Contravariant Tensor (original) (raw)

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A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 (a vector)

dr=dx_1x_1^^+dx_2x_2^^+dx_3x_3^^, (1)

for which

dx_i^'=(partialx_i^')/(partialx_j)dx_j. (2)

Now let A_i=dx_i, then any set of quantities A_j which transform according to

A_i^'=(partialx_i^')/(partialx_j)A_j, (3)

or, defining

a_(ij)=(partialx_i^')/(partialx_j), (4)

according to

A_i^'=a_(ij)A_j (5)

is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., a^mu.

Covariant tensors are a type of tensor with differing transformation properties, denoted a_nu. However, in three-dimensional Euclidean space,

(partialx_j)/(partialx_i^')=(partialx_i^')/(partialx_j)=a_(ij) (6)

for i,j=1, 2, 3, meaning that contravariant and covariant tensors are equivalent. Such tensors are known as Cartesian tensor. The two types of tensors do differ in higher dimensions, however.

Contravariant four-vectors satisfy

a^mu=Lambda_nu^mua^nu, (7)

where Lambda is a Lorentz tensor.

To turn a covariant tensor a_nu into a contravariant tensor a^mu (index raising), use themetric tensor g^(munu) to write

g^(munu)a_nu=a^mu. (8)

Covariant and contravariant indices can be used simultaneously in a mixed tensor.

See also

Cartesian Tensor, Contravariant Vector, Covariant Tensor, Four-Vector,Index Raising, Lorentz Tensor, Metric Tensor, Mixed Tensor, Tensor

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References

Arfken, G. "Noncartesian Tensors, Covariant Differentiation." ยง3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158-164, 1985.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-46, 1953.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

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Contravariant Tensor

Cite this as:

Weisstein, Eric W. "Contravariant Tensor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ContravariantTensor.html

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