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_B11.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_B11.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. The easiest examples are a sphere and a cylinder.

> with(plots):
> f1:=x^2+y^2+z^2=49;
> g1:=rho=7;
> implicitplot3d(f1,x=-7..7,y=-7..7,z=-7..7,axes=boxed,scaling=comstrained);
> implicitplot3d(g1,rho=0..7.5,theta=0..2*Pi,phi=0..Pi,coords=spherical,
numpoints=5000,axes=boxed);
> f2:=x^2+y^2=49;
> h2:=r=7;
> implicitplot3d(f2,x=-7..7,y=-7..7,z=-8..8,axes=boxed);
> implicitplot3d(h2,r=0..7.5,theta=0..2*Pi,z=-8..8,coords=cylindrical,
numpoints=3000,axes=boxed);

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. As an example, the equation of an ellipsoid in rectangular coordinates is

$$\frac{x^2}{23}+\frac{y^2}{23}+\frac{z^2}{122}=1$$

> f3:=x^2/23+y^2/23+z^2/122=1;
>implicitplot3d(f3,x=-5..5,y=-5..5,z=-12..12,scaling=constrained,axes=boxed);

Changing to sherical coordinates:

> g3:=simplify(subs({x=rho*sin(phi)*cos(theta),y=rho*sin(phi)*sin(theta),
z=rho*cos(phi)},f3));
> implicitplot3d(g3,rho=0..12,theta=0..2*Pi,phi=0..Pi,coords=spherical,axes=boxed,
scaling=constrained,numpoints=2000);

Exercises

1. Given the rectangular equation for a hyperboloid of one sheet:
   
   $$2(x^2+y^2)-z^2=2$$
   
   

   A)

   Graph the equation using the domain values of $-6 \leq x \leq 6$, $-6 \leq y \leq 6$ and the range values $-6 \leq z \leq 6$.

   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):
   
   $$36(x^2+y^2)=(x^2+y^2+z^2+5)^2$$
   
   
   Hint: In all graphs below, use $\texttt{scaling=constrained}$.

   A)

   Graph the equation using the domain values $-6 \leq x \leq 6$, $-6 \leq y \leq 6$ and the range values $-2 \leq z \leq 2$.

   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. Last week, you proved that the volume of a sphere of radius $R$ is $$\frac{4}{3} \pi R^3$$ using a double integral. Prove this again using a triple integral in all 3 coordinate systems. Use uppercase $R$ for the radius to distinguish from the variable $r$ in cylindrical coordinates. Remember to use the command $\texttt{value}$ instead of $\texttt{evalf}$. If you had to calculate one of these triple integrals by hand, which one would be the simplest?