Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 4

Take any differentiable curve $r(s)$ on the level surface. The chain rule givesIts tangent vector $r^{\prime }(s)$ can be any tangent direction at a regular point. Thus a nonzero gradient is orthogonal to the entire tangent plane, giving the unit normal $n=\pm \nabla f/|\nabla f|$. Nonvanishing is essential: at a critical point, the zero gradient alone does not determine a normal or guarantee a smooth level surface.

The relevant projection is the disk $x^2+y^2\le 9$. The surface area of a graph is $\iint \sqrt{1+z_x^2+z_y^2}dxdy$. Here $z_x=-2x$ and $z_y=-2y$, so polar coordinates giveThe boundary circle has zero surface area, so including the points at height zero does not change the result.