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 α and any form β element ℑ, then 1.α⋀β element ℑ, and 2.d β element ℑ. For example, ℑ = 〈x d y, d x⋀d y〉 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.
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