See my question on MO.
For $i=1,\ldots, r$, let $Z_i$ be $r$ linearly independent vector fields defined on an open subset $U$ of $\mathbb{R}^n$, $r Question: What hypothesis do we need to assure the local existence of a solution $f$? According to Bryant answer (remark at the end): it can be shown the existence using the involutivity of $\{Z_i\}$, by using Frobenius theorem. But also it can be proven in general, not only for linear inhomogeneous PDEs, but for first order PDEs whenever they give rise to a "involutive submanifold" $\Sigma$ of the first order jet bundle. I don't know yet what is an involutive manifold. ________________________________________ ________________________________________ ________________________________________ Author of the notes: Antonio J. Pan-Collantes INDEX: