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.

> 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:

> 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);

- Given the rectangular equation for a hyperboloid of one sheet:

**A)**- Graph the equation using the domain values of , and the range values .
**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?

- Given the equation of a torus (a.k.a. donut):

Hint: In all graphs below, use .**A)**- Graph the equation using the domain values , and the range values .
**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?

- Last week, you proved that the volume of a sphere of radius is using a double integral. Prove this again using a triple integral in all 3 coordinate systems. Use uppercase for the radius to distinguish from the variable in cylindrical coordinates. Remember to use the command instead of . If you had to calculate one of these triple integrals by hand, which one would be the simplest?

2011-12-06