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.
The disappearance of  is equivalent to the disappearance of , and this also demonstrates that there are six independent components for the significant case of three dimensions. Although it still involves two derivatives rather than 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.