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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:
![{\displaystyle v_{j}=\sum _{k}\chi _{jk}w_{k}\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9a338b22e42b98e1f95d20df0cfdd7a2c6670d)
or, introducing unit vectors êj along the coordinate axes:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/457699fa39e511739cde7912dc0285aaa4f58d7f)
where v is a vector with components {vj} and w is another vector with components {wj} and the quantity
= {χij} 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
= {χij}.
A rotation of the coordinate axes will alter the components of v and w. Suppose the rotation labeled A is described by the equation:
![{\displaystyle \mathbf {{\hat {e}}'_{i}} =\Sigma _{j}A_{ij}\mathbf {{\hat {e}}_{j}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aec91b91b177008e0c195d1982fa4cde9b74205)
![{\displaystyle \mathbf {{\hat {e}}_{i}} =\Sigma _{j}A_{ij}^{-1}\mathbf {{\hat {e}}'_{j}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/454810bd263f31efe44abede3da8cb8c8b6f16c8)
Then:
![{\displaystyle \mathbf {v} =\sum _{i}v_{i}\mathbf {{\hat {e}}_{i}} =\sum _{j}v'_{j}\mathbf {{\hat {e}}'_{j}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e38a3fc288cef38fb819dad4f53a04fec62524)
and
![{\displaystyle \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}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b018747b73877832c102247003b8551f46e3f3c)
![{\displaystyle \mathbf {w} =\sum _{m}w'_{m}\sum _{k}A_{mk}\mathbf {{\hat {e}}_{k}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22631f43b7ceabff9b78fa4a96fc9de9beadd3e9)
![{\displaystyle \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}} \ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/450e6ea330d33b5acf2756412fcb301b9390f511)
so, to be a tensor, the components of
transform as:
![{\displaystyle \chi '_{jm}=\sum _{i}\sum _{k}\chi _{ik}A_{mk}A_{ij}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1e9cf981caed04f31baa080988e37c2613bc77)
More directly:
![{\displaystyle \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} '\ ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47ff196f264e102ccb3b4ae2c5d2b15c4cf9b4f5)
where v' = v because v is a vector representing some physical quantity, say the velocity of a particle. Likewise, w' = w. The new equation represents the same relationship provided:
![{\displaystyle {\overleftrightarrow {\boldsymbol {\mathrm {X} }}}=A{\overleftrightarrow {\boldsymbol {\mathrm {X} }}}A^{-1}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92e6983761c480677dcb588d2e5cbbb98b9410f5)
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:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046175a45f6838df4cb03130414121db39425466)
Young, p 308
Akivis p. 55
p1
p6
tensor algebra p. 1
intro
p. 427; ch 14
Weyl
What is a tensor tensor as operator