User:John R. Brews/Sandbox

Tensor

In physics a tensor in its simplest form is a proportionality factor between two vector quantities that may differ in both magnitude and direction. Mathematically this relationship is:

$v_{j}=\sum _{k}\chi _{jk}w_{k}\ ,$

where v is a vector with components {v_j} and w is another vector with components {w_j} and the quantity Χ = {χ_ij} is a tensor. This example is a second rank tensor. The idea is extended to third rank tensors that relate a vector to a second rank tensor, as when electric polarization is related to stress in a crystal, and to fourth rank tensors that relate two second rank tensors, and so on.

Tensors can relate vectors of different dimensionality, as in the relation:

${\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}&T_{13}&T_{14}&T_{15}\\T_{21}&T_{22}&T_{23}&T_{24}&T_{25}\\T_{31}&T_{32}&T_{33}&T_{34}&T_{35}\\T_{41}&T_{42}&T_{43}&T_{44}&T_{45}\\T_{51}&T_{52}&T_{53}&T_{54}&T_{55}\end{pmatrix}}\ {\begin{pmatrix}q_{1}\\q_{2}\\q_{3}\\q_{4}\\q_{5}\end{pmatrix}}$