# Math 21a: Multivariable Calculus

One way to imagine the idea of parameterizing a surface is that it gives us a way to put the parameter domain (which is actually in $$\mathbb{R}^2$$) into $$\mathbb{R}^3$$ and stretch it to become the surface. In class, you might have looked at the parameterization $$\vec{r}(u, v) = \langle u, v, u^2 + v^2 + 1 \rangle$$ with parameter domain $$u^2 + v^2 < 16$$ (a disk); here's one way you could visualize it: