A contravariant tensor of tensor rank 
is a tensor of type 
, whose components have 
upper indices and transform like a product
of 
vector components. The rank-1 case is an ordinary vector,
such as a tangent vector to a curve.
On a finite-dimensional vector space 
, a contravariant tensor is an element of the tensor
product 
. On a manifold,
is the tangent space at the point under consideration.
Rank 0 gives a scalar.
For a tangent vector expressed in a coordinate vector basis,
 |
(1)
|
repeated upper and lower indices are summed using Einstein summation. Under a smooth invertible change of coordinates
from 
to 
, the chain rule
gives
 |  |  |
(2)
|
 |  |  |
(3)
|
Thus the components change inversely to the vector basis, leaving the vector itself unchanged. For example, under
the one-dimensional rescaling 
, the component becomes 
, while the vector
basis element 
is half of 
.
Writing 
for the Jacobian matrix, the rank-1 transformation is
 |
(4)
|
For a contravariant tensor of tensor rank 
, there is one such factor for each index,
 |
(5)
|
In particular, contravariant four-vectors transform under a Lorentz transformation as
 |
(6)
|
A covector instead has lower-index components and is a rank-1 covariant tensor. A nondegenerate metric tensor identifies covectors with vectors by index raising,
 |
(7)
|
where 
is the matrix inverse of 
. This identification uses the chosen metric
tensor and is not part of the definition of a contravariant tensor.
In an orthonormal basis of Euclidean space, 
, so corresponding upper- and lower-index
components have equal numerical values. Transformations between such orthonormal
bases are orthogonal transformations,
for which the matrix inverse is the transpose.
This is the setting of Cartesian tensors in any
dimension. In general coordinates, the distinction
remains important even in two or three dimensions. Both kinds of indices occur in
a mixed tensor.
