Computing the Normal to a Surface

Consider a surface $S$ defined as

$$g(x,y,z)=0$$

A path on the surface given by $g$ will be of the form

$$\vec{r}(t)=⟨x(t),y(t),z(t)⟩$$

where

$$g(x(t),y(t),z(t))=0$$

then

$$\begin{array}{rl}0& =f^{\prime }(t)\\ & =\frac{\partial g}{\partial x}{x}^{\prime }(t)+\frac{\partial g}{\partial y}{y}^{\prime }(t)+\frac{\partial g}{\partial z}{z}^{\prime }(t)\\ & =(\nabla g)\cdot \vec{v}\end{array}$$

Thus, as $\nabla g$ is orthogonal to the vector along the surface, then $\nabla g$ gives a normal vector to the surface $S$.