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# Affine Tensor

An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, , where the determinant of is nonzero. This transformation takes the rectangular coordinate system into the coordinate system having oblique axes. In this way an affine tensor can be seen as a special kind of Cartesian tensor.

These tensors have the Jacobians,

 (1) (2) (3) (4)

The transformation laws for affine contravariant (tangent) tensors are

 (5) (6) (7)

and so on, and the transformation laws for affine covariants (covectors) tensors are

 (8) (9) (10)

and so on.

The transformation laws for mixed affine tensors are

 (11) (12)

Cartesian Tensor, Tensor

This entry contributed by George Hrabovsky

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## References

Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 580, 1980.Kay, D. Schaum's Outline of Tensor Calculus. New York: McGraw-Hill, 1988.Lovelock, D. and Rund, H. Tensors, Differential Forms, and Variational Principles. New York: Dover, 1989.

Affine Tensor

## Cite this as:

Hrabovsky, George. "Affine Tensor." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AffineTensor.html