(Issued Wed, Oct 18)

Due Wed, Oct 25

Reading: Boas Ch. 6.11, 2.1-2.5 Blachman 2.11, Chapter 10

1a) What does Stokes theorem tell us about , for the surface S on this sketch, which differs from the surfaces S we've talked about in class because it has both an outside boundary and an inside boundary ?

1b) If everywhere on the surface S, what can you say about the circulation of F around and ?

2. Boas 6.11.21 Come up with at least two (preferably infinitely many) solutions, which differ by more than just a constant.

3. Use the formula (valid at any point, if the loop is small), to show that , i.e. the component of the curl, in cylindrical coordinates, is given by

Hint: The formula implies you should draw a small loop which is perpendicular to the direction in cylindrical coordinates. This loop is the one formed when z varies by dz, and [[theta]] varies by d[[theta]]. Its height is dz, its breadth is in the direction. But how long? Draw a careful picture, and calculate the circulation as you run around the 4 sides...

4. Using formulas from class (or Boas Ch. 10.9) or the handout, find the following, all in spherical coordinates: , , , .

Check the 1st two of them in Cartesian: show that your results agree.

5 Boas 2.5.1 b) Boas 2.5.6 c) Boas 2.5.9

6. Use Mma to check your answers to problem #5. Note: I (a capital i) is Mma's symbol for . Mma's builtin functions Conjugate, Re, Im, Abs, and Arg, do the obvious things with complex arguments.

Mma is comfortable with complex numbers. Let Mma try to solve the quadratic equation x^2+x+1=0 (which has no real roots.) Does it choke? (Recall the Solve[] function from the first HW set.) Next, ask Mma for Exp[I Pi], and Exp[3 I Pi]. (Mma can deal with complex arguments to many (most?) of its functions, including Sin, Cos, Log, etc.) Try finding Log[-1]. Given the answers to the two complex exponentials above, did you get what you expected? What do you think the most general solution to ln(-1) should be?

7. We often define functions by power series, and this becomes especially useful to help us understand fns of complex numbers. E.g, we can use as a definition of sin(z).

a) Define a function s[n_,z_] in Mma which adds up the first n terms of the series for sin(z). (Note: Mma understands the exclamation point as a symbol for factorials! You might want to read about the Sum[] function, too.) For z fixed at [[pi]]/6, show the "partial sums" s[n,z], for n=1,2,3,..., however far you need to go to get the correct answer to 5 decimal places. (Note: designing a rigorous test that you have reached the desired precision is tricky. Do whatever you think is reasonable. One simple choice is to check that successive approximations differ by less than some precision p, where you set p to be, say, ) Repeat this all for z=[[pi]]. Roughly how many terms did it take in each case to converge?

Hint: You might need to learn about the functions For[] or While[], possibly Block[], and Print[] described in Blachman 10.2 and 10.3. Then you can write a program to do all this for you. (If you can't figure out how to do this, you could always just make a table of values of successive approximations (Columnform[] might at least make this look nicer, Blachman 6.13) but then you'll have to use trial and error to see how many terms to run, which isn't so nice.

Writing a program will give you full credit, trial and error will give you partial credit.)

b) Now repeat this exercise for the function , with z=i[[pi]]/2 and 2[[pi]]i.

Roughly how many terms did it take in each case to converge?

Note: The power series is