When is a given matrix-valued function the Jacobian of something?

Question

Let $F$ be an $n\times m$ matrix of real-valued functions which are defined and smooth on a neighborhood of a point $p\in \mathbb{R}^m$. Under what conditions is it possible to find a smooth function $f\colon U \to \mathbb{R}^n$ defined on a (possibly smaller) neighborhood $U$ of $p$ such such that $F= J_f$ on $U$, where $J_f$ is the Jacobian matrix of $f$?

Answer 1

The $i^{\text{th}}$ row vector $F_i$ of $F$ needs to be $df_i$ for some smooth function $f_i\colon U\to\Bbb R$, so, viewing $F_i$ as a $1$-form, it is exact on a simply connnected $U$ if and only if it is closed. That is, we need $\dfrac{\partial F_{ij}}{\partial x_k} = \dfrac{\partial F_{ik}}{\partial x_j}$ for all $1\le i\le n$, $1\le j,k\le m$.