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Let $\mathbf{r}(s)$ be any smooth curve in the level surface $\varphi =c$. Differentiating the constant value along it givesThe curve tangent can be any tangent vector to a regular level set, soThe nonzero-gradient condition matters for determining a normal direction. For example, $\varphi =z^3$ 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.

Let a differentiable curve $\mathbf{r}(s)$ lie on the level surface $\varphi =c$, and pass through the point under consideration. Differentiating the constant relation along the curve givesEvery 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 $\nabla \varphi \ne 0$, it therefore defines a normal direction, with unit normalIf $\nabla \varphi =0$, 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, $\varphi =z^3$ defines the plane $z=0$ at level zero but has vanishing gradient everywhere on that plane.

Take any differentiable curve $r(s)$ on the level surface. The chain rule givesIts tangent vector $r^{\prime }(s)$ can be any tangent direction at a regular point. Thus a nonzero gradient is orthogonal to the entire tangent plane, giving the unit normal $n=\pm \nabla f/|\nabla f|$. Nonvanishing is essential: at a critical point, the zero gradient alone does not determine a normal or guarantee a smooth level surface.