4 02/05 F

1. 1st Ed: p103, q44, 2nd Ed: p123, q44. Do it with Stokes.

2. 1st Ed: p104, q62, 2nd Ed: p124, q62. (20 points) Do both directly and using the divergence theorem. Make sure to include the base of the cone. Use the Cartesian expression for $\vec{n} {\rm d}S$ to formulate the surface integral, then switch to polar to do it.

3. 1st Ed: p132, q50, 2nd Ed: p154, q50. Instead of $M_y=N_x$, show that the curl of the vector is zero and discuss Stokes after doing the integral directly.

4. 1st Ed: p133, q56, 2nd Ed: p155, q56.

5. Read through subsection 9.4 of QMFE and write a half-page summary.

6. 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 as $r=f(\theta,\phi)$. Use that:
   
   $$\vec r = r \hat\imath_r \quad \frac{\partial \hat\imat... ...al \hat\imath_r}{\partial\phi} = \sin\theta \hat\imath_\phi$$
   
   
   as will be derived in class. Next generalize the result to the case that the surface is given by the implicit expression $F(r,\theta,\phi)=0$. One way to do so is to find $\partial r/\partial\theta$ and $\partial r/\partial\phi$ 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. Compare it to the Cartesian case in which $x$ and $y$ are the integration variables.

7. 1st Ed: p160, q38, 2nd Ed: p183, q38.