> 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 spherical coordinates and then graph the equation.
**C)**- Write the equation in cylindrical 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):

**A)**- Graph the equation using the domain values , and the range values .
**B)**- Write the equation in spherical coordinates and graph it.
**C)**- Write the equation in cylindrical 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?

- A sphere centered at the origin (as seen in the background) is a very simple equation in spherical coordinates. Now let's look at a sphere moved away from the origin.

**A)**- Graph the equation using the domain values , and the range values .
**B)**- Write the equation in spherical coordinates and graph it. (You may want to use
`axes=normal`instead of`boxed`) **C)**- Write the equation in cylindrical coordinates and graph it. (You may want to use
`axes=normal`instead of`boxed`) **D)**- Looking at the three equations, which coordinates appear to give the simplest equation?
**E)**- Looking at the rectangular domain values, a hemisphere can be graphed two ways. Changing the values to (or ). The second way would be to change the y values to . Graph a hemisphere four ways:
**i.**- In spherical coordinates change the values.
**ii.**- In spherical coordinates change the values.
**iii.**- In cylindrical coordinates change the values.
**iv.**- In cylindrical coordinates change the values.

2006-02-15