= Solution
A $C^1$ embedded <hypersurface> $M\subseteq\mathbb R^n$ is locally carried by a $C^1$ coordinate diffeomorphism to an open subset of $\mathbb R^{n-1}\times\{0\}$. Equivalently, near each of its points it has the form $F^{-1}(0)$ for a $C^1$ function with nonzero <gradient> there.
The <Lagrange multiplier> theorem states that if a <differentiable> function $g$ has a local extremum on such a <hypersurface> at $x_0$, then $dg(x_0)$ annihilates the <tangent space>. In a regular local level-set description this is
$$
\boxed{\nabla g(x_0)=\lambda\nabla F(x_0)}
$$
for some real $\lambda$. The regularity condition $\nabla F(x_0)\ne0$ is essential; it is what identifies the <normal line> and permits the local constraint chart.
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