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# Contravariant Tensor

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)

 (1)

for which

 (2)

Now let , then any set of quantities which transform according to

 (3)

or, defining

 (4)

according to

 (5)

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

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

 (6)

for , 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

 (7)

where is a Lorentz tensor.

To turn a covariant tensor into a contravariant tensor (index raising), use the metric tensor to write

 (8)

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

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." §3.8 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