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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 êj along the coordinate axes:

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:



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