Tensor

Tensors are generalizations of vectors. These manifestate as

1. product of vector spaces
2. product of linear maps
3. multi-indexed arrangements of numbers or scalar functions

In the first case we take two vector spaces $V$, $W$ and construct a new one $V\otimes W$ by mean of their basis: if V has as basic vectors $b_1,b_2,...$ and W has $c_1,c_2,...$ then

$V\otimes W$

is the vector space generated by the symbols $b_1\otimes c_1,b_1\otimes c_2,...$ which are all the linear combinations of pairs $b_i\otimes c_j$.

In another hand, if we want to construct a bilinear map $B:V\times W\to F$ then we pick up a pair of covectors $f:V\to F$ and $g:W\to F$ then we manufacture

$B=f\otimes g:V\times W\to F$ defined via $B(v,w)=f\otimes g(v,w)=f(v)g(w)$

It is also know that for finite dimensional vector spaces the tensor of rank one are the elements of $V$ and its dual space $V^*$

In the multi-indexed versions tensors are expressions like $g_{ij}$, $\delta_i^j$, $\varepsilon_{ijkl}$, ${{\Gamma^i}_j}_k, {R^i}_{jkl},...$ which actualy are the components of tensors and are in nature scalars or scalar functions. In the multi-indexed version tensor are arrays of quantities that obeys certain rules of transformations when we change systems of coordinates.

Comments on how to learn this

• First of all you have to be sure that you understand linear algebra
  • perfectly distingishing among vector, vectorspace, vectorfunction and vectorproducts
  • relations bewteen matrix, basis, linear transformation and change of coordinates (or coordinates change)
• A key concept is that the set of all linear transformations among vector spaces it is also a vector space
• This vector space is representable thru matrices
• Some good experience on calculus on several variables are ideal
• Tensors are objects in the branch of math called multilinear algebra which can be developed in several levels
• It is better to understand multilinear algebra over the real numbers at the beginnings
• At the same time you are studying multilinear algebra, study differential geometry at least of curves and of surfaces
• There are discrepancies among the many notations of many different authors, so compare, contrast
• You shouldn't be scare of the intense use of symbols and relations inter symbols

External links

tensors in wikipedia