Parametric Surfaces
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.6: “Parametric Surfaces and Their Areas” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-6-parametric-surfaces-and-their-areas |
| Textbook used in class | Stewart, Calculus, Section 16.6: “Parametric Surfaces and Their Areas” (Examples 1, 2, 3) |
Quick Reference
Parametric surface: $\mathbf{r}(u,v) = \langle x(u,v), y(u,v), z(u,v)\rangle$, $(u,v) \in D$.
Tangent vectors: $\mathbf{r}_u = \langle x_u, y_u, z_u\rangle$ and $\mathbf{r}_v = \langle x_v, y_v, z_v\rangle$.
Normal vector to surface: $\mathbf{N} = \mathbf{r}_u\times\mathbf{r}_v$.
Motivation
Curves in $\mathbb{R}^3$ are parametrized by one variable $t$. Surfaces in $\mathbb{R}^3$ are parametrized by two variables $u$ and $v$. The parametrization allows surfaces to be described without necessarily being graphs $z = f(x,y)$ -- for example, the sphere $x^2+y^2+z^2 = 1$ cannot be expressed as a single graph, but it can be parametrized by spherical angles.
The tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ at a point span the tangent plane to the surface there, and their cross product $\mathbf{r}_u\times\mathbf{r}_v$ gives the normal to the surface.
Key Concepts
1. Grid Curves
Fixing $v = v_0$ and varying $u$ gives a curve on the surface (a $u$-grid curve). Fixing $u = u_0$ and varying $v$ gives a $v$-grid curve. These two families of curves form a “grid” on the surface.
2. Standard Examples
Sphere $\rho = a$: $\mathbf{r}(\phi,\theta) = \langle a\sin\phi\cos\theta, a\sin\phi\sin\theta, a\cos\phi\rangle$, $(u,v) = (\phi,\theta)$.
Cylinder $r = a$: $\mathbf{r}(\theta,z) = \langle a\cos\theta, a\sin\theta, z\rangle$.
Graph $z = f(x,y)$: $\mathbf{r}(x,y) = \langle x, y, f(x,y)\rangle$. Then $\mathbf{r}_x = \langle 1,0,f_x\rangle$ and $\mathbf{r}_y = \langle 0,1,f_y\rangle$, so $\mathbf{r}_x\times\mathbf{r}_y = \langle -f_x,-f_y,1\rangle$.
Worked Example
Identify the parametric surface $\mathbf{r}(u,v) = \langle 2\sin u, 3\cos u, v\rangle$, $0 \leq u \leq 2\pi$, $0 \leq v \leq 1$. (Adapted from Stewart 16.6.)
$x = 2\sin u$, $y = 3\cos u$: note that $x^2/4 + y^2/9 = \sin^2 u + \cos^2 u = 1$.
So the surface lies on the elliptic cylinder $x^2/4 + y^2/9 = 1$, and $z = v$ ranges from 0 to 1. The parametric surface is a section of an elliptic cylinder of height 1.
a surface can only be parametrized one way. A surface has infinitely many valid parametrizations. The sphere can be parametrized by spherical angles, or by other angle coordinates. Different parametrizations give different $\mathbf{r}_u\times\mathbf{r}_v$ (which may point in opposite directions), but they all describe the same geometric surface.
Common Misconceptions
a parametric surface $\mathbf{r}(u,v)$ is the same kind of object as a parametric curve $\mathbf{r}(t)$, just with one extra variable.
This is the input-output-confusion error. A parametric curve maps one real parameter $t$ to a point in space, tracing a one-dimensional path. A parametric surface maps two parameters $(u,v)$ to a point in space, sweeping out a two-dimensional sheet. The tangent structure also differs: a curve has one tangent vector $\mathbf{r}'(t)$, while a surface has two partial tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ at each point, and their cross product $\mathbf{r}_u \times \mathbf{r}_v$ is required to define the surface normal and the area element.
Leveled Practice
Problem 1. Find the normal vector $\mathbf{N} = \mathbf{r}_x\times\mathbf{r}_y$ for the graph $z = x^2+y^2$ (the paraboloid).
Show answer
$\mathbf{r}(x,y) = \langle x, y, x^2+y^2\rangle$.
$\mathbf{r}_x = \langle 1, 0, 2x\rangle$, $\mathbf{r}_y = \langle 0, 1, 2y\rangle$.
$\mathbf{N} = \mathbf{r}_x\times\mathbf{r}_y = \langle 0\cdot 2y - 2x\cdot 1, 2x\cdot 0 - 1\cdot 2y, 1\cdot 1 - 0\cdot 0\rangle = \langle -2x, -2y, 1\rangle$.