Recovering vector-valued function from its Jacobian Matrix

Question

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!

Answer 1

Your question is very closely related to:

• Frobenius integrability theorem
• Integrability conditions for differential systems

I suspect that the first reference will be of most use.

Answer 2

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 $$\frac{\partial^2f}{\partial x_i\partial x_j}=\frac{\partial^2f}{\partial x_j\partial x_i}$$