
Here is a hint to the homework to parametrize the torus.
We keep the angle theta as one of the parameters and let r the distance of a
point on the torus to the zaxis. This distance is r=2+cos(phi) if phi
is the angle you see on the animated figure below to the left. Note that phi
has no relations with the angle phi in spherical coordinates. The blue
segment you see has the length r. You can read off from the same (left) picture
also that z=sin(phi).
To finish the parametrization problem, you
have to translate back from cylindrical coordinates
(r,theta,z)=(2+cos(phi),theta,sin(phi)) to Cartesian coordinates (x,y,z).
Write down your result in the form r(theta,phi)= (x(theta,phi),y(theta,phi),
z(theta,phi)).
