Past exam of the mathematical methods II course of the University of Oxford / 2015 / 12 / d / Solution

Both paths start at $(0,0,0)$ and end at $(1,2,1)$. The potential values are $\varphi (0,0,0)=C$ and $\varphi (1,2,1)=4+C$, soThis is the fundamental theorem for line integrals: an exact one-form integrates to the difference of its single-valued potential at the endpoints. The agreement is guaranteed for every piecewise smooth connecting path, rather than being an accident of these two examples. In particular the integral around a closed path assembled from one route and the reverse of the other is zero.