Editing Tensor contraction (section)

=== 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"/>