Some Common Parametrized Surfaces — Quick Reference

Graphs of Functions

A parametrization for the graph of a function \(z = f(x,y)\) in Cartesian Coordinates:

\[\vec{r}(x,y) = x \, \hat{\imath} + y \, \hat{\jmath} + f(x,y) \, \hat{k}\]

Sample Graphs:

The plane \(z = 1\), parametrized in Cartesian coordinates \((x,y)\).
Parameter domain: \(-2 \leq x,y \leq 2\).

The plane \(x + y + z = 1\), or \(z = 1 - x - y\), parametrized in Cartesian coordinates \((x,y)\).
Parameter domain: \(-2 \leq x,y \leq 2\).

The graph of the function \(z = \cos^2 x + \sin^2 y\), parametrized in Cartesian coordinates \((x,y)\).
Parameter domain: \(-2 \leq x,y \leq 2\).

The graph of the function \(x = y^2 + z^2\), parametrized in Cartesian coordinates \((y,z)\).
Parameter domain: \(-2 \leq y,z \leq 2\).

A Cone

A parametrization for the cone \(z = \sqrt{x^2 + y^2}\) in Cartesian and polar/cylindrical coordinates:

\[\begin{align*}
\vec{r}(x,y) &= x \, \hat{\imath} + y \, \hat{\jmath} + \sqrt{x^2 + y^2} \, \hat{k}\\
\vec{r}(r,\theta) &= r \cos\theta \, \hat{\imath} + r \sin\theta \, \hat{\jmath} + r \, \hat{k}
\end{align*}\]

Sample Graphs:

The cone \(z = \sqrt{x^2 + y^2}\), parametrized in Cartesian coordinates \((x,y)\).
Parameter domain: \(-2 \leq x,y \leq 2\).

The cone \(z = r\), parametrized in polar/cylindrical coordinates \((r,\theta)\).
Parameter domain: \(0 \leq r \leq 2\), \(0 \leq \theta < 2\pi\).

Cylinders

A parametrization for a cylinder \(x^2 + y^2 = R^2\) (radius \(r = R\)) in cylindrical coordinates:

\[\vec{r}(\theta, z) = R \cos\theta \, \hat{\imath} + R \sin\theta \, \hat{\jmath} + z \, \hat{k}\]

Sample Graph:

The cylinder \(x^2 + y^2 = 4\), parametrized in cylindrical coordinates \((\theta,z)\).
Parameter domain: \(0 \leq \theta < 2\pi\), \(-2 \leq z \leq 2\).

Spheres

A parametrization for a sphere \(x^2 + y^2 + z^2 = P^2\) (radius \(\rho = P\)) in spherical coordinates:

\[\vec{r}(\phi,\theta) = P \sin\phi \cos\theta \, \hat{\imath} + P \sin\phi \sin\theta \, \hat{\jmath} + P \cos\phi \, \hat{k}\]

Sample Graph:

The sphere \(x^2 + y^2 + z^2 = 4\), parametrized in spherical coordinates \((\phi,\theta)\).
Parameter domain: \(0 < \phi < \pi\), \(0 \leq \theta < 2\pi\).

Parametrizing Surfaces — General Principles & Examples

Begin with the general position function in Cartesian coordinates:

\[\vec{r}(x,y,z) = x \, \hat{\imath} + y \, \hat{\jmath} + z \, \hat{k}\]

Then find a way to reduce from the three variables \(x, y, z\) to two parameters.

Graphs of Functions

The graph \(z = f(x,y)\) is a surface that can be parametrized by \(x\) and \(y\).

Replace \(z\) in the general position function:

\[\vec{r}(x,y,z) = x \, \hat{\imath} + y \, \hat{\jmath} + z \, \hat{k}\]

with \(f(x,y)\):

\[\vec{r}(x,y) = x \, \hat{\imath} + y \, \hat{\jmath} + f(x,y) \, \hat{k}\]

Example: The graph of the function:

\[z = \sqrt{x^2 + y^2}\]

is a cone, which can be parametrized:

\[\vec{r}(x,y) = x \, \hat{\imath} + y \, \hat{\jmath} + \sqrt{x^2 + y^2} \, \hat{k}\]

Surfaces in Polar/Cylindrical Coordinates

Convert the general position function:

\[\vec{r}(x,y,z) = x \, \hat{\imath} + y \, \hat{\jmath} + z \, \hat{k}\]

to cylindrical coordinates using the change-of-coordinate functions:

\[\begin{align*}
x &= r \cos\theta\\
y &= r \sin\theta\\
z &= z\\
\vec{r}(r,\theta,z) &= r \cos\theta \, \hat{\imath} + r \sin\theta \, \hat{\jmath} + z \, \hat{k}
\end{align*}\]

Then find a way to reduce from the three parameters \((r,\theta,z)\) to two parameters.

Example: In cylindrical coordinates, the cone \(z = \sqrt{x^2 + y^2}\) is given by:

\[z = r\]

Replacing \(z\) with \(r\) leads to the parametrization:

\[\vec{r}(r,\theta) = r \cos\theta \, \hat{\imath} + r \sin\theta \, \hat{\jmath} + r \, \hat{k}\]

Example: In cylindrical coordinates, the cylinder \(x^2 + y^2 = 25\) is given by:

\[r = 5\]

Replacing \(r\) with 5 leads to the parametrization:

\[\vec{r}(\theta, z) = 5 \cos\theta \, \hat{\imath} + 5 \sin\theta \, \hat{\jmath} + z \, \hat{k}\]

Surfaces in Spherical Coordinates

Convert the general position function:

\[\vec{r}(x,y,z) = x \, \hat{\imath} + y \, \hat{\jmath} + z \, \hat{k}\]

to cylindrical coordinates using the change-of-coordinate functions:

\[\begin{align*}
x &= \rho \sin\phi \cos\theta\\
y &= \rho \sin\phi \sin\theta\\
z &= \rho \cos\phi\\
\vec{r}(\rho, \phi, \theta) &= \rho \sin\phi \cos\theta \, \hat{\imath} + \rho \sin\phi \sin\theta \, \hat{\jmath} + \rho \cos\phi \, \hat{k}
\end{align*}\]

Then find a way to reduce from the three parameters \((\rho,\phi, \theta)\) to two parameters.

Example: In spherical coordinates, the sphere \(x^2 + y^2 + z^2 = 25\) is given by:

\[\rho = 5\]

Replacing \(\rho\) with 5 leads to the parametrization:

\[\vec{r}(\phi,\theta) = 5 \sin\phi \cos\theta \, \hat{\imath} + 5 \sin\phi \sin\theta \, \hat{\jmath} + 5 \cos\phi \, \hat{k}\]