Physics 263: Problem Set #12

1. (BTM 7.4.2) If you follow the example from Equation (7.4.3) through (7.4.10), you'll have no problem here. For example, for the first part take r(t) = t ihat + t jhat + t khat with 0 < t < 1 so the dr/dt = ihat + jhat + khat.
2. (BTM 7.4.3) The same suggestion as in the previous problem applies. Here we use two different parameterizations in terms of t. For the second part, think of t as an angle. When you get the integral, look for symmetries that give you the answer without an actual calculation.
3. (BTM 7.4.4) The example starting with Equation (7.4.18) shows you the way here. Note the symmetry of the vector V. Pick two of the sides and figure out dS for each, dot with V, and draw your conclusion!
4. (BTM 7.5.1) Think of the height in terms of the polar coordinate r first (and also the radius R), and then convert to x and y. Your "expectations" should be about what direction the height of the hemisphere increases most rapidly at any single point, and which points is the increase steepest.
5. (BTM 7.5.7) This is the problem we did in 1094 Session 6. For the second part, you are applying Equation (7.5.2), where you know the gradient and dr is determined by the problem. Remember that if you are stepping 1/10 on a diagonal, dx and dx are smaller than this by 1/sqrt(2).
6. (BTM 7.5.4) You'll want to apply equation (7.5.20) and recall the results of Section 3.2.