In chapter 12 we describe surfaces in $\mathbb{R}^3$ with single Cartesian equations that depended on some combination of $x$, $y$, and $z$. If the equation of the surface is expressed as $g(x,y,z)=0$ (or any constant) then the gradient of the function $g(x,y,z)$ is normal to, or perpendicular to, the surface.

Both tangent and normal vectors are used in the final weeks of the semester, when we integrate the tangential component of some vector field along a curve, or the normal component of another vector field over a surface.