Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / a / ii

The $z$ component of the curl is ${\partial }_x{F}_y-{\partial }_y{F}_x=e^{x+y}-x{e}^{xy}$, which is not identically zero; at the origin it is one. Hence the field is not conservative on ${\mathbb{R}}^3$ and has no global scalar potential.