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

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.

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.