Subsections

• Getting Started
• Background
• Exercises

Cylindrical and Spherical Coordinates

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples and even solutions to some of the exercises. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window for example, not in Maple.

cp /math/calclab/MA1024/Coords_start.mws ~/My_Documents

You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read to the exercises, start up Maple, load the worksheet Coords_start.mws, and go through it carefully. Then you can start working on the exercises.

Background

Defining surfaces with rectangular coordinates often times becomes more complicated than necessary. A change in coordinates can simplify things. Cylindrical coordinates can simplify plotting a region in space that is symmetric with respect to the $z$-axis such as paraboloids and cylinders. The paraboloid $z=x^2+y^2$ would become $z=r^2$ and the cylinder $x^2+y^2=1$ would become $r=1$. Spherical coordinates would simplify the equation of a sphere, such as $x^2+y^2+z^2=1$, to $\rho=1$. The conversion tables below show how to make the change of coordinates.

To change to cylindrical coordinates from rectangular coordinates use the conversion:

$x=r\cos(\theta)$

$y=r\sin(\theta)$

$z=z$

Where $r$ is the radius in the x-y plane and $\theta$ is the angle in the x-y plane.

To change to spherical coordinates from rectangular coordinates use the conversion:

$x=\rho\sin(\phi)\cos(\theta)$

$y=\rho\sin(\phi)\sin(\theta)$

$z=\rho\cos(\phi)$

Where $\theta$ is the angle in the x-y plane; $\rho$ is the radius from the origin in any direction; and $\phi$ is the angle in the x-z plane.

Exercises

1. Given the rectangular equation for a circular paraboloid:
   
   $$\frac{x^2}{9}+\frac{y^2}{9}=\frac{z}{2}$$
   
   

   A)

   Graph the equation using the domain values of $-10 \leq x \leq 10$, $-10 \leq y \leq 10$ and the range values $0 \leq z \leq 10$.

   B)

   Write the equation in cylindrical coordinates and then graph the equation.

   C)

   Write the equation in spherical coordinates and graph it.

   D)

   Looking at the three equations, which coordinates appears to give the simplest equation?

2. Given the equation of a torus (a.k.a. donut):
   
   $$16(x^2+y^2)=(x^2+y^2+z^2+3)^2$$
   
   

   A)

   Graph the equation using the domain values $-4 \leq x \leq 4$, $-4 \leq y \leq 4$ and the range values $-1 \leq z \leq 1$.

   B)

   Write the equation in cylindrical coordinates and graph it.

   C)

   Write the equation in spherical coordinates (hint: use the factor command outside the simplify command to simplify even more). Then graph your equation.

   D)

   Looking at the three equations, which coordinates appear to give the simplest equation?

3. A problem from your text describes a bumpy sphere or an exaggerated representation of waves on the surface of a very small planet that is covered by a very deep ocean. Such a bumpy or wrinkly sphere may also be used to model tumors. Use Maple to plot the spherical-coordinate surface
   
   $$\rho = a + b \cos(m \theta) \sin(n \phi)$$
   
   
   with values of the positive number $a$ and $b$ and the positive integers $m$ and $n$ given below. How does the surface depend on the value of each of these four parameters?

   A)

   $a=10$, $b=2$, $m=4$, $n=4$

   B)

   $a=5$, $b=2$, $m=4$, $n=4$

   C)

   $a=10$, $b=0.2$, $m=4$, $n=4$

   D)

   $a=10$, $b=2$, $m=20$, $n=4$

   E)

   $a=10$, $b=2$, $m=2$, $n=4$

   F)

   $a=10$, $b=2$, $m=2$, $n=10$