Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 26F / c / i / Solution

LetIts gradient isIf this vanishes, the first and third coordinates give $x=y=z$, and the middle coordinate then gives $y=0$. Thus the origin is the only critical point of $F$. It has been removed from $V$, so $0$ is a regular value of the restriction under consideration. The regular level set theorem shows thatis a smooth surface.