Multivariable calculus and vector analysis

Parametrized surface examples

Example 1

Parametrize the half-cone $z=\sqrt{x^2+y^2}$.

Solution: For a fixed $z$, the cross section is a circle with radius $z$. So, if $z=u$, the parameterization of that circle is $x=ucosv$, $y=usinv$, for $0\le v\le 2\pi$.

The parameterization of whole surface is

$$\begin{array}{lll}\hfill \left(x,y,z\right)=\mathbf{\Phi }\left(u,v\right)=\left(ucosv,usinv,u\right)& & \hfill \end{array}$$

for $0\le v\le 2\pi$, $0\le u\le \infty$.

Of course, there’s nothing sacred about $u$ and $v$. Could also use

$$\begin{array}{lll}\hfill \left(x,y,z\right)=\mathbf{\Phi }\left(r,\theta \right)=\left(rcos\theta ,rsin\theta ,r\right).& & \hfill \end{array}$$

Example 2

What happens if fix the radius of the circle to $x=3cos\theta$, $y=3sin\theta$?

Solution. The parameterization becomes

$$\begin{array}{lll}\hfill \left(x,y,z\right)=\mathbf{\Phi }\left(u,\theta \right)=\left(3cos\theta ,3sin\theta ,u\right)& & \hfill \end{array}$$

This is a right circular cylinder of radius 3.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

4b. Parametrized curves

4c. Parametrized surfaces

• Parametrized surfaces
• Parametrized surface examples
• Surface area of parametrized surfaces
• Surface area calculation
• Normal vector of parametrized surfaces
• Orienting surfaces
• A surface that is not orientable
• Surface area example

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material