< User:John R. BrewsRevision as of 23:46, 16 December 2010 by imported>John R. Brews
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 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:
where v’ = v because v is a vector representing some physical quantity, say the velocity of a particle.
Then:
which 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