3 01/30 W

1. Derive $\vec n {\rm d}S$ in terms of ${\rm d}\theta$ and ${\rm d}\phi$, where $(r,\theta,\phi)$ are spherical coordinates, assuming that the surface is given in the implicit form $F(r,\theta,\phi)=0$ Use that:
   
   $$\vec r = r \hat\imath_r \quad \frac{\partial \hat\imat... ...al \hat\imath_r}{\partial\phi} = \sin\theta \hat\imath_\phi$$
   
   
   and that $\partial r/\partial\theta$ and $\partial r/\partial\phi$ can be found from the total differential
   
   $$\frac{\partial F}{\partial r} {\rm d}r + \frac{\partial F}... ...m d}\theta + \frac{\partial F}{\partial\phi} {\rm d}\phi = 0$$
   
   
   Express the vector part of the final expression in terms of vector calculus.
2. p104, q62 (Use the Cartesian expression for $\vec{n} {\rm d}S$ to formulate the integral, then switch to polar to do it.)
3. p132, q42 (use Stokes)
4. p132, q44 (use Stokes with $\vec v=(-y,x,0)$, then $x=a\cos^3\alpha$, $y=a\sin^3\alpha$.)
5. p133, q56