Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 16 / b / i / Solution

Parametrize the straight path by $\mathbf{r}(t)=(t,t,2t)$ for $0\le t\le 1$. The conservative field along it is $(6t^5,4t^5,t^5)$ and the tangent is $(1,1,2)$. ThusThe associated potential has ${\varphi }_1(B)-{\varphi }_1(A)=2$, in agreement with the fundamental theorem for line integrals.