## Covariant Tensor

A covariant tensor is a Tensor having specific transformation properties (c.f., a Contravariant Tensor). To examine the transformation properties of a covariant tensor, first consider the Gradient

 (1)

for which
 (2)

where . Now let
 (3)

then any set of quantities which transform according to
 (4)

or, defining
 (5)

according to
 (6)

is a covariant tensor. Covariant tensors are indicated with lowered indices, i.e., .

Contravariant Tensors are a type of Tensor with differing transformation properties, denoted . However, in 3-D Cartesian Coordinates,

 (7)

for , 2, 3, meaning that contravariant and covariant tensors are equivalent. The two types of tensors do differ in higher dimensions, however. Covariant Four-Vectors satisfy
 (8)

where is a Lorentz Tensor.

To turn a Contravariant Tensor into a covariant tensor, use the Metric Tensor to write

 (9)

Covariant and contravariant indices can be used simultaneously in a Mixed Tensor.

See also Contravariant Tensor, Four-Vector, Lorentz Tensor, Metric Tensor, Mixed Tensor, Tensor

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.