Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 10 / b

Write the fixed point as $\mathbf{a}=(a_1,a_2,a_3)$ and the variable point as $\mathbf{r}=(x_1,x_2,x_3)$. Then $\mathbf{R}=\mathbf{r}-\mathbf{a}$ andSince the fixed point is not on the surface, $R>0$ at every point being considered. Componentwise differentiation gives $\partial R/\partial x_j=(x_j-a_j)/R$, henceThus the gradient points from the fixed point towards the variable point and is a unit vector. Keeping the fixed point fixed is important: differentiating a distance between two simultaneously moving points would include both endpoint derivatives.