On Saint Venant's compatibility conditions and Poincare's lemma

Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in L-2 by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in H-1. Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincare's lemma.