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)
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for which
(2)
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Now let , then any set of quantities which transform according to
(3)
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or, defining
(4)
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according to
(5)
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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)
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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)
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where is a Lorentz tensor.
To turn a covariant tensor into a contravariant tensor (index raising), use the metric tensor to write
(8)
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Covariant and contravariant indices can be used simultaneously in a mixed tensor.