Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 12 / a

The two coefficients are the corresponding partial derivatives of $e^{x+y}$. Therefore the differential form is exact, withThe arbitrary constant does not change the total differential.

For a smooth exact differential form $Mdx+Ndy$, its coefficients must satisfy $M_y=N_x$ by Clairaut's theorem. HereThese are not equal as functions on the plane, so the form is not exact. Equality at isolated points or along some lines is insufficient: an exactness test must hold throughout an open domain.

The coefficients follow by differentiating the same monomial:Thus this exact differential form hasExhibiting a global scalar potential proves exactness directly, without needing an additional domain argument.