# Physics 834: Problem Set #1

Here are some hints, suggestions, and comments on the assignment. Remember to keep track of the amount of time you spend doing the (entire) assignment and record this number on your problem solution.

• 22-Sep-2011 --- Original version.

1. Practice with δij and εijk
1. The implicit assumption here is that the vector components here all commute, so that AiBj = BjAi, but you should state this assumption explicitly.
2. It is easiest to first manipulate the expressions with two cross products, because we can use the "very useful identity" from the δ-ε handout that eliminates two ε tensors in favor of Kronecker δ's. Be careful about moving the del's when doing this: in this case the implicit assumption is that both a and b depend on x (but that a and b commute). Note that when you simplify one of the terms, the other follows immediately by switching all a's and b's. Try combining the two terms while you still have it in indices, remembering the product rule for derivatives!
3. Similar manipulations. Here the magnetic momentum is constant, so what does that imply about derivatives of it?
2. Vector operations in cylindrical and spherical coordinates. Here the idea is to apply the formulas on the back cover of Jackson (see handouts).
• For the divergences and curls, this requires identifying A1, A2, and A3 in each case (they should be very simple and mostly zero for unit vectors), then simply plugging into the formulas. That is all you need to do in your written solution.
• For the gradient and Laplacians, note that the functions only depend on ρ and r, so derivatives with respect to angles vanish. Assume that ρ and r are greater than zero; we'll consider later what happens when they equal zero.
• Use units as a check of your results.
3. Point charge.
• Do part b) first so you know what answer you should get. You might want to use rhat = xhat (x/r) + yhat (y/r) + zhat (z/r).
• As problem 2, identify A1, A2, and A3 and use the back cover of Jackson. Note how easy this is!
• I'm not expecting that you know about delta functions in this part. Just comment on why r=0 is a special point.
4. Continuity equation. By conservation of mass, the problem means that mass is neither created nor destroyed. So if the mass changes within a volume, what must have happened (look up "flux"!)? Don't just write equations in your solution to this problem; explain the physics. If an integral over a volume is zero for any volume, then the integrand itself must be zero.
5. Practice with Stokes's theorem. Assume a direction for going around C in the x-y plane. If you go counterclockwise, what is nhat? What is an easy surface to choose?
• It may be simplest to evaluate the surface integral in polar coordinates.
• This one should be easy!
• You'll have to do a double integral.
6. Practice with the Divergence theorem.
• Pick a convenient coordinate system for the volume integral.
• What is the easiest coordinate system for a volume integral over a hemisphere? You are welcome (and encouraged!) to use Mathematica to evaluate the resulting integral, but write down the command used. There is a new Mathematica page linked from the homepage, which includes an example relevant to this problem.