A covariant tensor, denoted with a lowered index (e.g., ) is a tensor having specific
transformation properties. In general, these transformation properties differ from
those of 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.

Contravariant tensors are a type of tensor with differing transformation properties, denoted . To turn a contravariant
tensor
into a covariant tensor (index lowering ), use
the metric tensor to write

(7)

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

In Euclidean spaces , and more generally in flat Riemannian
manifolds , a coordinate system can be found where the metric tensor is constant,
equal to Kronecker delta

(8)

Therefore, raising and lowering indices is trivial, hence covariant and contravariant tensors have the same coordinates, and can be identified. Such tensors are known
as Cartesian tensors .

A similar result holds for flat pseudo-Riemannian manifolds , such as Minkowski space , for which
covariant and contravariant tensors can be identified. However, raising and lowering
indices changes the sign of the temporal components of tensors,
because of the negative eigenvalue
in the Minkowski metric .

See also Cartesian Tensor ,

Contravariant Tensor ,

Four-Vector ,

Index
Lowering ,

Lorentz Tensor ,

Metric
Tensor ,

Mixed Tensor ,

Tensor
Portions of this entry contributed by Manuel
F. González Lázaro

Explore with Wolfram|Alpha
References Arfken, G. "Noncartesian Tensors, Covariant Differentiation." §3.8 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158-164,
1985. Lichnerowicz, A. Elements
of Tensor Calculus. New York: Wiley, 1962. 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 Covariant Tensor
Cite this as:
Lázaro, Manuel F. González and Weisstein, Eric W. "Covariant Tensor."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/CovariantTensor.html

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