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Affine Tensor

An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, T:x^_^i=a^i_jx^j, where the determinant of a^i_j is nonzero. This transformation takes the rectangular coordinate system (x^i) into the coordinate system (x^_^i) having oblique axes. In this way an affine tensor can be seen as a special kind of Cartesian tensor.

These tensors have the Jacobians,

J=|(partialx^_^1)/(partialx^1) ... (partialx^_^1)/(partialx^n); | ... |; (partialx^_^n)/(partialx^1) ... (partialx^_^n)/(partialx^n)|
(1)
=(a^i_j)
(2)
J^(-1)=|(partialx^1)/(partialx^_^1) ... (partialx^1)/(partialx^_^n); | ... |; (partialx^n)/(partialx^_^1) ... (partialx^n)/(partialx^_^n)|
(3)
=(a_i^j).
(4)

The transformation laws for affine contravariant (tangent) tensors are

T^_^i=a^i_qT^q
(5)
T^_^(ij)=a^i_qa^j_rT^(qr)
(6)
T^_^(ijk)=a^i_qa^j_ra^k_sT^(qrs),
(7)

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

T^__i=a_i^qT_q
(8)
T^__(ij)=a_i^qa_j^rT_(qr)
(9)
T^__(ijk)=a_i^qa_j^ra_k^sT_(qrs),
(10)

and so on.

The transformation laws for mixed affine tensors are

T^_^i_j=a^i_qa_j^rT^q_r
(11)
T^_^i_j^k=a^i_qa_j^ra^k_sT^q_r^s.
(12)

See also

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.

Referenced on Wolfram|Alpha

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

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