Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 2 / b / iii / Solution

Parametrize the straight segment by $\mathbf{r}(t)=(t,t,t)$, $0\le t\le 1$. Then the line integral isThe three different answers are possible because this vector field is not a conservative vector field; for example ${\partial }_y{F}_x=6$ whereas ${\partial }_x{F}_y=0$.