gives heat flow.

$\operatorname{grad}(f) = \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} = \mathbf{F}$

For static (unchanging) fields the Gradient of the electric potential is the electric field^w itself. Image below shows the potential of a single point charge.

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Its gradient gives the electric field which is shown in the 2 images below. In the image on the left the field strength is proportional to the length of the vectors. In the image on the right the field strength is proportional to the density of the ^*flux lines. The image is 2 dimensional and therefore the flux density in the image follows an inverse first power law but in reality the field lines from a real proton or electron spread outward in 3 dimensions and therefore follow an inverse square law. Inverse square means that at twice the distance the field is four times weaker.

The field of 2 point charges is simply the linear sum of the separate charges.

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Divergence

The Divergence^w of a vector field is a scalar.

The divergence of the electric field is non-zero wherever there is electric charge^w and zero everywhere else.

^*Field lines begin and end at charges because the charges create the electric field.

$\operatorname{div}\,\mathbf{F} = {\color{red} \nabla\cdot\mathbf{F} } = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \cdot (F_x,F_y,F_z) = \frac{\partial F_x}{\partial x} +\frac{\partial F_y}{\partial y} +\frac{\partial F_z}{\partial z}.$

The Laplacian^w is the divergence of the gradient of a function:

$\Delta f = \nabla^2 f = (\nabla \cdot \nabla) f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.$

^*elliptic operators generalize the Laplacian.

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Curl

See also: ^*Biot–Savart law

The Curl^w of a vector field describes how much the vector field is twisted.

(The field may even go in circles.)

The curl at a certain point of a magnetic field^w is the current^w vector at that point because current creates the magnetic field^w.

In 3 dimensions the dual of the current vector is a bivector.

$\text{curl} (\mathbf{F}) = {\color{blue} \nabla \times \mathbf{F} } = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ F_x & F_y & F_z \end{vmatrix}$

$\text{curl}( \mathbf{F}) = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}$

In 2 dimensions this reduces to a single scalar

$\text{curl}( \mathbf{F}) = \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)$

The curl of the gradient of any scalar field is always zero.

The curl of a vector field in 4 dimensions would no longer be a vector. It would be a bivector. However the curl of a bivector field in 4 dimensions would still be a vector.

See also: ^*differential forms.

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Gradient of vector field

The Gradient^w of a vector field is a tensor field. Each row is the gradient of the corresponding scalar function:

$\nabla \mathbf{F} = \begin{bmatrix} \frac{\partial}{\partial x} \mathbf{e}_x, \frac{\partial}{\partial y} \mathbf{e}_y, \frac{\partial}{\partial z} \mathbf{e}_z \end{bmatrix} \begin{bmatrix} f_x \mathbf{e}_x \\ f_y \mathbf{e}_y \\ f_z \mathbf{e}_z \end{bmatrix} = \begin{bmatrix} {\color{red} \frac{\partial f_x}{\partial x} \mathbf{e}_{xx} } & {\color{blue} \frac{\partial f_x}{\partial y} \mathbf{e}_{xy} } & {\color{blue} \frac{\partial f_x}{\partial z} \mathbf{e}_{xz} } \\ {\color{blue} \frac{\partial f_y}{\partial x} \mathbf{e}_{yx} } & {\color{red} \frac{\partial f_y}{\partial y} \mathbf{e}_{yy} } & {\color{blue} \frac{\partial f_y}{\partial z} \mathbf{e}_{yz} } \\ {\color{blue} \frac{\partial f_z}{\partial x} \mathbf{e}_{zx} } & {\color{blue} \frac{\partial f_z}{\partial y} \mathbf{e}_{zy} } & {\color{red} \frac{\partial f_z}{\partial z} \mathbf{e}_{zz} } \end{bmatrix}$

Remember that $\mathbf{e}_{xy} = - \mathbf{e}_{yx}$ because rotation from y to x is the negative of rotation from x to y.

Partial differential equations can be classified as ^*parabolic, ^*hyperbolic and ^*elliptic.

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Green's theorem

The line integral^w along a 2-D vector field is:

$\int (V_1 \cdot dx + V_2 \cdot dy) = \int_a^b \bigg [V_1(x(t),y(t)) \frac{dx}{dt} + V_2(x(t),y(t)) \frac{dy}{dt} \bigg ] dt$

$\int (V_1 \cdot dx + V_2 \cdot dy) = \iint \bigg ( \frac{\partial V_2}{\partial x} - \frac{\partial V_1}{\partial y} \bigg ) \cdot dx \cdot dy = \iint \bigg ( {\color{blue} \nabla \times \mathbf{V} } \bigg ) \cdot dx \cdot dy$

Divergence is zero everywhere except at the origin where a charge is located. A line integral around any of the red circles will give the same answer because all the circles contain the same amount of charge.

You can think of each field line as ending in a single unit of charge.

Green's theorem^w states that if you want to know how many field lines exit a region then you can either count how many lines cross the boundary (perform a line integral) or you can simply count the number of charges (or the amount of current) within that region. See Divergence theorem^w.

\oiint${\scriptstyle S }$ $\vec{F} \cdot \ \mathrm{d} \vec{s} = \iiint_D \nabla \cdot \vec{F} \,\mathrm{d}V = \iiint_D \nabla^2 f \,\mathrm{d}V$

In 2 dimensions this is

$\oint_S \vec{F} \cdot \vec{n} \ \mathrm{d} s = \iint_D \nabla \cdot \vec{F} \ \mathrm{d} A= \iint_D \nabla^2 f \ \mathrm{d} A$

A version of Green's theorem also works for curl.

Green's theorem is perfectly obvious when dealing with vector fields but is much less obvious when applied to complex valued functions in the complex plane.

See also Kelvin–Stokes theorem^w

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The complex plane

Highly recomend: Fundamentals of complex analysis with applications to engineering and science by Saff and Snider

External link: http://www.solitaryroad.com/c606.html

The formula for the derivative of a complex function f at a point z₀ is the same as for a real function:

$f'(z_0) = \lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }.$

Every complex function can be written in the form $f(z)=f(x+iy)=f_x(x,y)+i f_y(x,y)$

Because the complex plane is two dimensional, z can approach z₀ from an infinite number of different directions.

However, if within a certain region, the function f is holomorphic^w (that is, complex differentiable^w) then, within that region, it will only have a single derivative whose value does not depend on the direction in which z approaches z₀ despite the fact that f_x and f_y each have 2 partial derivatives. One in the x and one in the y direction..

${df \over dz} \quad = \quad {\part f_x \over \part x} + i {\part f_y \over \part x} \quad = \quad {\part f_y \over \part y} - i {\part f_x \over \part y} \quad = \quad {\part f_x \over \part x} - i {\part f_x \over \part y} \quad = \quad {\part f_y \over \part y} + i {\part f_y \over \part x}$

${d^2f \over dz^2} \quad = \quad {\part^2 f_x \over \part x^2} + i {\part^2 f_y \over \part x^2} \quad = \quad {\part^2 f_y \over \part y \part x} - i {\part^2 f_x \over \part y \part x}$

This is only possible if the Cauchy–Riemann conditions^w are true.

$\frac{\part f_x}{\part x}=\frac{\part f_y}{\part y}\ ,\ \quad \frac{\part f_y}{\part x}=-\frac{\part f_x}{\part y}$

An ^*entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

As with real valued functions, a line integral of a holomorphic function depends only on the starting point and the end point and is totally independant of the path taken.

$\int f(z) \cdot dz = \int (f_x \cdot dx - f_y \cdot dy) + i \int (f_y \cdot dx + f_x \cdot dy)$

$\int f(z) \cdot dz = F(z) = \int_0^t f(z(t)) \cdot \frac{dz}{dt} \cdot dt$

$\int_a^b f(z) \cdot dz = F(b) - F(a)$

The starting point and the end point for any loop are the same. This, of course, implies Cauchy's integral theorem^w for any holomorphic function f:

$\oint f(z) \, dz = \iint \left( \frac{- \partial f_x}{\partial y} + \frac{- \partial f_y}{\partial x} \right) dx \, dy + i \iint \left( \frac{\partial f_x}{\partial x} + \frac{- \partial f_y}{\partial y} \right) \, dx \, dy = 0$

$\oint f(z) \, dz = \iint \left( {\color{blue} \nabla \times \bar{f}} + i {\color{red} \nabla \cdot \bar{f}} \right) \, dx \, dy = 0$

Therefore curl and divergence must both be zero for a function to be holomorphic.

Green's theorem^w for functions (not necessarily holomorphic) in the complex plane:

$\oint f(z) \, dz = 2i \iint \left( df/d\bar{z} \right) \, dx \, dy = i \iint \left( \nabla f \right) \, dx \, dy = i \iint \left( 1 {\partial f \over \partial x} + i {\partial f \over \partial y} \right) \, dx \, dy$

Computing the residue^w of a monomial

$\begin{align} \oint_C (z-z_0)^n dz = \int_0^{2\pi} e^{in \theta} \cdot i e^{i \theta} d \theta = i \int_0^{2\pi} e^{i (n+1) \theta} d\theta = \begin{cases} 2\pi i & \text{if } n = -1 \\ 0 & \text{otherwise} \end{cases} \end{align}$

where $C$ is the circle with radius $1$ therefore $z \to e^{i\theta}$ and $dz \to d(e^{i\theta}) = ie^{i\theta}d\theta$

$\oint_{C_r}\frac{f(z)}{z-z_0}dz = \oint_{C_r}\frac{f(z_0)}{z-z_0}dz + \oint_{C_r}\frac{f(z)-f(z_0)}{z-z_0}dz = f(z_0)2\pi i + 0$

The last term in the equation above equals zero when r=0. Since its value is independent of r it must therefore equal zero for all values of r.

$\bigg | \int_\Gamma f(z) \cdot dz \bigg | \leq Max(|f(z)|) \cdot length(\Gamma)$

Cauchy's integral formula^w states that the value of a holomorphic function within a disc is determined entirely by the values on the boundary of the disc.

Divergence can be nonzero outside the disc.

Cauchy's integral formula can be generalized to more than two dimensions.

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$f^{(0)}(z_0)=\dfrac{1}{2\pi i}\oint_\gamma f(z)\frac{1}{z-z_0}dz$

Which gives:

$f'(z_0)=\dfrac{1}{2\pi i}\oint_\gamma f(z)\frac{1}{(z-z_0)^2}dz$

$f''(z_0)=\dfrac{2}{2\pi i}\oint_\gamma f(z)\frac{1}{(z-z_0)^3}dz$

$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_\gamma f(z)\frac{1}{(z-z_0)^{n+1}}\, dz$

Note that n does not have to be an integer. See ^*Fractional calculus.

The Taylor series becomes:

$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n \quad \text{where} \quad a_n=\frac{1}{2\pi i} \oint_\gamma \frac{f(z)\,\mathrm{d}z}{(z-z_0)^{n+1}} = \frac{f^{(n)}(z_0)}{n!}$

The ^*Laurent series for a complex function f(z) about a point z₀ is given by:

$f(z)=\sum_{n=-\infty}^\infty a_n(z-z_0)^n \quad \text{where} \quad a_n=\frac{1}{2\pi i} \oint_\gamma \frac{f(z)\,\mathrm{d}z}{(z-z_0)^{n+1}} = \frac{f^{(n)}(z_0)}{n!}$

The positive subscripts correspond to a line integral around the outer part of the annulus and the negative subscripts correspond to a line integral around the inner part of the annulus. In reality it makes no difference where the line integral is so both line integrals can be moved until they correspond to the same contour gamma. See also: ^*Z-transform

The function $\frac{1}{(z-1)(z-2)}$ has poles at z=1 and z=2. It therefore has 3 different Laurent series centered on the origin (z₀ = 0):

For 0 < |z| < 1 the Laurent series has only positive subscripts and is the Taylor series.

For 1 < |z| < 2 the Laurent series has positive and negative subscripts.

For 2 < |z| the Laurent series has only negative subscripts.

^*Cauchy formula for repeated integration:

$f^{(-n)}(a) = \frac{1}{(n-1)!} \int_0^a f(z) \left(a-z\right)^{n-1} \,\mathrm{d}z$

For every holomorphic function^w $f(z)=f(x+iy)=f_x(x,y)+i f_y(x,y)$ both f_x and f_y are harmonic functions^w.

Any two-dimensional harmonic function is the real part of a complex analytic function^w.

See also: complex analysis^w.

f_y is the ^*harmonic conjugate of f_x.

Geometrically f_x and f_y are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which f_x and f_y are constant (^*equipotentials and ^*streamlines) cross at right angles.

In this regard, f_x+if_y would be the complex potential, where f_x is the ^*potential function and f_y is the ^*stream function.

f_x and f_y are both solutions of Laplace's equation^w $\nabla^2 f = 0$ so divergence of the gradient is zero

^*Legendre function are solutions to Legendre's differential equation.

This ordinary differential equation is frequently encountered when solving Laplace's equation (and related partial differential equations) in spherical coordinates.

A harmonic function^w is a scalar potential function therefore the curl of the gradient will also be zero.

See ^*Potential theory

Harmonic functions are real analogues to holomorphic functions.

All harmonic functions are analytic, i.e. they can be locally expressed as power series.

This is a general fact about ^*elliptic operators, of which the Laplacian is a major example.

The value of a harmonic function at any point inside a disk is a ^*weighted average of the value of the function on the boundary of the disk.

$P[u](x) = \int_S u(\zeta)P(x,\zeta)d\sigma(\zeta).\,$

The ^*Poisson kernel gives different weight to different points on the boundary except when x=0.

The value at the center of the disk (x=0) equals the average of the equally weighted values on the boundary. See The_mean_value_property.

All locally integrable functions satisfying the mean-value property are both infinitely differentiable and harmonic.

The kernel itself appears to simply be 1/r^n shifted to the point x and multiplied by different constants.

For a circle (K = Poisson Kernel):

$f(re^{i\phi})= \oint_0^{2\pi} f(Re^{i\theta}) K(R,r,\theta-\phi) d\theta$

$\frac{d(a(x,y)+ib(x,y))}{d(x+iy)} = \frac{da+idb}{dx+idy} = \frac{(da+idb)(dx-idy)}{dx^2+dy^2} = \frac{dadx+dbdy+i(dbdx-dady)}{dx^2+dy^2}$ $\frac{d(a(x,y)+ib(x,y))}{d(x+iy)} = \frac{da}{dx} +\frac{db}{dy} +i \bigg(\frac{db}{dx}- \frac{da}{dy} \bigg) = {\color{red} \nabla \cdot f} + i {\color{blue} \nabla \times f}$

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Discrete mathematics

Set theory

See also: ^*Naive set theory, ^*Zermelo–Fraenkel set theory, Set theory^w, ^*Set notation, ^*Set-builder notation, Set^w, ^*Algebra of sets, ^*Field of sets, and ^*Sigma-algebra

$\varnothing$ is the empty set (the additive identity)

$\mathbf{U}$ is the universe of all elements (the multiplicative identity)

$a \in A$ means that a is a element^w (or member) of set A. In other words a is in A.

$\{ x \in \mathbf{A} : x \notin \mathbb{R} \}$ means the set of all x's that are members of the set A such that x is not a member of the real numbers^w. Could also be written $\{ \mathbf{A} - \mathbb{R} \}$

A set^w does not allow multiple instances of an element. $\{1,1,2\} = \{1,2\}$

A multiset^w does allow multiple instances of an element. $\{1,1,2\} \neq \{1,2\}$

A set can contain other sets. $\{1,\{2\},3\} \neq \{1,2,3\}$

$A \subset B$ means that A is a proper subset^w of B

$A \subseteq A$ means that a is a subset^w of itself. But a set is not a proper subset^w of itself.

$A \cup B$ is the Union^w of the sets A and B. In other words, $\{A+B\}$

$\{1,2\}+\{2,3\}=\{1,2,3\}$

$A \cap B$ is the Intersection^w of the sets A and B. In other words, $\{A \cdot B\}$ All a's in B.

Associative: $A \cdot \{B \cdot C\} = \{A \cdot B\} \cdot C$

Distributive: $A \cdot \{B + C\}=\{A \cdot B\} + \{A \cdot C\}$

Commutative: $\{A \cdot B\} =\{B \cdot A\}$

$A \setminus B$ is the Set difference^w of A and B. In other words, $\{A - A \cdot B\}$

$\overline{A}$ or $A^c = \{U - A\}$ is the complement^w of A.

$A \bigtriangleup B$ or $A \ominus B$ is the Anti-intersection^w of sets A and B which is the set of all objects that are a members of either A or B but not in both.

$A \bigtriangleup B = (A + B) - (A \cdot B) = (A - A \cdot B) + (B - A \cdot B)$

$A \times B$ is the Cartesian product^w of A and B which is the set whose members are all possible ordered pairs^w (a, b) where a is a member of A and b is a member of B.

The Power set^w of a set A is the set whose members are all of the possible subsets of A.

A ^*cover of a set X is a collection of sets whose union contains X as a subset.

A subset A of a topological space X is called ^*dense (in X) if every point x in X either belongs to A or is arbitrarily "close" to a member of A.

A subset A of X is ^*meagre if it can be expressed as the union of countably many nowhere dense subsets of X.

^*Disjoint union of sets $A_0$ = {1, 2, 3} and $A_1$ = {1, 2, 3} can be computed by finding:

$\begin{align} A^*_0 & = \{(1, 0), (2, 0), (3, 0)\} \\ A^*_1 & = \{(1, 1), (2, 1), (3, 1)\} \end{align}$

so

$A_0 \sqcup A_1 = A^*_0 \cup A^*_1 = \{(1, 0), (2, 0), (3, 0), (1, 1), (2, 1), (3, 1)\}$

Let H be the subgroup of the integers (mZ, +) = ({..., −2m, −m, 0, m, 2m, ...}, +) where m is a positive integer.

Then the ^*cosets of H are the mZ + a = {..., −2m+a, −m+a, a, m+a, 2m+a, ...}.

There are no more than m cosets, because mZ + m = m(Z + 1) = mZ.

The coset (mZ + a, +) is the congruence class^w of a modulo m.

Cosets are not usually themselves subgroups of G, only subsets.

$\exists$ means "there exists at least one"

$\exists!$ means "there exists one and only one"

$\forall$ means "for all"

$\and$ means "and" (not to be confused with wedge product^w)

$\lor$ means "or" (not to be confused with antiwedge product^w)

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Probability

$\vert A \vert$ is the cardinality^w of A which is the number of elements in A. See measure^w.

$P(A) = {\vert A \vert \over \vert U \vert}$ is the unconditional probability^w that A will happen.

$P(A \mid B) = {\vert A \cdot B \vert \over \vert B \vert}$ is the conditional probability^w that A will happen given that B has happened.

$P(A + B) = P(A) + P(B) - P(A \cdot B)$ means that the probability that A or B will happen is the probability of A plus the probability of B minus the probability that both A and B will happen.

$P(A \cdot B) = P(A \cdot B \mid B)P(B) = P(A \cdot B \mid A)P(A)$ means that the probability that A and B will happen is the probability of "A and B given B" times the probability of B.

$P(A \cdot B \mid B) = \frac{P(A \cdot B \mid A) \, P(A)}{P(B)},$ is ^*Bayes' theorem

If you dont know the certainty then you can still know the probability. If you dont know the probability then you can always know the Bayesian probability. The Bayesian probability is the degree to which you expect something.

Even if you dont know anything about the system you can still know the ^*A priori Bayesian probability. As new information comes in the ^*Prior probability is updated and replaced with the ^*Posterior probability by using ^*Bayes' theorem.

From Wikipedia:Base rate fallacy:

In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software. 99% of the time it behaves correctly. 1% of the time it behaves incorrectly, ringing when it should not and failing to ring when it should. Suppose now that an inhabitant triggers the alarm. What is the chance that the person is a terrorist? In other words, what is P(T | B), the probability that a terrorist has been detected given the ringing of the bell? Someone making the 'base rate fallacy' would infer that there is a 99% chance that the detected person is a terrorist. But that is not even close. For every 1 million faces scanned it will see 100 terrorists and will correctly ring 99 times. But it will also ring falsely 9,999 times. So the true probability is only 99/(9,999+99) or about 1%.

permutation^w relates to the act of arranging all the members of a set^w into some sequence^w or ^*order.

The number of permutations of n distinct objects is n!^w.

A derangement is a permutation of the elements of a set, such that no element appears in its original position.

In other words, derangement is a permutation that has no ^*fixed points.

The number of ^*derangements of a set of size n, usually written ^*!n, is called the "derangement number" or "de Montmort number".

The ^*rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements.

a combination^w is a selection of items from a collection, such that the order of selection does not matter.

For example, given three numbers, say 1, 2, and 3, there are three ways to choose two from this set of three: 12, 13, and 23.

More formally, a k-combination of a set^w S is a subset of k distinct elements of S.

If the set has n elements, the number of k-combinations is equal to the binomial coefficient^w

$\binom nk = \textstyle\frac{n!}{k!(n-k)!}.$ Pronounced n choose k. The set of all k-combinations of a set S is often denoted by $\textstyle\binom Sk$.

The central limit theorem (CLT) establishes that, in most situations, when ^*independent random variables are added, their properly normalized sum tends toward a normal distribution^w (informally a "bell curve") even if the original variables themselves are not normally distributed.

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A plot of normal distribution^w (or bell-shaped curve) where each band has a width of 1 standard deviation – See also: ^*68–95–99.7 rule

In statistics^w, the standard deviation (SD, also represented by the Greek letter sigma σ^w or the Latin letter s) is a measure that is used to quantify the amount of variation or ^*dispersion of a set of data values.

A low standard deviation indicates that the data points tend to be close to the mean^w (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The ^*hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure.

In contrast, the ^*binomial distribution describes the probability of k successes in n draws with replacement.

^*Extreme value theory is used to model the risk of extreme, rare events, such as the 1755 Lisbon earthquake. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any previously observed.

See also ^*Dirichlet distribution, ^*Rice distribution, ^*Benford's law

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Logic

From Wikipedia:Inductive reasoning

Given that "if A is true then the laws of cause and effect would cause B, C, and D to be true",

An example of deduction would be

"A is true therefore we can deduce that B, C, and D are true".

An example of induction would be

"B, C, and D are observed to be true therefore A might be true".

A is a reasonable explanation for B, C, and D being true.

For example:

A large enough asteroid impact would create a very large crater and cause a severe impact winter that could drive the dinosaurs to extinction.

We observe that there is a very large crater in the Gulf of Mexico dating to very near the time of the extinction of the dinosaurs

Therefore this impact is a reasonable explanation for the extinction of the dinosaurs.

The Temptation is to jump to the conclusion that this must therefore be the true explanation. But this is not necessarily the case. The Deccan Traps in India also coincide with the disappearance of the dinosaurs and could have been the cause their extinction.

A classical example of an incorrect inductive argument was presented by John Vickers:

All of the swans we have seen are white.

Therefore, we know that all swans are white.

The correct conclusion would be, "We expect that all swans are white". As a logic of induction, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to a prior probability for a hypothesis based on logic or previous experience, and when faced with evidence, we adjust the strength of our belief (expectation) in that hypothesis in a precise manner using Bayesian logic.

Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true. Instead of being valid or invalid, inductive arguments are either strong or weak, which describes how probable it is that the conclusion is true.

It is often said that when thinking subjectively you will see whatever you want to see. In fact this is always the case. It's just that if you truly want to see what the facts say when they are allowed to speak for themselves then you will see that too. This is called "objectivity". It is man's capacity for objective reasoning that separates him from the animals. None of us are all-human though. All of us have a little bit of ego that tries to think subjectively. (Reality is not the egos native habitat.)

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Morphisms

See also: Higher category theory^w and ^*Multivalued function (misnomer)

Injectionw	Invertible function Bijectionw	Surjectionw
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Every function^w, f(x), has exactly one output for every input.

If its inverse function^w, f⁻¹(x), has exactly one output for every input then the function is ^*invertible.

$f^{-1}(f(x))=x$

If it isn't invertible then it doesn't have an inverse function.

An ^*involution which is a function that is its own inverse function. Example: f(x)=x/(x-1)

$f(f(x))=x$

A morphism^w is exactly the same as a function but in Category theory^w every morphism has an inverse which is allowed to have more than one value or no value at all.

^*Categories consist of:

Objects (usually Sets^w)

Morphisms^w (usually maps^w) possessing:

one source object (domain)

one target object (codomain)

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a morphism is represented by an arrow:

$f(x)=y$ is written $f : x \to y$ where x is in X and y is in Y.

$g(y)=z$ is written $g : y \to z$ where y is in Y and z is in Z.

The ^*image of y is z.

The ^*preimage (or ^*fiber) of z is the set of all y whose image is z and is denoted $g^{-1}[z]$

A picture is worth 1000 words

A space Y is a ^*covering space (a fiber bundle) of space Z if the map $g : y \to z$ is locally homeomorphic^w.

A covering space is a ^*universal covering space if it is ^*simply connected.

The concept of a universal cover was first developed to define a natural domain for the ^*analytic continuation of an analytic function^w.

The general theory of analytic continuation and its generalizations are known as ^*sheaf theory.

The set of ^*germs can be considered to be the analytic continuation of an analytic function.

A topological space is ^*(path-)connected if no part of it is disconnected.

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Not simply connected

A space is ^*simply connected if there are no holes passing all the way through it (therefore any loop can be shrunk to a point)

See ^*Homology

Composition of morphisms:

$g(f(x))$ is written $g \circ f$

f is the ^*pullback of g

f is the ^*lift of $g \circ f$

? is the ^*pushforward of ?

A ^*homomorphism is a map from one set to another of the same type which preserves the operations of the algebraic structure:

$f(x \cdot y) = f(x) \cdot f(y)$

$f(x + y) = f(x) + f(y)$

See ^*Cauchy's functional equation

A ^*Functor is a homomorphism with a domain in one category and a codomain in another.

A ^*group homomorphism from (G, ∗) to (H, ·) is a ^*function h : G → H such that

$h(u*v) = h(u) \cdot h(v) = h(c)$ for all u*v = c in G.

For example $log(a*b) = log(a) + log(b)$

Since log is a homomorphism that has an inverse that is also a homomorphism, log is an ^*isomorphism of groups. The logarithm^w is a ^*group isomorphism of the multiplicative group of ^*positive real numbers $\R^+$ to the ^*additive group of real numbers, $\R$.

See also ^*group action and ^*group orbit

A ^*Multicategory has morphisms with more than one source object.

A ^*Multilinear map $f(v_1,\ldots,v_n) = W$:

$f\colon V_1 \times \cdots \times V_n \to W\text{,}$

has a corresponding Linear map^w:$F(v_1\otimes \cdots \otimes v_n) = W$:

$F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,}$

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Numerical methods

See also: ^*Explicit and implicit methods

From Wikipedia:Numerical analysis:

One of the simplest problems is the evaluation of a function at a given point.

The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient.

For polynomials, a better approach is using the ^*Horner scheme, since it reduces the necessary number of multiplications and additions.

Generally, it is important to estimate and control ^*round-off errors arising from the use of ^*floating point arithmetic.

^*Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?

^*Extrapolation is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points.

A line through 20 points

^*Regression is also similar, but it takes into account that the data is imprecise.

Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function.

The ^*least squares-method is one popular way to achieve this.

Much effort has been put in the development of methods for solving ^*systems of linear equations.

Standard direct methods, i.e., methods that use some ^*matrix decomposition

^*Gaussian elimination, ^*LU decomposition, ^*Cholesky decomposition for symmetric^w (or hermitian^w) and positive-definite matrix^w, and ^*QR decomposition for non-square matrices.

^*Iterative methods

^*Jacobi method, ^*Gauss–Seidel method, ^*successive over-relaxation and ^*conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a ^*matrix splitting.

^*Root-finding algorithms are used to solve nonlinear equations.

If the function is differentiable^w and the derivative is known, then Newton's method^w is a popular choice.

^*Linearization is another technique for solving nonlinear equations.

Optimization^w problems ask for the point at which a given function is maximized (or minimized).

Often, the point also has to satisfy some ^*constraints.

Wind direction in blue, true trajectory in black, Euler method in red.

Wind direction in blue, true trajectory in black, Euler method in red.

Differential equation^w: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens?

The feather will follow the air currents, which may be very complex.

One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again.

This is called the ^*Euler method for solving an ordinary differential equation.

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Information theory

From Wikipedia:Information theory:

Information theory studies the quantification, storage, and communication of information.

Communications over a channel—such as an ethernet cable—is the primary motivation of information theory.

From Wikipedia:Quantities of information:

Shannon derived a measure of information content called the ^*self-information or "surprisal" of a message m:

$I(m) = \log \left( \frac{1}{p(m)} \right) = - \log( p(m) ) \,$

where $p(m) = \mathrm{Pr}(M=m)$ is the probability that message m is chosen from all possible choices in the message space $M$. The base of the logarithm only affects a scaling factor and, consequently, the units in which the measured information content is expressed. If the logarithm is base 2, the measure of information is expressed in units of ^*bits.

Information is transferred from a source to a recipient only if the recipient of the information did not already have the information to begin with. Messages that convey information that is certain to happen and already known by the recipient contain no real information. Infrequently occurring messages contain more information than more frequently occurring messages. This fact is reflected in the above equation - a certain message, i.e. of probability 1, has an information measure of zero. In addition, a compound message of two (or more) unrelated (or mutually independent) messages would have a quantity of information that is the sum of the measures of information of each message individually. That fact is also reflected in the above equation, supporting the validity of its derivation.

An example: The weather forecast broadcast is: "Tonight's forecast: Dark. Continued darkness until widely scattered light in the morning." This message contains almost no information. However, a forecast of a snowstorm would certainly contain information since such does not happen every evening. There would be an even greater amount of information in an accurate forecast of snow for a warm location, such as Miami. The amount of information in a forecast of snow for a location where it never snows (impossible event) is the highest (infinity).

The more surprising a message is the more information it conveys. The message "LLLLLLLLLLLLLLLLLLLLLLLLL" conveys exactly as much information as the message "25Ls". The first message which is 25 bytes long can therefore be "compressed" into the second message which is only 4 bytes long.

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Early computers

Python 3.6 for beginners, https://repl.it/languages, https://www.wolframalpha.com

See also: ^*Time complexity

• ^*Antikythera mechanism

• ^*Mechanical computer

• ^*Analog computer

• ^*Abacus

• ^*Napier's bones

• ^*Slide rule

• ^*Pascal's calculator

• ^*Difference engine

• ^*Analytical Engine

• ^*Pinwheel calculator

• ^*Differential analyser

• ^*Ball-and-disk integrator

• ^*Curta

• ^*Lehmer sieve

• ^*Z2 (computer)

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Tactical thinking

^*The prisoner's dilemma
A non-zero-sum game

	Tactic X (Cooperate)	Tactic Y (Defect)
Tactic A (Cooperate)	1, 1	5, -5
Tactic B (Defect)	-5, 5	-5, -5

See also ^*Wikipedia:Strategy (game theory)

From Wikipedia:Game theory:

In the accompanying example there are two players; Player one (blue) chooses the row and player two (red) chooses the column.

Each player must choose without knowing what the other player has chosen.

The payoffs are provided in the interior.

The first number is the payoff received by Player 1; the second is the payoff for Player 2.

Tit for tat is a simple and highly effective tactic in game theory for the iterated prisoner's dilemma.

An agent using this tactic will first cooperate, then subsequently replicate an opponent's previous action.

If the opponent previously was cooperative, the agent is cooperative.

If not, the agent is not.

A zero-sum game

	X	Y
A	1,-1	-1,1
B	-1,1	1,-1

In zero-sum games the sum of the payoffs is always zero (meaning that a player can only benefit at the expense of others).

Cooperation is impossible in a zero-sum game.

John Forbes Nash proved that there is a Nash equilibrium (an optimum tactic) for every finite game.

In the zero-sum game shown to the right the optimum tactic for player 1 is to randomly choose A or B with equal probability.

Strategic thinking differs from tactical thinking by taking into account how the short term goals and therefore optimum tactics change over time.

For example the opening, middlegame, and endgame of chess require radically different tactics.

See also: ^*Reverse game theory

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Physics

See also: Wikisource:The Mathematical Principles of Natural Philosophy (1846) and ^*Galilean relativity

Reality is what doesnt go away when you arent looking at it.

Something is known Beyond a reasonable doubt if any doubt that it is true is unreasonable. A doubt is reasonable if it is consistent with the laws of cause and effect.

From Wikipedia:Philosophiæ Naturalis Principia Mathematica:

In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them.

Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes.

Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

Classical mechanics^w

Newtonian mechanics^w, Lagrangian mechanics^w, and Hamiltonian mechanics^w

The difference between the net kinetic energy and the net potential energy is called the “Lagrangian.”

The action is defined as the time integral of the Lagrangian.

The Hamiltonian is the sum of the kinetic and potential energies.

^*Noether's theorem states that every differentiable symmetry of the ^*action of a physical system has a corresponding ^*conservation law.

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^*Relativistic mechanics

^*Special relativity, and ^*General relativity

Energy is conserved in relativity and proper velocity is proportional to momentum at all velocities.

Quantum mechanics^w

https://www.eng.famu.fsu.edu/~dommelen/quantum/style_a/contents.html

Highly recommend:

• Thinking Physics Is Gedanken Physics by Lewis Carroll Epstein
• Understanding physics by Isaac Asimov
• The Feynman Lectures

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Relativity

Most confusion about relativity centers around a poor understanding of relativity of simultaneity.

Since the length of an object is the distance from head to tail at one simultaneous moment, it follows that if two observers disagree about what events are simultaneous then they will also disagree on the length of objects.

If a line of clocks appear synchronized to a stationary observer and appear to be out of sync to that same observer after accelerating to a certain velocity then it follows that during the acceleration the clocks ran at different speeds. Some may even run backwards. This line of reasoning leads to general relativity.

The gravitational time dilation at any point in a gravity well is equal to the time dilation that an object falling to that point would experience due its velocity (which never reaches "c") alone.

There might be a loophole to the law that you can't travel faster than light: If the distance between the front of rocket and the back can be made zero then it's conceivable that it could travel faster than light.

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Dimensional analysis

See also: Natural units

From Wikipedia:Dimensional analysis:

Any physical law that accurately describes the real world must be independent of the units (e.g. km or mm) used to measure the physical variables.

Consequently, every possible commensurate equation for the physics of the system can be written in the form

$a_0 \cdot D_0 = (a_1 \cdot D_1)^{p_1} (a_2 \cdot D_2)^{p_2}...(a_n \cdot D_n)^{p_n}$

The dimension, D_n, of a physical quantity can be expressed as a product of the basic physical dimensions length (L), mass (M), time (T), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J), each raised to a rational power.

Suppose we wish to calculate the ^*range of a cannonball when fired with a vertical velocity component $V_\mathrm{y}$ and a horizontal velocity component $V_\mathrm{x}$, assuming it is fired on a flat surface.

The quantities of interest and their dimensions are then

range as L_x

$V_\mathrm{x}$ as L_x/T

$V_\mathrm{y}$ as L_y/T

g as L_y/T²

The equation for the range may be written:

$range = (V_x)^a (V_y)^b (g)^c$

Therefore

$\mathsf{L}_\mathrm{x} = (\mathsf{L}_\mathrm{x}/\mathsf{T})^a\,(\mathsf{L}_\mathrm{y}/\mathsf{T})^b (\mathsf{L}_\mathrm{y}/\mathsf{T}^2)^c\,$

and we may solve completely as $a = 1$, $b=1$ and $c=-1$.

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Atoms

See also: Periodic table^w and Spatial_structure_of_the_electron

The first pair of electrons fall into the ground shell. Once that shell is filled no more electrons can go into it. Any additional electrons go into higher shells.

The nucleus however works differently. The first few neutrons form the first shell. But any additional neutrons continue to fall into that same shell which continues to expand until there are 49 pairs of neutrons in that shell.

The highest energy gamma rays emitted by nuclei are around 10 Mev which corresponds to a wavelength of 124 fm.

The electric force between two electrons is 4.166 * 10⁴² times stronger than the gravitational force. (12,242 * 2¹²⁸)

The energy required to assemble a sphere of uniform charge density = $\frac{3}{5}\frac{Q^2}{4 \pi \epsilon_0 r}$

For Q=1 electron charge and r=1.8506 angstrom thats 4.669 ev. That energy is stored in the electric field of the electron.

The energy per volume stored in an electric field is proportional to the square of the field strength so twice the charge has 4 times as much energy.

4*4.669 = 18.676.

Mass of electron = M_e = 510,999 ev

Mass of proton = M_p = 938,272,000 ev

Mass of neutron = M_n = 939,565,000 ev

M_n = M_p + M_e + 782,300 ev

Mass of muon = M_μ = 105.658 ev = 206.7683 * M_e

Mass of helium atom = 3,728,400,000 = 4*M_e+4*M_p -52.31 M_e

The missing 52.31 electron masses of energy is called the mass deficit or nuclear binding energy. Fusing hydrogen into helium releases this energy.

Iron can be fused into heavier elements too but doing so consumes energy rather than releases energy.

The total outward force for a solid 4-dimensional sphere of uniform density in Clifford rotation is $\frac{4}{5} \cdot \frac{m v^2}{r}$

The angular momentum of a solid 4-dimensional sphere of uniform density is $\frac{2}{3} \cdot mvr$

Empirically determined values for the size of atoms:

Diatomic Hydrogen (Z=2) = 1.9002 angstroms

Helium (Z=2) = 1.8506 angstroms

In 3 dimensions the force between 2 electrons is:

$F = \frac{1}{4\pi\varepsilon_0} { e_1 e_2 \over r^2}$

where m_e is the electron's mass, e₁ is the charge of the electron,

Failed to parse (syntax error): {\displaystyle \varepsilon_0 = \frac{1}{180.95} \frac{e^2}{\text{eV} Å}}

but in 4 dimensions:

$\varepsilon = \frac{2 \varepsilon_0}{\pi r}$

where r is the distance at which the inverse square law gives the same result as the inverse cube law. In other words, the distance at which the inverse square law of the macroscopic world gives way to the inverse cube law of the microscopic world.

The angular momentum is:

$\frac{2}{3} \cdot m_\mathrm{e} v r = \hbar$

where ħ is reduced Planck constant

$\hbar={{h}\over{2\pi}} = 1.054\ 571\ 800(13)\times 10^{-34}\text{J}{\cdot}\text{s}$

Therefore:

$v = \frac{3}{2} \cdot \frac{\hbar}{m_\mathrm{e}} \frac{1}{r}$

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Density and thermal expansion

Densities:

Crystalline solids: 1.2

Amorphous solids: 1.1

liquids: 1

Water ice is an exception. Ice has a density of 0.9167

From Wikipedia:Thermal expansion

Thermal expansion generally decreases with increasing bond energy, which also has an effect on the melting point of solids, so, high melting point materials are more likely to have lower thermal expansion. In general, liquids expand slightly more than solids. The thermal expansion of glasses is higher compared to that of crystals.

For an ideal gas, the volumetric thermal expansion (i.e., relative change in volume due to temperature change) depends on the type of process in which temperature is changed. Two simple cases are where the pressure is held constant (Isobaric process), or when the volume (Isochoric process) is held constant.

The derivative of the ideal gas law, $PV = T$, is

$P dV + V dP = dT$

where $P$ is the pressure, $V$ is the specific volume, and $T$ is temperature measured in energy units.

By definition of an isobaric thermal expansion, we have $dP=0$, so that $P dV=dT$, and the isobaric thermal expansion coefficient is

$\alpha_{P = C^{te}} \equiv \frac{1}{V} \left(\frac{d V}{d T}\right) = \frac{1}{V} \left(\frac{1}{P}\right) = \frac{1}{PV} = \frac{1}{T}$.

Similarly, if the volume is held constant, that is if $dV=0$, we have $V dP=dT$, so that the isovolumic thermal expansion is

$\alpha_{V=C^{te}} \equiv \frac{1}{P} \left(\frac{d P}{d T}\right) = \frac{1}{P} \left(\frac{1}{V}\right) = \frac{1}{P V} = \frac{1}{T}$.

For a solid, we can ignore the effects of pressure on the material, and the volumetric thermal expansion coefficient can be written:

$\alpha_V = \frac{1}{V}\,\frac{dV}{dT}$

where $V$ is the volume of the material, and $dV/dT$ is the rate of change of that volume with temperature.

This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 K. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 K, or 0.004% K⁻¹.

If we already know the expansion coefficient, then we can calculate the change in volume

$\frac{\Delta V}{V} = \alpha_V\Delta T$

where $\Delta V/V$ is the fractional change in volume (e.g., 0.002) and $\Delta T$ is the change in temperature (50 °C).

For common materials like many metals and compounds, the thermal expansion coefficient is inversely proportional to the melting point. In particular for metals the relation is:

$\alpha \approx \frac{0.020}{M_P}$

for halides and oxides

$\alpha \approx \frac{0.038}{M_P} - 7.0 \cdot 10^{-6} \, \mathrm{K}^{-1}$

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Quasiparticles

See also: Wikipedia:List of quasiparticles and Wikipedia:Quasiparticle

A hole is a region with a net surplus of positive charges.
An anti-hole is a region with a net surplus of negative charges.
Electricity is the flow of holes and anti-holes.

A flow of positive charges gives the same electric current, and has the same effect in a circuit, as an equal flow of negative charges in the opposite direction. Since current can be the flow of either positive or negative charges, or both, a convention is needed for the direction of current that is independent of the type of charge carriers. The direction of conventional current is arbitrarily defined as the same direction as positive charges flow.

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A p-type semiconductor only conducts holes.
An n-type semiconductor only conducts anti-holes.
Holes and anti-holes combine at the junction of a forward biased diode.

In the case of a light emitting diode the combining of electrons and holes results in the creation of light.

Electricity will not flow through a reverse biased diode because this would require holes and anti-holes to form at and move in opposite directions away from the junction.

However, in the case of a photodiode, current can be induced to flow by shining a light on the junction.

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A transistor can be thought of as two diodes placed end-to-end (i.e. in series). When there is a voltage drop between the collector and the emitter then one diode is forward biased and the other diode is reverse biased. Because one of the diodes is reverse biased current will not flow. However current will flow from the collector to the emitter when a small amount of current is allowed to pass through the base.

To understand why this happens it helps to imagine that the forward biased diode is a light emitting diode and the reverse biased diode is a photodiode. (Such a transistor is called a photon coupled transistor.) When current is allowed to pass through the base light is created by the light emitting diode. This light is then absorbed by the photodiode and therefore current is able to pass from collector to emitter. This in turn creates still more light which allows still more current to pass. If the photodiode absorbed 100% of the light emitted than the current would flow forever. Since not all the light created by the light emitting diode is absorbed by the photodiode the current will decay rapidly.

The total amount of current that flows from collector to emitter will be some multiple of the current that originally flowed through the base.

From Wikipedia:Exciton:

An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and in some liquids. The wavefunction of the bound state is said to be hydrogenic, an exotic atom state akin to that of a hydrogen atom. However, the binding energy is much smaller and the particle's size much larger than a hydrogen atom. This is because of both the screening of the Coulomb force by other electrons in the semiconductor (i.e., its dielectric constant), and the small effective masses of the excited electron and hole. Provided the interaction is attractive, an exciton can bind with other excitons to form a biexciton, analogous to a dihydrogen molecule.

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Tidal acceleration

See also: Formation_of_the_Solar_System^w

Image shows an approximation of the shape (^*Equipotentials) of a rapidly spinning planet. North pole is at the top. South pole is at the bottom. The equator reaches orbital velocity.

Orbital velocity:

$v_o = \sqrt{\frac{GM}{r}}$

Orbital period:

$T = 2\pi\sqrt{\frac{r^3}{GM}}$

Orbital angular momentum:

$mvr \quad = \quad m \Bigg ( \sqrt{\frac{GM}{r}} \Bigg ) r \quad = \quad m \sqrt{GMr}$

Rotational angular momentum of solid sphere:

$L=I \omega = \frac{2}{5}mr^2 \frac{v}{r} = \frac{2}{5}mvr$

where:

• r is the orbit's semi-major axis^w
• G is the gravitational constant^w,
• M is the mass of the more massive body.
• m is the mass of the less massive body.

Moons orbital angular momentum is 28.73 * 10^33 Js

Earths rotational angular momentum is 7.079 * 10^33 Js

Correcting for Earths uneven mass distribution: 3.85 + 0.44 + 0.25 + 0.073 = 4.6 * 10^33 Js

The total amount of angular momentum for the Earth-Moon system is 28.73 + 4.6 = 33.33 * 10^33 Js

Moons current orbit is 384,399 km. Its orbital period is 2.372 * 10⁶ seconds. (27 days, 10 hours, 50 minutes). Its orbital velocity is 1.022 km/s.

^*Roche limit for the moon is

Fluid: 18,381 km fluid

384,399 / 18,381 = 20.9

Orbital momentum of moon at fluid Roche limit = 28.73 * 10^33 Js / sqrt(20.9) = 6.3 * 10^33

Earth would spin (28.73-6.3+4.6)/4.6 = 5.876 times faster

Rigid: 9,492 km

384,399 / 9,492 = 40.5

Orbital momentum of moon at rigid Roche limit = 28.73 * 10^33 Js / sqrt(40.5) = 4.5 * 10^33

Earth would spin (28.73-4.5+4.6)/4.6 = 6.27 times faster

Orbital radius with period = 4 hours:

$\sqrt[3]{G \cdot 1 \text{ Earth mass} \cdot \Bigg ( \frac{4\text{ hours}}{2\pi} \Bigg )^2 }$ = 12,800 km

Alternately we can ask what the orbital period would be if Earth had a moon (not necessarily the moon) at 18,381 km.

$T = 2\pi\sqrt{\frac{(18,381 \text{ km})^3}{G*1 \text{ Earth mass} }}$ = 7.554 hours

Earth would spin 24/7.554 = 3.177 times faster

Earths angular momentum would be 3.177 * 4.6 * 10^33 Js = 14.6142 * 10^33 Js

Our current Moons angular momentum would be 28.73 - (14.6142 - 4.6) * 10^33 Js = 18.7158 * 10^33 Js

Thats 18.7158 / 28.73 = 0.65

So the current moons orbit would have been 0.65^2 * 384,399 km = 0.424 * 384,399 km = 162985 km

Tidal rhythmites are alternating layers of sand and silt laid down offshore from estuaries having great tidal flows. Daily, monthly and seasonal cycles can be found in the deposits. This geological record indicates that 620 million years ago there were 400±7 solar days/year

The motion of the Moon can be followed with an accuracy of a few centimeters by lunar laser ranging. Laser pulses are bounced off mirrors on the surface of the moon. The results are:

+38.08±0.04 mm/yr (384,399 km / 63.4 billion years)

1.42*10^24 Js/yr (33.33 * 10^33 Js / 23 billion years)

1.42*10^26 Js/century

The corresponding change in the length of the day can be computed:

(1.42*10^26)/(4.6 * 10^33) * 24 hours = 3.087*10^-8 * 24 hours = +2.667 ms/century

620 million yrs ago the Moon had 1.42*10^24 * 620*10^6 = 0.88*10^33 Js less angular momentum. The moons orbit was therefore 384,399 km * ((28.73-0.88)/28.73)^2 = 361,211 km. One month lasted 2.161 * 10⁶ seconds. (25 days, 16 minutes, 40 seconds)

The Earth spun (4.6+0.88)/4.6 = 1.19 times faster so the day was 24 hours / 1.19 = 20.1680672 hours

The year was 400 "days" * 20.1680672 hours per "day" = 336.135 24-hour periods

Earths orbit was therefore

$\sqrt[3]{G \cdot 1 \text{ Solar mass} \Bigg ( \frac{336.134453 \text{ days}}{2\pi} \Bigg )^2}$ = 0.9461 au

Therefore Earth must be receding from the sun at 13 m/yr

Thats 0.395 au / 4.543 billion years

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Planets

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^*Titius–Bode law.

#	Planet	g/cm^3	km	g's	au
1	Mercury	5.427	2,440	0.377	0.387
2	Venus	5.243	6,052	0.904	0.723
3	Earth	5.515	6,371	1	1.000
4	Mars	3.934	3,390	0.378	1.524
5	Ceres	2.093	476.2	0.028	2.766
6	Jupiter	1.326	69,911	2.528	5.203
7	Saturn	0.687	58,232	1.065	9.537
8	Ouranos	1.270	25,362	0.904	19.191
9	Neptune	1.638	24,622	1.137	30.069

From Wikipedia:16 Psyche:

16 Psyche is one of the ten most massive asteroids in the asteroid belt. It is over 200 km (120 mi) in diameter and contains a little less than 1% of the mass of the entire asteroid belt. It is thought to be the exposed iron core of a protoplanet.

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Brown dwarfs

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Hatzes & Rauer (2015), “A Definition for Giant Planets Based on the Mass-Density Relationship”, arXiv:1506.05097 [astro-ph.EP]

Hydrogen	Atomic radius	g/cm3	Jupiter volume	g's
Liquid	1	0.07085	0.053 MJup	0.14
Metallic	1/ 4	4.5344	3.400 MJup	9.00
Double	1/ 5.657	12.8250	9.669 MJup	25.56
Triple	1/ 8	36.2752	27.300 MJup	72.16
Quadruple	1/ 11.31	102.6	77.350 MJup	204.40
Quintuple	1/ 16	290.2016	219.000 MJup	578.80
Sextuple	1/ 22.63	820.8140	618.800 MJup	1636.00

As can be seen in the image to the right, all planets (Brown dwarfs) from 1 to 100 Jupiter masses are about 1 Jupiter radius which is 69,911 km. The largest "puffy" planets are 2 Jupiter radii. 1 Jupiter volume = 1.431×10¹⁵ km³

This suggests that the pressure an electron shell (in degenerate matter) can withstand without again becoming degenerate (^*Electron degeneracy pressure) is inversely proportional to the sixth power of its radius:

$P=\frac{1}{r^6}$

(This formula only applies to degenerate matter like metallic hydrogen. Non-degenerate matter can withstand far more pressure).

If so then the maximum size (radius) that a planet composed entirely of one (degenerate) element could grow would depend only on, and be inversely proportional to, the atomic mass of its atoms. (Use 2 for the atomic mass of diatomic hydrogen).

Simplified calculation of radius of brown dwarf as core grows from zero to 1 Jupiter radius:

plot (-(1.83*r³/x -1.83*r²)+(x²/2-r²/2)=0.5) for r=0 to 1 and x=0 to 1

r is radius of core with 2.83 (sqrt(2)³) times the density of overlying material

Rock floats on top of the metallic hydrogen but iron sinks to the Core. 0.1% of the mass of the brown dwarf is iron. Assuming iron density of 231.85 g/cm3 (as in Earths core), the gravity of the iron core will cause the brown dwarf to be about 3% smaller then it would be otherwise.

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Dark matter

From Wikipedia:Dark matter

Dark matter is a type of unidentified matter that may constitute about 80% of the total matter in the universe. It has not been directly observed, but its gravitational effects are evident in a variety of astrophysical measurements. The primary evidence for dark matter is that calculations show that many galaxies would fly apart instead of rotating if they did not contain a large amount of matter beyond what can be observed.

From Wikipedia:Gravitational microlensing

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An Einstein ring.

Microlensing allows the study of objects that emit little or no light. With microlensing, the lens mass is too low for the displacement of light to be observed easily, but the apparent brightening of the source may still be detected. In such a situation, the lens will pass by the source in seconds to years instead of millions of years.

The Einstein radius, also called the Einstein angle, is the angular radius of the Einstein ring in the event of perfect alignment. It depends on the lens mass M, the distance of the lens d_L, and the distance of the source d_S:

$\theta_E = \sqrt{\frac{4GM}{c^2} \frac{d_S - d_L}{d_S d_L}}$ (in radians).

For M equal to 60 Jupiter masses, d_L = 4000 parsecs, and d_S = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.00024 arcseconds (angle subtended by 1 au at 4000 parsecs). By comparison, ideal Earth-based observations have angular resolution around 0.4 arcseconds, 1660 times greater. One parsec is equal to about 3.26 light-years (30 trillion km).

Any brown dwarf surrounded by a circumstellar disk larger and thicker than 1 au would therefore be virtually completely undetectable.

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Stars

See also: ^*Stellar evolution, ^*Helium flash, ^*Schönberg–Chandrasekhar limit, Coronal_heating_problem^w

Image of spiral galaxy ^*M81.

Explanation of spiral galaxy arms.

Fusion of diatomic hydrogen begins around 60 Jupiter masses. Fusion of monatomic helium requires significantly more pressure.

Fusion releases energy that heats the star causing it to expand. The expansion reduces the pressure in the core which reduces the rate of fusion. So the rate of fusion is self limiting. A low mass star has a lifetime of billions of years. A high mass star has a lifetime of only a few tens of millions of years despite starting with more hydrogen.

Low mass stars are far more common than high mass stars. The masses of the two component stars of NGC 3603-A1, A1a and A1b, determined from the orbital parameters are 116 ± 31 M☉ and 89 ± 16 M☉respectively. This makes them the two most massive stars directly measured, i.e. not estimated from models.

The luminousity of a star is:

$L = 4 \pi R^2 \sigma T^4$

where σ is the ^*Stefan–Boltzmann constant:

$\sigma = \frac{2\pi^5k_{\rm B}^4}{15h^3c^2} = \frac{\pi^2k_{\rm B}^4}{60\hbar^3c^2} = 5.670373(21) \, \times 10^{-8}\ \textrm{J}\,\textrm{m}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}$

The luminosity of the sun at 5772 K and 695,700 km is 3.828×10^26 Watts

Thats 6,297,000 watts/m²

The brightness of sunlight at the surface of the Earth is 1400 watt/meter²

The plasma inside a star is non-relativistic. A relativistic plasma with a thermal ^*distribution function has temperatures greater than around 260 keV, or ^*3.0 * 10⁹ K. Those sorts of temperatures are only created in a supernova. The core of the sun is about 15 * 10⁶ K.

Plasmas, which are normally opaque to light, are transparent to light with frequency higher than the ^*plasma frequency. The plasma literally cant vibrate fast enough to keep up with the light. Plasma frequency is proportional to the square root of the electron density.

$\omega = \sqrt{\frac{n_\mathrm{e} q_e^{2}}{m_e \varepsilon_0}}$

where

n_e = number of electrons / volume.

See also: ^*Bremsstrahlung#Thermal_bremsstrahlung

From Wikipedia:Radiative zone

From 0.3 to 1.2 solar masses, the region around the stellar core is a radiative zone. (The light frequency is higher than the plasma frequency). The radius of the radiative zone increases monotonically with mass, with stars around 1.2 solar masses being almost entirely radiative.

From Wikipedia:Convective zone

In main sequence stars of less than about 1.3 solar masses, the outer envelope of the star contains a region of relatively low temperature which causes the frequency of the light to be lower than the plasma frequency which causes the opacity to be high enough to produce a steep temperature gradient. This produces an outer convection zone. The Sun's convection zone extends from 0.7 solar radii (500,000 km) to near the surface.

From Wikipedia:Cepheid variable

A Cepheid variable is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude.

A strong direct relationship between a Cepheid variable's luminosity and pulsation period allows one to know the true luminosity of a Cepheid by simply observing its pulsation period. This in turn allows one to determine the distance to the star, by comparing its known luminosity to its observed brightness.

From Wikipedia:Variable star

The pulsation of cepheids is known to be driven by oscillations in the ionization of helium. From fully ionized (more opaque) He++ to partially ionized (more transparent) He+ and back to He++. See ^*Kappa mechanism.

In the swelling phase. Its outer layers expand, causing them to cool. Because of the decreasing temperature the degree of ionization also decreases. This makes the gas more transparent, and thus makes it easier for the star to radiate its energy. This in turn will make the star start to contract. As the gas is thereby compressed, it is heated and the degree of ionization again increases. This makes the gas more opaque, and radiation temporarily becomes captured in the gas. This heats the gas further, leading it to expand once again. Thus a cycle of expansion and compression (swelling and shrinking) is maintained.

From Wikipedia:Instability strip

In normal A-F-G stars He is neutral in the stellar photosphere. Deeper below the photosphere, at about 25,000–30,000K, begins the He II layer (first He ionization). Second ionization (He III) starts at about 35,000–50,000K.

Recombination and Reionization

From Wikipedia:Reionization The first phase change of hydrogen in the universe was recombination due to the cooling of the universe to the point where electrons and protons form neutral hydrogen. The universe was opaque before the recombination, due to the scattering of photons (of all wavelengths) off free electrons, but it became increasingly transparent as more electrons and protons combined to form neutral hydrogen atoms. The Dark Ages of the universe start at that point, because there were no light sources. The second phase change occurred once objects started to condense in the early universe that were energetic enough to re-ionize neutral hydrogen. As these objects formed and radiated energy, the universe reverted to once again being an ionized plasma. (See *Warm–hot intergalactic medium). At this time, however, matter had been diffused by the expansion of the universe, and the scattering interactions of photons and electrons were much less frequent than before electron-proton recombination. Thus, a universe full of low density ionized hydrogen will remain transparent, as is the case today.

The Sun's photosphere has a temperature between 4,500 and 6,000 K. Negative hydrogen ions (H-) are the primary reason for the highly opaque nature of the photosphere.

As the star fuses hydrogen into heavier elements the heavier elements build up in the core. Eventually the outer layers of the star are blown away and all thats left is the core. We call whats left a white dwarf.

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A plot of 22000 stars from the Hipparcos Catalogue together with 1000 low-luminosity stars (red and white dwarfs) from the Gliese Catalogue of Nearby Stars. The ordinary hydrogen-burning dwarf stars like the Sun are found in a band running from top-left to bottom-right called the Main Sequence. Giant stars form their own clump on the upper-right side of the diagram. Above them lie the much rarer bright giants and supergiants. At the lower-left is the band of white dwarfs.

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White dwarfs

*Z	*A	Element	(ppm)	g/cm3	g/cm3	radius
1	1	*Hydrogen	739,000	0.07085	290.2	71,492
1	2	*Deuterium	100	0.1417	580.4	35,746
2	4	*Helium	240,000	0.125	512	35,746
4	8	*Beryllium	0	2	8,192	17,873
8	16	*Oxygen	10,400	32	131,072	8,936
6	12	*Carbon	4,600	10.125	41,472	11,915
10	20	*Neon	1,340	78.125	320,000	7,149
26	56	*Iron-56	1,090	4358	15,748,096	2,553
7	14	*Nitrogen	960	18.76	76,841	10,213
14	28	*Silicon	650	300.125	1,229,312	5,107
12	24	*Magnesium	580	162	663,552	5,958
16	32	*Sulfur	440	512	2,097,152	4,468

	1H 2D
		H	+	H	=	2D
		D	+	D	=	4He
	4He
		He	+	He	=	Unstable
		He	×	3	=	12C
		He	+	C	=	16O
	12C
		C	+	C	=	24Mg
		C	+	O	=	28Si
	16O
		O	+	O	=	32S
		O	+	Mg	=	40Ca
	24Mg
		Mg	+	S	=	56Fe
	28Si 32S
		Si	+	Si	=	56Fe
		Si	+	S	=	60Ni
		S	+	S	=	64Zn
	56Fe 60Ni
14N and 20Ne are produced when the outer layers become convective. 8Be, 18F, and 26Al are unstable.

A white dwarf is about the same size as the Earth but is far denser and far more massive. A typical temperature for a white dwarf is 25,000 K. That would make its surface brightness 350 times the surface brightness of the sun.

Simplified calculation of radius of White dwarf as core grows from zero to half the original radius:

plot (-(15*r^3/x -15*r^2)+(x^2/2-r^2/2)=0.5) for r=0 to 0.5 and x=0 to 1

r is radius of core. The core has 16 times the density (twice the atomic number) of the overlying material. The final state has half the radius and twice the mass of the original white dwarf.

A 0.6 solar mass White dwarf is 8900 km in radius which Is 8.03 times smaller than Jupiter which suggests a composition of oxygen. It has a surface gravity of

$\frac{G \cdot 0.6 \text{ solar mass}}{(8900 \text{ km})^2}$ = 103,000 g's

Its density is 404,000 g/cm3 which is 12,625 times denser than oxygen in its ground state. Thats 23.285³ times denser. Sqrt(2)⁹ = 22.63

A 1.13 solar mass White dwarf is 4500 km in radius which Is 15.9 times smaller than Jupiter which suggests a composition of sulfur. It has a surface gravity of

$\frac{G \cdot 1.13 \text{ solar mass}}{(4500 \text{ km})^2}$ = 755,000 g's

Its density is 5.887 * 10⁶ g/cm3 which is 11,498 times denser than sulfur in its ground state. Thats 22.57³ times denser.

For a white dwarf made of iron:

Radius: 2,553 km

Surface area: 8.2*10⁷ km²

Mass per surface area: 3.8 * 10¹³ g/mm²

Mass: 4.454 * 10⁷ g/cm3 * (4/3)*pi*(2553 km)³ in solar masses = 1.56 solar masses.

Surface gravity: 3.24 * 10⁶ g's

Density: (sqrt(2)⁹)³ * 3844.75 g/cm3 = 4.454 * 10⁷ g/cm3

Core pressure: 1.8 * 10¹⁹ bars

The core of a white dwarf with a mass greater than the ^*Chandrasekhar limit (1.44 solar masses) will undergo gravitational collapse and become a neutron star.
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Neutron stars

See also: ^*Gravitoelectromagnetism

Assuming a solid honeycomb array of neutron pairs with radius 1 fm, a sheet of ^*neutronium (if such a thing existed) would have a density of 1.2893598 g/mm².

Density of a liquid neutron star made of neutron pairs with radius 1 fm would be 479.8×10¹² g/cm³

The maximum observed mass of neutron stars is about 2.01 M_☉.

At that density a 2 solar mass neutron star would have a radius of 12.5544 km

Its gravitational binding energy would be 0.282 solar masses of energy

The ^*Tolman–Oppenheimer–Volkoff limit (or TOV limit) is an upper bound to the mass of cold, nonrotating neutron stars, analogous to the Chandrasekhar limit for white dwarf stars. Observations of GW170817 suggest that the limit is close to 2.17 solar masses.

The equation of state^w for a neutron star is not yet known.

A 2 solar mass neutron star with radius of 12.5544 km would have a surface gravity of:

$\frac{G \cdot 2 \text{ solar mass}}{(12.5544 \text{ km})^2}$ = 1.717 10¹¹ g's

The pressure in its core would be $\frac{3}{8 \pi} \frac{G \cdot Mass_{active} \cdot Mass_{passive}}{r^4} =$ 5.072 * 10²⁸ bar = 5.071 * 10²⁸ bar

Its moment of inertia is: 0.4*2 solar masses*(12.5544 km)^2 = 2.507×10^38 kg m^2

From Wikipedia:Glitch (astronomy)

A glitch (See ^*Global_resurfacing_event) is a sudden increase of up to 1 part in 10⁶ in the rotational frequency of a rotation-powered pulsar. Following a glitch is a period of gradual recovery, lasting 10-100 days, where the observed periodicity slows to a period close to that observed before the glitch.

If mass is constant then

$radius = 1/\sqrt[3]{density}$

If angular momentum is constant then

$frequency = v/r = 1/r^2 = 1/(1/\sqrt[3]{density})^2 = density^{\frac{2}{3}}$

The moment of inertia of a solid crust 1 cm thick is: 0.666*((1.2*479.8×10^12 g/cm3)*1 cm*4*pi*(12.5544 km)^2)*(12.5544 km)^2 = 1.197×10^33 kg m^2. Thats 1/209,440 of the total moment of inertia. 1 cm doesn't seem like much but if each Neutron were the size of an atom then that one centimeter would be one or two km.

External link: Pulsar glitches and their impact on neutron-star astrophysics

From Wikipedia:Supermassive black hole

A supermassive black hole (SMBH or SBH) is the largest type of ^*black hole, on the order of hundreds of thousands to billions of ^*solar masses (M_☉), and is found in the centre of almost all currently known massive galaxies.

The mass of the SMBH in a galaxy is often close to the combined mass of the galaxy's globular clusters.

The mean ratio of black hole mass to bulge mass is now believed to be approximately 1:1000.

The most massive galaxy known is 30 trillion solar masses.

Some supermassive black holes appear to be over 10 billion solar masses.

From Wikipedia:Quasar:

A quasar is an active galactic nucleus of very high luminosity. A quasar consists of a supermassive black hole surrounded by an orbiting accretion disk of gas. The most powerful quasars have luminosities exceeding 2.6×10¹⁴ ℒ_☉ (10⁴¹ W or 17.64631 M_☉/year), thousands of times greater than the luminosity of a large galaxy such as the Milky Way.

Growing at a rate of 17.6/1.4^2 solar mass per year a 60 billion solar mass Black hole would take 6.66 billion years to reach full size. See ^*TON 618

Growing at a rate of 17.6/2.8^2 solar mass per year a 240 billion solar mass Black hole would take 107 billion years to reach full size.

Masses of supermassive black holes in billions of solar masses:

1. 240? (Hypothetical based on Zipf's law)
2. 120? (Hypothetical)
3. 80? (Hypothetical)
4. 66
6. 40
7. 33
8. 30
10. 23
11. 21
12. 20
13. 19.5
14. 18
15. 17
16. 15
17. 14
18. 14
19. 13.5
20. 13
21. 12
22. 12.4
23. 11
24. 11
25. 10
26. 10
27. 9.8
28. 9.7
29. 9.1
30. 7.8
31. 7.2
32. 7.2
33. 6.9

From Wikipedia:Eddington luminosity

The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers.

For pure ionized hydrogen

$\begin{align}L_{\rm Edd}&=\frac{4\pi G M m_{\rm p} c} {\sigma_{\rm T}}\\ &\cong 1.26\times10^{31}\left(\frac{M}{M_\odot}\right){\rm W} = 1.26\times10^{38}\left(\frac{M}{M_\odot}\right){\rm erg/s} = 3.2\times10^4\left(\frac{M}{M_\odot}\right) L_\odot \end{align}$

where $M_{\odot}$ is the mass of the Sun and $L_\odot$ is the luminosity of the Sun.

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Gamma-ray bursts

From Wikipedia:Gamma-ray burst

Gamma-ray bursts (GRBs) are extremely energetic explosions that have been observed in distant galaxies. They are the brightest electromagnetic events known to occur in the universe. Bursts can last from ten milliseconds to several hours. After an initial flash of gamma rays, a longer-lived "afterglow" is usually emitted at longer wavelengths (X-ray, ultraviolet, optical, infrared, microwave and radio).

Assuming the gamma-ray explosion to be spherical, the energy output of ^*GRB 080319B would be within a factor of two of the rest-mass energy of the Sun (the energy which would be released were the Sun to be converted entirely into radiation).

No known process in the universe can produce this much energy in such a short time.

From Wikipedia:GRB 111209A

GRB 111209A is the longest lasting gamma-ray burst (GRB) detected by the Swift Gamma-Ray Burst Mission on December 9, 2011. Its duration is longer than 7 hours.

On average two long gamma ray burst occurs every 3 days and have average redshift of 2. Making the simplifying assumption that all long gamma ray bursts occur at exactly redshift 2 (9.2 * 10⁹ light years) we get one gamma ray burst per (1,635,000 light years)³

There are 12 galaxies per cubic megaparsec. Thats 1 galaxy per (1,425,000 light years)³

One short grb per 3 days at average redshift of 0.5 (4.6 * 10⁹ light years) gives 1 grb per (1,300,000 light years)³

Duration Type Radius Mass? Escape velocity 0.3 sec *Short 0.129 $R^{N}_{\odot}$ 18.535 MJup 0.0007622 c 3 sec long 899,377 km 1.76928 M☉ 0.002410 c 30 sec 8,994,000 km 176.928 M☉ 0.007622 c 5 min 89,940,000 km 17,692.8 M☉ 0.02410 c 50 min 899,400,000 km 1,769,280 M☉ 0.07622 c 8.33 hours Ultra-long 8,994,000,000 km 176,928,000 M☉ 0.2410 c 3.472 days Hypothetical 89,940,000,000 km 17,692,800,000 M☉ 0.7622 c Super massive black hole 89,940,000,000 km 60,000,000,000 M☉ 1.4 c Zipf's law 89,940,000,000 km 240,000,000,000 M☉ 2.807 c

Surface gravity = 29.6 g's, Density = 3.461099×108 g/mm^2

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Ultra-high-energy Cosmic rays

External links: http://hires.physics.utah.edu/reading/uhecr.html, The cosmic ray energy spectrum as measured using the Pierre Auger Observatory

The highest energy gamma rays ever detected from space were around 16 TeV which corresponds to a wavelength of 1/13,000 fm.

From Wikipedia:Cosmic ray

^*Cosmic rays are high-energy radiation, mainly originating outside the Solar System and even from distant galaxies. Upon impact with the Earth's atmosphere, cosmic rays can produce showers of secondary particles that sometimes reach the surface. Composed primarily of high-energy protons and atomic nuclei, they are of uncertain origin. Data from the Fermi Space Telescope (2013) have been interpreted as evidence that a significant fraction of primary cosmic rays originate from the supernova explosions of stars. Active galactic nuclei are also theorized to produce cosmic rays.

From Wikipedia:Ultra-high-energy cosmic ray

In ^*astroparticle physics, an ultra-high-energy cosmic ray (UHECR) is a cosmic ray particle with a kinetic energy greater than than 1×10^{18 *}eV, far beyond both the ^*rest mass and energies typical of other cosmic ray particles.

An extreme-energy cosmic ray (EECR) is an UHECR with energy exceeding 5×10¹⁹ eV (about 8 joule), the so-called ^*Greisen–Zatsepin–Kuzmin limit (GZK limit). This limit should be the maximum energy of cosmic ray protons that have traveled long distances (about 160 million light years), since higher-energy protons would have lost energy over that distance due to scattering from photons in the ^*cosmic microwave background (CMB). However, if an EECR is not a proton, but a nucleus with $A$ nucleons, then the GZK limit applies to its nucleons, each of which carry only a fraction $1/A$ of the total energy.

These particles are extremely rare; between 2004 and 2007, the initial runs of the ^*Pierre Auger Observatory (PAO) detected 27 events with estimated arrival energies above 5.7×10¹⁹ eV, i.e., about one such event every four weeks in the 3000 km² area surveyed by the observatory.

At that rate 5.46 * 10¹⁸ particles will fall onto a star with radius 1 million kilometers every hundred million years.

From Wikipedia:Oh-My-God particle:

The Oh-My-God particle was an ultra-high-energy cosmic ray detected on the evening of 15 October 1991 by the Fly's Eye Cosmic Ray Detector. Its observation was a shock to astrophysicists, who estimated its energy to be approximately 3×10²⁰ eV. It was probably a cluster of 6 ultra-high-energy cosmic ray particles.

mv² = (205,887*128^2*2 neutron mass * (2.807*c)^2) = 5×10¹⁹ eV

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Expansion of the universe

From Wikipedia:Expansion of the universe

The expansion of the universe is the increase of the distance between two distant parts of the universe with time.

The expansion of space is often illustrated with conceptual models. In the "balloon model" a spherical balloon is inflated from an initial size of zero (representing the big bang).

From Wikipedia:Scale factor (cosmology)

Some insight into the expansion can be obtained from a Newtonian expansion model which leads to a simplified version of the Friedman equation. It relates the proper distance (which can change over time, unlike the comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time $t$ to their distance at some reference time $t_0$. The formula for this is:

$d(t) = a(t)d_0,\,$

where $d(t)$ is the proper distance at epoch $t$, $d_0$ is the distance at the reference time $t_0$ and $a(t)$ is the scale factor. Thus, by definition, $a(t_0) = 1$.

The scale factor is dimensionless, with $t$ counted from the birth of the universe and $t_0$ set to the present age of the universe: $13.799\pm0.021\,\mathrm{Gyr}$ giving the current value of $a$ as $a(t_0)$ or $1$.

The evolution of the scale factor is a dynamical question, determined by the equations of general relativity, which are presented in the case of a locally isotropic, locally homogeneous universe by the ^*Friedmann equations.

The Hubble parameter is defined:

$H \equiv {\dot{a}(t) \over a(t)}$

where the dot represents a time derivative. From the previous equation $d(t) = d_0 a(t)$ one can see that $\dot{d}(t) = d_0 \dot{a}(t)$, and also that $d_0 = \frac{d(t)}{a(t)}$, so combining these gives $\dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}$, and substituting the above definition of the Hubble parameter gives $\dot{d}(t) = H d(t)$ which is just Hubble's law.

From Wikipedia:Hubble's law

A variety of possible recessional velocity vs. redshift functions including the simple linear relation v = cz; a variety of possible shapes from theories related to general relativity; and a curve that does not permit speeds faster than light in accordance with special relativity. All curves are linear at low redshifts. See Davis and Lineweaver.

The discovery of the linear relationship between redshift and distance, coupled with a supposed linear relation between recessional velocity and redshift, yields a straightforward mathematical expression for Hubble's Law as follows:

$v = H_0 \, D$

where

• $v$ is the recessional velocity, typically expressed in km/s.
• H₀ is Hubble's constant and corresponds to the value of $H$ (often termed the Hubble parameter which is a value that is ^*time dependent and which can be expressed in terms of the ^*scale factor) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the Universe for a given comoving time.
• $D$ is the proper distance (which can change over time, unlike the comoving distance, which is constant) from the galaxy to the observer, measured in mega parsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just v = dD/dt).

Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted, and is not established except for small redshifts.

For distances D larger than the radius of the Hubble sphere r_HS , objects recede at a rate faster than the speed of light:

$r_{HS} = \frac{c}{H_0} \ .$

Its radius is the Hubble radius and its volume is the Hubble volume.

The Hubble constant $H_0$ has units of inverse time; the Hubble time t_H is simply defined as the inverse of the Hubble constant, i.e. $t_H \equiv {1 \over H_0} = {1 \over 67.8\textrm{(km/s)/Mpc}} = 4.55\cdot 10^{17}\textrm{s}$ = 14.4 billion years. The Hubble time is the age it would have had if the expansion had been linear.

The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called deceleration parameter $q$, which is defined by

$q = -\left(1+\frac{\dot H}{H^2}\right).$

In a universe with a deceleration parameter equal to zero, it follows that H = 1/t, where t is the time since the Big Bang.

The age of the universe is thought to be 13.8 billion years.

1/13.8 billion years = 70.9 (km/s)/Mpc

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Weather

File:Example of a cold front.svg

Main article: Weather

A cold front is the leading edge of a cold dense mass of air, replacing (at ground level) a warmer mass of air. Like a hot air balloon, the warm air rises above the cold air. The rising warm air expands and therefore cools. This causes the moisture within it to condense into droplets and releases the latent heat of condensation which causes the warm air to rise even further. If the warm air is moist enough, rain can occur along the boundary. A narrow line of thunderstorms often forms along the front. Temperature changes across the boundary can exceed 30 °C (54 °F).

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An idealised view of three large circulation cells showing surface winds

The polar front is a cold front that arises as a result of cold polar air meeting warm subtropical air at the boundary between the polar cell and the Ferrel cell in each hemisphere.

Earth's weather is primarily driven by rising air in three Low-pressure areas.

• The northern hemisphere polar front.
  • Extratropical cyclones form along the front and move eastward at 12-15 m/s (43-54 km/h) and last 3-5 days. (3600-6000 km)
• The Intertropical Convergence Zone. Sometimes, a double ITCZ forms, with one north and one south of the Equator.
  • Tropical cyclones (Hurricanes) form here and move westward.
• The southern hemisphere polar front.
  • Extratropical cyclones form along the front and move eastward

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Tropical air is far warmer than air outside the tropics and therefore holds far more moisture and as a result thunderstorms in the tropics are much taller. Nevertheless severe thunderstorms are not common in the tropics because the storms own downdraft shuts off the inflow of warm moist air killing the thunderstorm before it can become severe. Severe thunderstorms tend to occur further north because of the polar jet stream. The jet stream pushes against the top of the thunderstorm displacing the downdraft so that it can no longer shut off the inflow of warm moist air. As a result severe thunderstorms can continue to feed and grow for many hours whereas normal thunderstorms only last 30 minutes.

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Over a 30 minute period a normal thunderstorm releases 10¹⁵ Joules of energy equivalent to 0.24 megatons of TNT. A storm that lasted 24 hours would release 48 times as much energy (48 x 10¹⁵ Joules). A hurricane (a tropical cyclone) releases 52 x 10¹⁸ Joules/day equivalent to 1000 continuous thunderstorms.

From Wikipedia:Scale height

Scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by

$H = \frac{kT}{Mg}$

where:

• k = ^*Boltzmann constant = 1.38 x 10⁻²³ J·K⁻¹
• T = mean atmospheric temperature in kelvins = 250 K for Earth
• M = mean mass of a molecule (units kg)
• g = acceleration due to gravity on planetary surface (m/s²)

Approximate atmospheric scale heights for selected Solar System bodies follow.

• Venus: 15.9 km
• Earth: 8.5 km
• Mars: 11.1 km
• Jupiter: 27 km
• Saturn: 59.5 km

• Titan: 21 km

• Uranus: 27.7 km
• Neptune: 19.1–20.3 km
• Pluto: ~60 km

If all of Earths atmosphere were at 1 bar then the atmosphere would be 8.5 km thick.

See also:

precipitable water

Madden-Julian Oscillation monitoring

National Weather Service

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Life

See also: ^*Emergence and ^*Nanobe

External link: Molecular biology of the cell

Did life begin with ^*nucleic acids or amino acids? Maybe it began with a molecule that was both a nucleic acid and an amino acid.

3-Aminobenzoic-acid:

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Creating the monomers in the ^*Primordial soup is easy but getting the monomers to bond into a polymer is hard. So maybe it wasnt a polymer at all. Maybe it was a one dimensional liquid crystal. See ^*Mesogen.

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Unexplained phenomena

See also: Wikipedia:List of unsolved problems in physics

Books published by ^*William R. Corliss include:

• Mysteries of the Universe (1967)
• Mysteries Beneath the Sea (1970)
• Strange Phenomena: A Sourcebook of Unusual Natural Phenomena (1974)
• Strange Artifacts: A Sourcebook on Ancient Man (1974)
• The Unexplained (1976)
• Strange Life (1976)
• Strange Minds (1976)
• Strange Universe (1977)
• Handbook of Unusual Natural Phenomena (1977)
• Strange Planet (1978)
• Ancient Man: A Handbook of Puzzling Artifacts (1978)
• Mysterious Universe: A Handbook of Astronomical Anomalies (1979)
• Unknown Earth: A Handbook of Geological Enigmas (1980)
• Incredible Life: A Handbook of Biological Mysteries (1981)
• The Unfathomed Mind: A Handbook of Unusual Mental Phenomena (1982)
• Lightning, Auroras, Nocturnal Lights, and Related Luminous Phenomena (1982)
• Tornados, Dark Days, Anomalous Precipitation, and Related Weather Phenomena (1983)
• Earthquakes, Tides, Unidentified Sounds, and Related Phenomena (1983)
• Rare Halos, Mirages, Anomalous Rainbows, and Related Electromagnetic Phenomena (1984)
• The Moon and the Planets (1985)
• The Sun and Solar System Debris (1986)
• Stars, Galaxies, Cosmos (1987)
• Carolina Bays, Mima Mounds, Submarine Canyons (1988)
• Anomalies in Geology: Physical, Chemical, Biological (1989)
• Neglected Geological Anomalies (1990)
• Inner Earth: A Search for Anomalies (1991)
• Biological Anomalies: Humans I (1992)
• Biological Anomalies: Humans II (1993)
• Biological Anomalies: Humans III (1994)
• Science Frontiers: Some Anomalies and Curiosities of Nature (1994)
• Biological Anomalies: Mammals I (1995)
• Biological Anomalies: Mammals II (1996)
• Biological Anomalies: Birds (1998)
• Ancient Infrastructure: Remarkable Roads, Mines, Walls, Mounds, Stone Circles: A Catalog of Archeological Anomalies (1999)
• Ancient Structures: Remarkable Pyramids, Forts, Towers, Stone Chambers, Cities, Complexes: A Catalog of Archeological Anomalies (2001)
• Remarkable Luminous Phenomena in Nature: A Catalog of Geophysical Anomalies (2001)
• Scientific Anomalies and other Provocative Phenomena (2003)
• Archeological Anomalies: Small Artifacts (2003)
• Archeological Anomalies: Graphic Artifacts I (2005)

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Intermediate mathematics/Psychology

 

See also

• Introductory mathematics
• Advanced mathematics
• ^*Impostor syndrome
• List of free mathematics books

External links

• MIT open courseware
• Cheat sheets
• http://mathinsight.org
• https://math.stackexchange.com
• https://www.eng.famu.fsu.edu/~dommelen/quantum/style_a/IV._Supplementary_Informati.html
• http://www.sosmath.com
• https://webhome.phy.duke.edu/~rgb/Class/intro_math_review/intro_math_review/node1.html
• Wikiversity:Mathematics
• w:c:4chan-science:Mathematics