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)
 | (1) |
for which
 | (2) |
Now let 
, then any set of quantities 
which transform according to
 | (3) |
or, defining
 | (4) |
according to
 | (5) |
is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., 
.
Covariant tensors are a type of tensor with differing transformation properties, denoted 
. However, in three-dimensional Euclidean space,
 | (6) |
for 
, 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
 | (7) |
where 
is a Lorentz tensor.
To turn a covariant tensor 
into a contravariant tensor 
(index raising), use the metric tensor 
to write
 | (8) |
Covariant and contravariant indices can be used simultaneously in a mixed tensor.
