TITLE: General solution for the system of PDEs from the curl of a vector field equaling another
QUESTION [6 upvotes]: In my vector calculus class, when we were introduced to the curl operator the professor gave us this example:
Is it possible to find a vector field $\mathbf{G}$ such that 
$$\mathbf{F} = \nabla \times {\mathbf{G}}?$$
As a motivation for the identity that the divergence of a curl of a vector field is $0$. 
My question is:

Given that the identity holds true, how do you solve for a general solution for the component functions of $\mathbf{G}$, given $\mathbf{F}$?

The best I can do is find one or two solutions. For example, if $\mathbf{F}$ = $\langle-y,-z,-x\rangle$, a solution I find is $\mathbf{G}$ = $\langle xy,0,-\frac{1}{2}y^2+xz\rangle$. I had to go through a system of partial differential equations and made it work, but can the general solution be written explicitly?
Thanks!

REPLY [2 votes]: Since we know that the solution to the Maxwell equation
$\nabla \times B$ = j
is given by the Biot-Savart formula
$B(r) = \int \frac{\bf{j(r')} \times {(\bf{r}-\bf{r'})}}{|r-r'|^3} d^3\bf{r'}$
this gives a convenient shortcut (and the derivation/proof is in any E&M book).