Exterior Differential Systems and Partial Differential Equations
An exterior differential system on a manifold M is a differential ideal
- .
One can express any partial differential equation system as an exterior differential system with independence condition. Say that we have kth order partial differential equation systems for maps, given by
- .
The solution of this partial differential equation system is the submanifold of the jet space consisting of integral manifolds of the pullback of the contact system to .
This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply Cartan's method on partial differential equation systems by writing down the exterior differential system associated with it.
Read more about this topic: Differential Ideal
Famous quotes containing the words exterior, differential, systems and/or partial:
“The competent leader of men cares little for the niceties of other peoples’ characters: he cares much—everything—for the exterior uses to which they may be put.... These are men to be moved. How should he move them? He supplies the power; others simply the materials on which that power operates.”
—Woodrow Wilson (1856–1924)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“People stress the violence. That’s the smallest part of it. Football is brutal only from a distance. In the middle of it there’s a calm, a tranquility. The players accept pain. There’s a sense of order even at the end of a running play with bodies stewn everywhere. When the systems interlock, there’s a satisfaction to the game that can’t be duplicated. There’s a harmony.”
—Don Delillo (b. 1926)
“And meanwhile we have gone on living,
Living and partly living,
Picking together the pieces,
Gathering faggots at nightfall,
Building a partial shelter,
For sleeping and eating and drinking and laughter.”
—T.S. (Thomas Stearns)