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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. Mathematically this relationship is:

where v is a vector with components {vj} and w is another vector with components {wj} and the quantity Χ = {χij} is a tensor. 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