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!