Editing Tensor contraction (section)

== Application to tensor fields ==

Contraction is often applied to [[tensor fields]] over spaces (e.g. [[Euclidean space]], [[manifold]]s, or [[scheme (mathematics)|schemes]]{{fact|date=April 2015}}). Since contraction is a purely algebraic operation, it can be applied pointwise to a tensor field, e.g. if ''T'' is a (1,1) tensor field on Euclidean space, then in any coordinates, its contraction (a scalar field) ''U'' at a point ''x'' is given by

: <math>U(x) = \sum_{i} T^{i}_{i}(x)</math>

Since the role of ''x'' is not complicated here, it is often suppressed, and the notation for tensor fields becomes identical to that for purely algebraic tensors.

Over a [[Riemannian manifold]], a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory. For example, the [[Ricci tensor]] is a non-metric contraction of the [[Riemann curvature tensor]], and the [[scalar curvature]] is the unique metric contraction of the Ricci tensor.

One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold<ref name="o'neill"/> or the context of sheaves of modules over the structure sheaf;<ref name="hartshorne">{{cite book |first=Robin |last=Hartshorne |author-link=Robin Hartshorne |title=Algebraic Geometry |location=New York |publisher=Springer |year=1977 |isbn=0-387-90244-9 }}</ref> see the discussion at the end of this article.

=== Tensor divergence ===

As an application of the contraction of a tensor field, let ''V'' be a [[vector field]] on a [[Riemannian manifold]] (for example, [[Euclidean space]]). Let <math> V^\alpha {}_{\beta}</math> be the [[covariant derivative]] of ''V'' (in some choice of coordinates). In the case of [[Cartesian coordinates]] in Euclidean space, one can write

: <math> V^\alpha {}_{\beta} = {\partial V^\alpha \over \partial x^\beta}. </math>

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:

: <math> V^\alpha {}_{\alpha} = V^0 {}_{0} + \cdots + V^n {}_{n}, </math>

which is the [[divergence]] div ''V''. Then

: <math> \operatorname{div} V = V^\alpha {}_{\alpha} = 0 </math>

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''.<ref name="o'neill"/>