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

or, introducing unit vectors **ê _{j}** along the coordinate axes:

where **v** is a vector with components {v_{j}} and **w** is another vector with components {w_{j}} and the quantity = {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 = {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:

Then:

and

so, to be a tensor, the components of transform as:

More directly:

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:

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:

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