Past exam of the mathematical methods II course of the University of Oxford / 2013 / 11 / a

Let $\mathbf{r}(s)$ be any smooth curve in the level surface $\varphi =c$. Differentiating the constant value along it givesThe curve tangent can be any tangent vector to a regular level set, soThe nonzero-gradient condition matters for determining a normal direction. For example, $\varphi =z^3$ has the plane $z=0$ as a level set, but its gradient vanishes there and supplies no direction, even though the geometric plane has a well-defined normal.