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)


for which


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


or, defining


according to


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,


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


where Lambda is a Lorentz tensor.

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


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

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Contravariant Tensor

Cite this as:

Weisstein, Eric W. "Contravariant Tensor." From MathWorld--A Wolfram Web Resource.

Subject classifications