= Solution
If $u\,dx+v\,dy=dF$ with continuous second <partial derivatives> of $F$, then $u=F_x$ and $v=F_y$. The <symmetry of second derivatives> gives the necessary condition for an <exact differential form>:
$$
\boxed{\left(\frac{\partial u}{\partial y}\right)_x=\left(\frac{\partial v}{\partial x}\right)_y.}
$$
This condition is also locally sufficient on a small rectangle for continuously <differentiable> coefficients; global sufficiency additionally needs suitable topology of the domain.
Back to article page
