Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 2 / Section B / 5 / a / Solution

If $udx+vdy=dF$ 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:This condition is also locally sufficient on a small rectangle for continuously differentiable coefficients; global sufficiency additionally needs suitable topology of the domain.