Mathematics 528

METHODS OF APPLIED MATHEMATICS

SPRING 2007

NOTE: Solutions to homework assignments are no longer available through this web page.

Final Exam:

• The final exam will be held on Thursday, May 3, from 4:00 to 7:00 P.M., in Hill 124. The exam is cumulative.
• Office hours during reading period and exam week (all in Hill 520): Note change (5/1) in office hours for Thursday 5/3.
  • Tuesday, May 1, 3:20-4:40 PM;
  • Wednesday, May 2, 10:30-11:30 AM;
  • Thursday, May 3, 9:30-10:30 AM, 3:10-3:50 PM;
  • Monday, May 7, 1:30-2:30 PM;
  • Tuesday, May 8, 3:20-4:40 PM.
• Here is a tentative formula sheet for the final exam.

Exam 1 will be held in class on Tuesday, February 27. It will cover through our work on conformal mapping.

Homework assignments.

• Assignment 13. Hildebrand, Chapter 2: 59, 60, 61, 62, 63. This assignment will not be collected.
  • Solutions to assignment 13.
• Assignment 12. Hildebrand, Chapter 2: *8, *9, *13, *15. In problem 13, show that the solutions are great circles.
  • Here is Assignment 12 in pdf form.
  • Solutions to assignment 12.
• Assignment 11. Here is assignment 11 in pdf form. No problems from this assignment will be collected.
  • Solutions to assignment 11. As usual, there may be misprints or errors; [please let me know if you find them.
  • Section 24.5: 2 (b), (d), (h), (j); 3 (f), (i), (j); 4; 6 (a), (d), 7.
  • Obviously, there are many more problems in Section 24.5 which you may try; this seems a reasonable selection to practice on. We are omitting the inverse Laplace and Fourier transforms.
• Assignment 10, starred problems due 4/03. Here is assignment 10 in pdf form.
  • Solutions to Assignment 10. (Corrected 04/09.)
• Assignment 9, starred problems due 3/27. Here is assignment 9 in pdf form.
  • Solutions to Assignment 9.
  • Section 24.2: 6 (c), *(d), *(f), (g), (h); 8 (a), *(c), (d); *9; 11 (e), *(f), (j); 16 *(g).
  • In 6(d), also identify explicitly the function which is the sum of the series.
  • For extra credit, identify explicitly the function which is the sum of the series in 8(a).
• Assignment 8. This is due 3/20, and is here in pdf form.
  • Solutions to Assignment 8
• Assignment 7 was passed out in class Thursday, March 1. Its due date has been revised to Thursday, March 8.
  • Solutions to Assignment 7
  • Section 23.2: 1. *(a), (b), (d), *(f), 3. (f), 5. *(a)
  • Section 23.3: 1, 2, 4 (d), (f), (h), *(k), *(l), 5, *7
  • Note: the problems 23.3:1,2 are just thought problems to help you understand Cauchy's theorem. After that, there are a good many problems, none very hard, just so that you can practice doing these contour integrals. Use Cauchy's Theorem whenever possible in (Section 23.3)! For 23.3:4(f), see Example 3 in that section. Cauchy's Theorem could help you do 22.3.2:1(d) but of course you don't know it at that point.
• Assignment 6. Further problems on conformal mapping. These problems will not be collected.
  • Solutions to Assignment 6 (corrected version). Here is a Maple worksheet supporting the calculations for problem 22.6.6 in this assignment.
  • In the solutions posted above I did not provide a solution for problem 22.6.6(c). Yi Pan, one of the students in the course, has provided a solution to this problem in Microsoft Word format.
  • Section 22.4: 4 (c), (g), (j). Section 22.5: 3 (a). Section 22.6: 2 (a), 4 (b), 6.
• Assignment 5, due 2/20. Turn in only starred problems. Here is assignment 5 in pdf form.
  • Solutions to Assignment 5.
  • Section 22.3: 10 (a), (e), *(g), 11 (a), *(c), 14(a), *(d), *(e)
  • Supplementary exercise 5.1: In each case below, find a conformal mapping w=f(z) carrying the given region D onto the upper half plane v>0 (here w=u+iv and in describing D we always write z=x+iy). Give a brief explanation of your answer, but not a full proof.
    • (a) D is the right half plane x>0.
    • (b) D is the second quadrant x<0, y>0. Hint: think about z^2.
    • *(c) D is the intersection of the right half plane with the unit disk: x>0, x^2+y^2<1. Hint: start with a bilinear transfomation, then use the idea of (b).
    • *(d) D is the strip 1>x>0.
    • (e) D is the half strip 1>x>0, y>0. Hint: modify a homework problem.
• Assignment 4, due 2/13. Turn in only starred problems. Here is assignment 4 in pdf form.
  • Solutions to Assignment 4.
  • Section 21.5: 15 *(a), (b), *(d). For (d), work in polar coordinates. In these coordinates you can use equation (1) of section 20.3 to determine if a given function is harmonic, and the appropriate Cauchy-Riemann equations (equations (30) of Section 21.5) to construct the conjugate harmonic function of a given harmonic function.
  • Section 21.5: *16 (a), (c).
  • Section 22.2: *1.
• Assignment 3, due 2/06. All problems are from Section 21.5. Turn in only starred problems. Here is assignment 3 in pdf form.
  • Solutions to Assignment 3.
  • 2(a), *2(b), 5(a).
  • *5(b) and *10(e). Think about these two problems together. Show that if you calculate the derivative of 1/z (i) by the rule for powers (or equivalently the quotient rule), (ii) directly as a limit, from (8), as in Example 2, or (iii)by any of the formulas (19), the answer is the same. NOTE: There was an error on the copy of assignment 3 handed out in class: problem 10(c) was assigned. The correct assignment is 10(e), as above.
  • *9. Hint: this is a bit tricky. To calculate any partial derivatives of u(x,y) or v(x,y) at x=y=0, or to try to calculate the derivative of f at z=0, you must use the direct definition of the derivative as a limit, since there is a special formula for f(0). Calculate all the partial derivatives this way, then try to calculate f'(0) as z approaches 0 along the direction x=y.
    As an additional exercise (not to be turned in): From the results of the problem, and Theorem 21.5.1 (or more properly from the version of this theorem stated in class) we know that not all of the partial derivatives u_x, u_y, v_x, and v_y can be continuous at x=y=0. Be sure you understand why this is true, then show it by direct calculation.
  • *10(a), 10(g), *12(b), 14(a). Hint for 14(a): use the Cauchy-Riemann equations.
• Assignment 2, due 1/30. Turn in only starred problems.
  • Section 21.3: 2(b), *2(f), 9(e), 12, *13, 16(c), *18(c). Note: In problem 21.3.13, the hint referred to is in problem 12, not problem 11.
  • Section 21.4: 11(c), *11(e), 11(f), *13.
  • Solutions to assignment 2 will be distributed in class.
• Assignment 1, due 1/23. Greenberg 21.2: 9(d,h), 11(b,d); 21.4: 4(f), 5(c,f), 8(c,f).
• Solutions to Assignment 1.

Course handouts:

• Course organization and information.
• Electronic reserve material on calculus of variations.