Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 2 / Section B / 6 / b / Solution

Integrating ${\psi }_x=-(2xy+z)$ with respect to $x$ gives $\psi =-x^{2}y-xz+h(y,z)$. Matching ${\psi }_y=-(x^2+2y)$ requires $h=-y^2+j(z)$, and matching ${\psi }_z=-x$ requires $j$ constant. A scalar potential with the requested sign isThus the force field is a conservative vector field on all of ${\mathbb{R}}^3$.