Let $\displaystyle C$ be the circle in the x-z plane with radius $\displaystyle r>0$ and center (R,0,0). A torus of revolution is obtained by revolving $\displaystyle C$ about the z-axis. Show that the patch is given by $\displaystyle x(u,v)=((R+r cos u)cos v, (R+r cos u)sin v, r sin u)$.

Ahmmm, a patch is just a mapping x:$\displaystyle R^{2} to R^{3}$ that defines the whole curve.