Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 1 / a / ii / Solution

The $y$ component integrates to $\varphi =xy{e}^{z-x}+g(x,z)$. Matching the $x$ component gives $g_x=2x$, and matching the $z$ component then gives the remaining derivative zero. ConsequentlyDirect differentiation recovers all three components, proving this is a gradient field globally on ${\mathbb{R}}^3$.