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 α element I, then also d α element I, where d is the exterior derivative. For example, in R^3, the ideal I = {a_1 y d x + a_2 d x⋀d y + a_3 y d x⋀d z + a_4 d x⋀d y⋀d z}, where the a_i are arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form a_2 y d x⋀d y, then the ideal would not be integrable because it would not contain d(y d x) = - d x⋀d y.

differential k-form | integrable | partial differential equation | wedge product