Past exam of the short options course of the University of Oxford / 2014 / 4 / c / Solution

The regular level set theorem says that a $C^1$ map $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ with surjective derivative at every point of $f^{-1}(0)$ has a zero set that is an embedded $C^1$ submanifold of dimension $n-m$. At each point $a$ of that zero set,For a smooth map the submanifold is smooth. The full-rank hypothesis is essential for this kernel description when using a particular defining map.