The $z$ component of the <curl> is $\partial_xF_y-\partial_yF_x=e^{x+y}-xe^{xy}$, which is not identically zero; at the origin it is one. Hence \b[the field is not conservative on $\mathbb R^3$] and has no global <scalar potential>.

Solution

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 / Solution

 Solution (source code)