A covariant tensor of tensor rank 
on a vector space 
is a scalar-valued multilinear
map of 
vectors. Its components have 
lower indices, and it is a tensor
of type 
.
In finite dimensions it is an element of the tensor
product 
,
where 
is the dual vector space. On a manifold,
is the tangent
space at the point under consideration. Rank 0 gives a scalar.
The rank-1 case is a covector, which takes a vector as input and returns a scalar linearly. For example, the
differential of a smooth scalar-valued
function 
is the one-form
 |
(1)
|
Applied to a tangent vector 
, it gives the directional
derivative
 |
(2)
|
where repeated upper and lower indices are summed using Einstein summation.
Under a smooth invertible change of coordinates from 
to 
, the chain rule gives
 |
(3)
|
where 
.
Any covector 
transforms by the same rule,
 |
(4)
|
Writing 
for the inverse Jacobian matrix, this becomes
 |
(5)
|
The transformation ensures that the value of a covector on a vector does not depend on coordinates,
 |
(6)
|
For example, under 
in one dimension, 
and 
,
so their product is unchanged. The vector
is a rank-1 contravariant tensor.
For a covariant tensor of tensor rank 
, each lower index contributes an inverse Jacobian
matrix factor,
 |
(7)
|
A metric tensor 
is an example of a rank-2 covariant tensor. It acts on
two vectors to give the scalar 
.
A nondegenerate metric tensor identifies vectors with covectors by index lowering,
 |
(8)
|
Conversely, the gradient vector associated with 
is obtained by index raising,
 |
(9)
|
where 
is the matrix inverse of 
. Thus 
requires no metric tensor,
but its identification with a gradient vector
does.
In an orthonormal basis of Euclidean space of any dimension, the metric tensor is
the Kronecker delta, 
. Corresponding upper- and lower-index components
then agree numerically, as in the theory of Cartesian
tensors. On a flat manifold with a Riemannian
metric, such coordinates can be chosen locally.
The equality of components need not hold in general coordinates.
In Minkowski space, index lowering instead introduces the signs of the Minkowski
metric. For the convention 
in an inertial coordinate system,
 |
(10)
|
while the spatial components are unchanged. With the opposite sign convention, the spatial components change sign instead. Both kinds of indices occur in a mixed tensor.
