Physics 263: Problem Set #13

1. (BTM 7.5.5.) The expression for the gradient in polar coordinates is in section 7.5.1; you'll need to remember how to rewrite x and y in terms of r and theta. In part iii), you should change 2i+3j to a unit vector (i.e., divide by its magnitude). In part iv), you'll want to invoke Eq. (7.5.15).
2. (BTM 7.6.2.) Just apply the criterion given in Eq. (7.6.9), but make sure you've read that section!
3. (BTM 7.6.4.) If a field is conservative, it can be written as a gradient of some scalar field. So check if it is conservative (same criterion as the last problem). If it is the gradient of a scalar, Eq. (7.5.15) comes in to play. You can guess the scalar function pretty easily, or integrate the x component with respect to x and the y component with respect to y (in general there are integration constants).
4. (BTM 7.6.5.) Break up the integral into the four sides and do them in turn (be careful on the top segment that you don't include too many minus signs). If it could be written as the gradient of a scalar field, what would (7.5.15) say about integrating around a closed path (i.e., the endpoint is the same as the starting point)?
5. (BTM 7.6.10.) If you end up doing a complicated integral, you're working too hard on this one. Apply Stokes' theorem and do a little trig substitution. The integral in the end should be doable by inspection.
6. (BTM 7.6.13.) This one is like the last one, but you need to figure out the vector field, which when dotted into dr gives you the integrand. Be careful that you are going around the circle clockwise.