Physics 263: Problem Set #14

1. (BTM 7.6.9.) For the first part, the area integrated over is the entire square. The "worked example" mentioned in the second part is in section 7.4. To integrate over an area between curves, integrate over x from 0 to 1 but the limits for y are determined by the curve. For example, to integrate between the x-axis and the line y=x, the lower integration limit is 0 while the upper one is y. For the last part, ignore what it says to show and just show that Green's theorem works for that path.
2. (BTM 7.7.1.) Just apply the definition. Remember that the answer must be a scalar.
3. (BTM 7.7.3.) Both ways of calculating the integral should give you 1 as an answer. The volume integral is direct, it's the surface integral where you want to think before working too hard. The "symmetries" the problem hints at are either integration over odd functions or opposite sides contributing equal and opposite (so net zero).
4. (BTM 7.7.4.) This is much easier to do via the volume integral than the surface integral (assuming you calculate the divergence correctly!).
5. (BTM 7.10.1.) Simply write them out in cartesian coordinates; nothing tricky.
6. (BTM 7.11.4.) Just follow the instructions. You'll see the Dirac delta function a LOT in future classes.