Differential element

The differential element or just differential of a quantity refers to an infinitesimal change in said quantity, and is defined as the limit of a change in quantity as the change approaches zero.

$dx=\lim_{\Delta x\to0}\Delta x$

Differentials are useful in the definitions of both derivatives and integrals; for example, the derivative of $y$ with respect to $x$ is defined as

$\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}$

When transforming coordinates, the value of a differential element is computed using the determinant of the Jacobian matrix.

$\prod_{i=1}^n dx_i=\frac{\part(x_1,\ldots,x_n)}{\part(u_1,\ldots,u_n)}\prod_{i=1}^n du_i= \begin{vmatrix} \dfrac{\part x_1}{\part u_1}&\cdots&\dfrac{\part x_n}{\part u_1}\\ \vdots&\ddots&\vdots\\ \dfrac{\part x_1}{\part u_n}&\cdots&\dfrac{\part x_n}{\part u_n} \end{vmatrix}\prod_{i=1}^n du_i$

Formulae for differential elements

Line elements

• $ds=\|\vec{r'}\|dt=\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2 + \left(\tfrac{dz}{dt}\right)^2}dt$ (scalar line element)
• $d\vec r=\vec{r'}dt=\frac{\vec{r'}}{\|\vec{r'}\|}ds$ (tangential vector element)
• $d\vec n=\frac{\vec T'}{\left\|\vec T'\right\|}ds$ (normal vector element)

Area elements

• $dA=dx\cdot dy$ (two-dimensional Cartesian coordinates)
• $dA=r\cdot dr\cdot d\theta$ (polar coordinates)

Surface elements

• $dS=\big\|\vec{r}_u\times\vec{r}_v\big\|dA$ (scalar surface element)
• $(\vec{r}_u\times\vec{r}_v)dA=\frac{\vec{r}_u\times\vec{r}_v}{\big\|\vec{r}_u\times\vec{r}_v\big\|}dS$ (normal vector surface element)

Volume elements

• $dV=dx\cdot dy\cdot dz$ (three-dimensional Cartesian coordinates)
• $dV=r\cdot dr\cdot d\theta\cdot dz$ (cylindrical coordinates)
• $dV=\rho^2\sin(\phi)d\rho\cdot d\theta\cdot d\phi$ (spherical coordinates)