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 .
Read more about this topic: Saint-Venant's Compatibility Condition
Famous quotes containing the words higher and/or rank:
“The momentary charge at Balaklava, in obedience to a blundering command, proving what a perfect machine the soldier is, has, properly enough, been celebrated by a poet laureate; but the steady, and for the most part successful, charge of this man, for some years, against the legions of Slavery, in obedience to an infinitely higher command, is as much more memorable than that as an intelligent and conscientious man is superior to a machine. Do you think that that will go unsung?”
—Henry David Thoreau (1817–1862)
“I have not loved the world, nor the world me;
I have not flatter’d its rank breath, nor bow’d
To its idolatries a patient knee.”
—George Gordon Noel Byron (1788–1824)