2
$\begingroup$

Differential equations can be written as an ideal of n-forms. Solutions are manifolds where the forms pull back to zero. Is it possible, or useful, to represent the solution by multivectors? For example, can or should one describe solutions to the Laplace equation using alternating bivectors? In a second-order partial differential equation, would a bivector to some extent play the role that a Cauchy characteristic has in a first-order partial differential equation?

If so, is there a good reference for this type of construction? Multivectors seem to be neglected compared with alternating k-forms, to which they are "dual", so to speak. Maybe they are much less useful, but maybe not. Thanks for any pointers!

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ Since you ultimately want a submanifold on which the exterior forms vanish, it is more useful to focus on decomposable multivector solutions. In any case, you should consult a textbook on exterior differential systems, for example "Cartan for Beginners" by Ivey and Landsberg. $\endgroup$
    – Deane Yang
    Commented Apr 9, 2011 at 14:01
  • $\begingroup$ You mean, I should not bother with k-multivectors and simply think in terms of k vector fields? Yes, I read "Cartan for Beginners" on a regular basis, thanks! $\endgroup$
    – Pait
    Commented Apr 9, 2011 at 14:09
  • 1
    $\begingroup$ I don't know if the authors ever mention multivectors explicitly, but if you study what they do carefully you'll see that they are at least implicitly there. For example, they probably define an "integral element", which is a $k$-dimensional subspace on which an exterior differential system vanishes. This is equivalent to a $1$-dimensional subspace of decomposable $k$-vectors. And it is more or less the same as thinking of $k$ vector fields. $\endgroup$
    – Deane Yang
    Commented Apr 9, 2011 at 14:25

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .