Past exam of the mathematical methods II course of the University of Oxford / 2021 / 9 / ii

The position vector field is a gradient field:Along any piecewise smooth curve, $\mathbf{r}\cdot d\mathbf{r}=d(|\mathbf{r}{|}^2/2)$. Therefore a closed line integral has identical endpoint values andFor a loop bounding an admissible oriented surface, Stokes theorem gives the same result because $\nabla \times \mathbf{r}=0$. The gradient field proof also applies directly to arbitrary piecewise smooth closed loops without choosing a spanning surface.