Integrable Differential Ideal
A differential ideal is an ideal I in the ring of smooth forms on a manifold M. That is, it is closed under addition, scalar multiplication, and wedge product with an arbitrary form. The ideal I is called integrable if, whenever alpha in I, then also dalpha in I, where d is the exterior derivative.
For example, in R^3, the ideal
| I={a_1ydx+a_2dx ^ dy+a_3ydx ^ dz+a_4dx ^ dy ^ dz}, |
(1)
|
where the a_i are arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form a_2ydx ^ dy, then the ideal would not be integrable because it would not contain d(ydx)=-dx ^ dy.
Given an integral differential ideal I on M, a smooth map f:X->M is called integrable if the pullback of every form alpha vanishes on X, i.e., f^*alpha=0. In coordinates, an integral manifold solves a system of partial differential equations. For example, using I above, a map f=(f_1,f_2,f_3) from an open set in R^2 is integral if
| [画像: (partialf_1)/(partialx)(partialf_2)/(partialy)-(partialf_1)/(partialy)(partialf_2)/(partialx)=0 ] |
(4)
|
Conversely, any system of partial differential equations can be expressed as an integrable differential ideal on a jet bundle. For instance, partialf/partialx=g on R corresponds to I=<df-gdx> on R^2={(x,f)}.
See also
Differential k-Form, Integrable, Partial Differential Equation, Wedge ProductThis entry contributed by Todd Rowland
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Rowland, Todd. "Integrable Differential Ideal." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IntegrableDifferentialIdeal.html