For a symmetric rank 2 tensor field in n-dimensional Euclidean space the integrability condition takes the form of the vanishing of the Saint-Venant's tensor defined by The result that, on a simply connected domain W=0 implies that strain is the symmetric derivative of some vector field, was first described by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886. For non-simply connected domains there are finite dimensional spaces of symmetric tensors with vanishing Saint-Venant's tensor that are not the symmetric derivative of a vector field. The situation is analogous to de Rham cohomology The Saint-Venant tensor is closely related to the Riemann curvature tensor. Indeed the first variation about the Euclidean metric with a perturbation in the metric is precisely. Consequently the number of independent components of is the same as specifically for dimension n. Specifically for, has only one independent component where as for there are six. In its simplest form of course the components of must be assumed twice continuously differentiable, but more recent work proves the result in a much more general case. The relation between Saint-Venant's compatibility condition and Poincaré's lemma can be understood more clearly using a reduced form of the Kröner tensor where is the permutation symbol. For, is a symmetric rank 2 tensor field. The vanishing of is equivalent to the vanishing of and this also shows that there are six independent components for the important case of three dimensions. While this still involves two derivatives rather than the one in the Poincaré lemma, it is possible to reduce to a problem involving first derivatives by introducing more variables and it has been shown that the resulting 'elasticity complex' is equivalent to the de Rham complex. In differential geometry the symmetrized derivative of a vector field appears also as the Lie derivative of the metric tensorgwith respect to the vector field. where indices following a semicolon indicate covariant differentiation. The vanishing of is thus the integrability condition for local existence of in the Euclidean case. As noted above this coincides with the vanishing of the linearization of the Riemann curvature tensor about the Euclidean metric.
Generalization to higher rank tensors
Saint-Venant's compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincaré's lemma for skew-symmetric tensor fields. The result can be generalized to higher rank symmetric tensor fields. Let F be a symmetric rank-k tensor field on an open set in n-dimensional Euclidean space, then the symmetric derivative is the rank k+1 tensor field defined by where we use the classical notation that indices following a comma indicate differentiation and groups of indices enclosed in brackets indicate symmetrization over those indices. The Saint-Venant tensor of a symmetric rank-k tensor field is defined by with On a simply connected domain in Euclidean space implies that for some rank k-1 symmetric tensor field.
