Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 18 / b

For the printed coefficients, $p=2\sin x\cosh y$ and $q={\sin }^{2}x\sinh y$. Their cross derivatives areThey are not equal as functions; their difference is $2\sin x(1-\cos x)\sinh y$. Hence the form is not exact. Agreement on certain lines or isolated points is insufficient, because exactness requires the derivative identity throughout an open domain.

Let $w=x+iy$. Then $y-ix=-iw$, and the complex trigonometric identity $\sin (-iw)=-i\sinh w$ givesThus the form is exact, with complex-valued scalar potential $F(x,y)=\cosh (x+iy)$. Differentiating that potential with respect to the two real variables checks both coefficients individually.