Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12 / c

Integrate the first component of $\mathbf{F}=\nabla \varphi$ with respect to $x$:Matching the $y$ component gives $g_y=0$, and matching the $z$ component gives $g_z=0$. HenceThe fundamental theorem for line integrals now yieldsThe potential is a single-valued polynomial everywhere, so any piecewise smooth path with these endpoints has the same integral. The plus-gradient convention here should not be confused with the minus gradient of a mechanical potential energy.