= Solution
Let $\mathbf r(s)$ be any smooth curve in the <level surface> $\phi=c$. Differentiating the constant value along it gives
$$
0=\frac d{ds}\phi(\mathbf r(s))=\nabla\phi\cdot\frac{d\mathbf r}{ds}.
$$
The curve tangent can be any tangent vector to a <regular level set>, so
$$
\boxed{\nabla\phi\perp T_P\Sigma,\qquad \hat{\mathbf n}=\pm\frac{\nabla\phi}{|\nabla\phi|}\quad\text{when }\nabla\phi\ne0.}
$$
The nonzero-gradient condition matters for determining a normal direction. For example, $\phi=z^3$ has the plane $z=0$ as a level set, but its gradient vanishes there and supplies no direction, even though the geometric plane has a well-defined normal.
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