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

On the specified parametrized curve, $\mathbf{r}(t)=(t^2,2t,t^3)$ and ${\mathbf{r}}^{\prime }(t)=(2t,2,3t^2)$. Substitution into the field gives $\mathbf{F}(\mathbf{r}(t))=(4t^7,2t^8,8t^6)$. HenceThe line integral is thereforeBoth the field and the tangent vector must be evaluated along the curve; integrating one field component alone would miss two contributions.