Past exam of the mathematical methods II course of the University of Oxford / 2015 / 12 / c / ii

Interpret the axis-directed route as successive coordinate-axis-parallel segmentsOn the first segment, $y=z=0$, so the field vanishes. On the second, $z=0$, so it again vanishes. On the last, $x=1$, $y=2$ and only $dz$ is nonzero, giving $\mathbf{F}\cdot d\mathbf{r}=8zdz$. ThusThe second and third segments are parallel to the corresponding global axes; they do not return to the global $y$- or $z$-axis before reaching the endpoint.