User:Terry Richard Linter Cole/Sandbox

Copy of 'Tensor' page from John's sandbox, for playing
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 êj along the coordinate axes:



\begin{align} \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{align}

$$

where v is a vector with components {vj} and w is another vector with components {wj} and the quantity $$\overleftrightarrow\boldsymbol{ \Chi}$$ = {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 $$\overleftrightarrow\boldsymbol{ \Chi}$$ = {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:
 * $$ \mathbf {\hat {e}'_i} = \Sigma_j A_{ij} \mathbf {\hat {e}_j} \, $$


 * $$\mathbf {\hat {e}_i} = \Sigma_j A^{-1}_{ij} \mathbf {\hat {e}'_j} \, $$

Then:


 * $$\mathbf v = \sum_i v_i \mathbf {\hat {e}_i} = \sum_j v'_j \mathbf {\hat {e}'_j} \, $$

and
 * $$\mathbf v = \sum_i v_i \sum_j A^{-1}_{ij} \mathbf {\hat {e}'_j} = \sum_i \sum_k \chi_{ik} w_k \sum_j A^{-1}_{ij}  \mathbf {\hat {e}'_j} \, $$


 * $$\mathbf w = \sum_m w'_m \sum_k A_{mk} \mathbf {\hat {e}_k} \ ,$$


 * $$\mathbf v = \sum_i \sum_k \chi_{ik} \sum_m w'_m A_{mk}  \sum_j A^{-1}_{ij}  \mathbf {\hat {e}'_j} = \sum_m \chi'_{jm} w'_m \mathbf {\hat {e}'_j} \, $$

so, to be a tensor, the components of $$\overleftrightarrow\boldsymbol{ \Chi}$$ transform as:
 * $$\chi'_{jm}= \sum_i \sum_k \chi_{ik}  A_{mk} A^{-1}_{ij} $$

More directly:
 * $$ \mathbf v' = A \mathbf v =  A \overleftrightarrow\boldsymbol{ \Chi} \mathbf w = A  \overleftrightarrow{\boldsymbol {\Chi}}  A^{-1}  A \mathbf w =  A  \overleftrightarrow{\boldsymbol {\Chi}}  A^{-1} \mathbf w' \ ,$$

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:


 * $$\overleftrightarrow\boldsymbol{ \Chi} = A  \overleftrightarrow {\boldsymbol {\Chi}}A^{-1} \ .$$

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 tensor as operator