Solution
= 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,
$$
\boxed{T_a(f^{-1}(0))=\ker Df_a.}
$$
For a smooth map the <submanifold> is smooth. The full-rank hypothesis is essential for this kernel description when using a particular defining map.