Exterior differential ideal ($d\mathcal{I}\subseteq \mathcal{I}$)

An exterior differential ideal on a smooth manifold is a graded ideal $\mathcal{I}$ in the exterior algebra ${\Omega }^{\ast }(M)$ of differential forms that is closed under the exterior derivative:Being an ideal also requires $\alpha \wedge \beta \in \mathcal{I}$ for every $\beta \in {\Omega }^{\ast }(M)$. Thus stability under differentiation is an additional condition, not a consequence of wedge closure.

For example, the ideal generated by $dz$ in ${\mathbb{R}}^3$ is differential because $d(fdz)=df\wedge dz$. Its integral surfaces include the planes $z=\text{constant}$. In contrast, the ideal generated by $\alpha =dz-xdy$ is not differential: $d\alpha =-dx\wedge dy$ and $\alpha \wedge d\alpha =-dz\wedge dx\wedge dy\ne 0$, whereas wedging $\alpha$ with any element of its generated ideal gives zero. The Frobenius theorem interprets this obstruction as failure of the associated tangent-plane distribution to be integrable.