Physics 263: Problem Set #5

1. (BTM 3.1.2.) Not much to say here. You can use MATLAB to check your answers.
2. (BTM 3.1.5.) Draw a picture; then you will have an idea roughly what the answer should be. The constraint is specified by the equation for the line. The other ingredient is the function to be minimized. Here is where you translate the problem statement into this function. Since you want the shortest distance, you need a function of x and y for the distance from the origin to the point (x,y). What is it?
3. (BTM 3.1.6.) As with any Lagrange multiplier problem, there is a quantity to be extremized (maximized or minimized) and a constraint. Identify each of these first (after drawing a picture and introducing variables). We know the answer from an alternative method used in the first chapter, so there's no excuse for algebra mistakes here!
4. (BTM 3.2.2.) As always, draw a picture first so that you can see the symmetries. With most integration problems, you would like to identify symmetries that mean you can integrate over a more restricted area or volume (or whatever) and simply multiply by a factor. What can you do in this case? Determine your limits and then carry out the integrations sequentially. The second integral follows using one of your favorite trig substitutions.
5. (BTM 3.2.3.) If you are confused about what to do, try the two-dimensional, polar coordinates version first. In that case, in your sketch identify a small area obtained by varying r from r to r + Delta r, and theta from theta to theta plus Delta theta. The factors h_r and h_theta are what you need to convert Delta r and Delta theta to the sides of that area. For r, the side IS Delta r, so h_r=1. For theta, the length is r*Delta theta (to first order in Delta theta), so h_theta = r. Now do the same in three dimensions with r, theta, and phi. Vary each one in a separate picture to make it easier to visualize.
6. (Bonus) Using a Lagrange multiplier is the way to go here. As in the two-dimensional version, start with a sketch where you introduce labels for the relevant variables (what do you need to specify for a three-dimensional, rectangular box?). What is your constraint equation in terms of those variables? What is the quantity you want to extremize in terms of those variables? When you get an answer, check whether it is a natural generalization of the 2D result.