Surface

For ${\mathbb{R}}^3$

A surface is a set of points in ${\mathbb{R}}^3$ that can be locally described as the graph of a function or as a parametric equation. A smoother surface is one where these functions are continuously differentiable.

Explicit Form

A surface can be given in the form

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

for example, a sphere of radius $R$ is defined by

$$x^2+y^2+z^2-R^2=0$$

Parametric Form

A surface can be described by a vector valued function

$$\vec{r}(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩$$