Differential Ideal

A differential ideal I on a manifold M is an ideal in the exterior algebra of differential k-forms on M which is also closed under the exterior derivative d. That is, for any differential k-form alpha and any form beta in I, then

1. alpha ^ beta in I, and

2. dbeta in I.

For example, I=<xdy,dx ^ dy> is a differential ideal on M=R^2.

A smooth map f:X->M is called an integral of I if the pullback map of all forms in I vanish on X, i.e., f^*(I)=0.

See also

Differential k-Form, Form Envelope, Integrable Differential Ideal, Manifold

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Differential Ideal." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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