Recovering vector-valued function from its Jacobian Matrix Consider a function $f:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$, for which the Jacobian matrix 
$J_f(x_1,...,x_n)= \left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1}  & ... & \frac{\partial f_1}{\partial x_n} \\
\vdots &  & \vdots \\
\frac{\partial f_m}{\partial x_1} & ... & \frac{\partial f_m}{\partial x_n} \end{array} \right) $ is given. 
Also, assume the component functions of $J_f$ are continuously differentiable on $\Omega$, and $\Omega$ is simply connected. If $m=1$ and $n=2$, it is well known that the function $f$ can be recovered from $J_f$ (in this case the gradient) if and only if $\frac{\partial}{\partial x_2}\frac{\partial f_1}{\partial x_1}=\frac{\partial}{\partial x_1}\frac{\partial f_1}{\partial x_2}$.
So my question is whether there is a generalization of this result for arbitrary values of $m$ and $n$. I would appreciate any references!
Thank you!

Yes there is. For one thing, the mapping $f$ is recoverable if and only if each of its component $f_i$ is recoverable. So we can only consider the case where $m=1$. Then we know its all $n$ partial derivatives, or
$$df=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i$$
Frobenius integrability theorem tells us that $f$ can be recovered if
$$\frac{\partial^2f}{\partial x_i\partial x_j}=\frac{\partial^2f}{\partial x_j\partial x_i}$$
