= Solution
Let a differentiable curve $\mathbf r(s)$ lie on the <level surface> $\phi=c$, and pass through the point under consideration. Differentiating the constant relation along the curve gives
$$
0=\frac{d}{ds}\phi(\mathbf r(s))=\nabla\phi\cdot\mathbf r'(s).
$$
Every surface <tangent vector> arises as the <velocity> of such a surface curve, so the <gradient> is orthogonal to every tangent direction. At a regular point, where $\nabla\phi\ne0$, it therefore defines a normal direction, with unit normal
$$
\boxed{\hat{\mathbf n}=\pm\frac{\nabla\phi}{|\nabla\phi|}}.
$$
If $\nabla\phi=0$, the dot-product identity is still true but the zero vector does not determine a normal direction; a regularity hypothesis or another defining function is needed. For example, $\phi=z^3$ defines the plane $z=0$ at level zero but has vanishing <gradient> everywhere on that plane.
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