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

Let a differentiable curve $\mathbf{r}(s)$ lie on the level surface $\varphi =c$, and pass through the point under consideration. Differentiating the constant relation along the curve givesEvery surface tangent vector arises as the velocity of such a surface curve, so the gradient is orthogonal to every tangent direction. At a regular point, where $\nabla \varphi \ne 0$, it therefore defines a normal direction, with unit normalIf $\nabla \varphi =0$, the dot-product identity is still true but the zero vector does not determine a normal direction; a regularity hypothesis or another defining function is needed. For example, $\varphi =z^3$ defines the plane $z=0$ at level zero but has vanishing gradient everywhere on that plane.