Saint-Venant's Compatibility Condition - Generalization To Higher Rank Tensors

Generalization To Higher Rank Tensors

Saint-Vanant's compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincare's lemma for skew-symmetric tensor fields (differential forms). 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 .