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 / ii

For $\mathbf{r}(t)=(t,t,2t^3)$, the conservative field is $(6t^7,4t^7,t^5)$ and the tangent is $(1,1,6t^2)$. ThereforeThe two different paths give the same value because their line integrals depend only on the potential difference ${\varphi }_1(B)-{\varphi }_1(A)=2$. There is no scalar potential for ${\mathbf{F}}_2$ to which the conditional request could be applied.