In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression.
The pairing is the linear map from the tensor product of these two spaces to the field k: corresponding to the bilinear form where f is in V∗ and v is in V. The map C defines the contraction operation on a tensor of type (1, 1), which is an element of
and the space of linear maps from V to V,[1] one obtains a basis-free definition of the trace.
[2] By analogy with the (1, 1) case, the general contraction operation is sometimes called the trace.
In tensor index notation, the basic contraction of a vector and a dual vector is denoted by which is shorthand for the explicit coordinate summation[4] (where vi are the components of v in a particular basis and fi are the components of f in the corresponding dual basis).
As in the previous example, contraction on a pair of indices that are either both contravariant or both covariant is not possible in general.
One uses the metric to raise or lower one of the indices, as needed, and then one uses the usual operation of contraction.
One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold[5] or the context of sheaves of modules over the structure sheaf;[6] see the discussion at the end of this article.
In the case of Cartesian coordinates in Euclidean space, one can write Then changing index β to α causes the pair of indices to become bound to each other, so that the derivative contracts with itself to obtain the following sum: which is the divergence div V. Then is a continuity equation for V. In general, one can define various divergence operations on higher-rank tensor fields, as follows.
If T is a tensor field with at least one contravariant index, taking the covariant differential and contracting the chosen contravariant index with the new covariant index corresponding to the differential results in a new tensor of rank one lower than that of T.[5] One can generalize the core contraction operation (vector with dual vector) in a slightly different way, by considering a pair of tensors T and U.
Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field.
More generally, let OX be a sheaf of commutative rings over a topological space X, e.g. OX could be the structure sheaf of a complex manifold, analytic space, or scheme.
Let M be a locally free sheaf of modules over OX of finite rank.
Then the dual of M is still well-behaved[6] and contraction operations make sense in this context.