Suppose we find, somewhere in the wild, a Jacobian matrix $J(x)$ (so, $J_{i,j}=\frac{\partial f_i}{\partial x_j}$). Let us assume the matrix function is square and continuous and maps to a real space of square matrices. Can we then always find the corresponding vector field $f$?
And the follow-up question would be, given any square ($n\times n$) and continuous matrix function $K(x)$ (with $x\in\mathbb{R}^n$), can we always see this as a Jacobian, and find a function $g:\mathbb{R}^n\to\mathbb{R}^n$ such that $\frac{\partial g (x)}{\partial x}=K(x)$?