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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, and which is a relation that remains the same under changes in the coordinate system. Mathematically this relationship in some particular coordinate system is:

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

or, introducing unit vectors ê_j along the coordinate axes:

${\begin{aligned}\mathbf {v} &=v_{1}\mathbf {{\hat {e}}_{1}} +v_{2}\mathbf {{\hat {e}}_{2}} +...\\&=\left(\chi _{11}w_{1}+\chi _{12}w_{2}...\right)\mathbf {{\hat {e}}_{1}} +\left(\chi _{21}w_{1}+\chi _{22}w_{2}...\right)\mathbf {{\hat {e}}_{2}} ...\end{aligned}}$

where v is a vector with components {v_j} and w is another vector with components {w_j} and the quantity ${\overleftrightarrow {\boldsymbol {\mathrm {X} }}}$ = {χ_ij} is a tensor. Because v and w are vectors, they are physical quantities independent of the coordinate axes chosen to find their components. Likewise, if this relation between vectors constitutes a physical relationship, then the above connection between v and w expresses some physical fact that transcends the particular coordinate system where ${\overleftrightarrow {\boldsymbol {\mathrm {X} }}}$ = {χ_ij}.

A rotation of the coordinate axes will alter the components of v and w. Suppose the rotation labeled A is described by the equation:

$\mathbf {{\hat {e}}'_{i}} =\Sigma _{j}A_{ij}\mathbf {{\hat {e}}_{j}} \ ,$

$\mathbf {{\hat {e}}_{i}} =\Sigma _{j}A_{ij}^{-1}\mathbf {{\hat {e}}'_{j}} \ ,$

Then:

$\mathbf {v} =\sum _{i}v_{i}\mathbf {{\hat {e}}_{i}} =\sum _{j}v'_{j}\mathbf {{\hat {e}}'_{j}} \ ,$

and

$\mathbf {v} =\sum _{i}v_{i}\sum _{j}A_{ij}^{-1}\mathbf {{\hat {e}}'_{j}} =\sum _{i}\sum _{k}\chi _{ik}w_{k}\sum _{j}A_{ij}^{-1}\mathbf {{\hat {e}}'_{j}} \ ,$

$\mathbf {w} =\sum _{m}w'_{m}\sum _{k}A_{mk}\mathbf {{\hat {e}}_{k}} \ ,$

$\mathbf {v} =\sum _{i}\sum _{k}\chi _{ik}\sum _{m}w'_{m}A_{mk}\sum _{j}A_{ij}^{-1}\mathbf {{\hat {e}}'_{j}} =\sum _{m}\chi '_{jm}w'_{m}\mathbf {{\hat {e}}'_{j}} \ ,$

so, to be a tensor, the components of ${\overleftrightarrow {\boldsymbol {\mathrm {X} }}}$ transform as:

$\chi '_{jm}=\sum _{i}\sum _{k}\chi _{ik}A_{mk}A_{ij}^{-1}$

More directly:

$\mathbf {v} '=A\mathbf {v} =A{\overleftrightarrow {\boldsymbol {\mathrm {X} }}}\mathbf {w} =A{\overleftrightarrow {\boldsymbol {\mathrm {X} }}}A^{-1}A\mathbf {w} =A{\overleftrightarrow {\boldsymbol {\mathrm {X} }}}A^{-1}\mathbf {w} '\ ,$

where v’ = v because v is a vector representing some physical quantity, say the velocity of a particle. The new equation represents the same relationship provided:

${\overleftrightarrow {\boldsymbol {\mathrm {X} }}}=A{\overleftrightarrow {\boldsymbol {\mathrm {X} }}}A^{-1}\ .$

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:

${\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}&T_{13}&T_{14}&T_{15}\\T_{21}&T_{22}&T_{23}&T_{24}&T_{25}\\T_{31}&T_{32}&T_{33}&T_{34}&T_{35}\end{pmatrix}}\ {\begin{pmatrix}q_{1}\\q_{2}\\q_{3}\\q_{4}\\q_{5}\end{pmatrix}}$

Young, p 308 Akivis p. 55 p1 p6 tensor algebra p. 1 intro p. 427; ch 14 Weyl What is a tensor