## Partial Differential Equations Math 110B, Winter 2013:

under construction

Office hours: MW 3:30-4:30 and by appointment (just talk to me after class or send me an email)

Office: APM 5256, tel. 534-2734

Email: hwenzl@ucsd.edu

Course book: Walter A. Strauss: Partial Differential Equations, Wiley & Sons, 2nd edition.

Teaching assistant: Michael White, APM 5712, T:10am-12pm

Computation of grade: (tentative) The grade is computed from your scores in the final (50%), 1 midterm (25%) and homework (25%), with passing grades for exams required. Although homework counts comparatively little, it is extremely important that you do it, as most of the exam problems will be very similar to homework problems. It is OK to compare homework notes or to discuss problems with other students; just copying someone else's homework, however, will not count.

Dates of exams: No make-up exams!

Midterm: 2/13 in class; for further information click at link for homework below

Homework assignments Homework is to be turned in in class on the given date at the latest. The TA also got us a drop box.

for 1/16: Section 9.1: 1, 4(a)-(c), 5, Section 9.2: 3, 6(a)-(d) (you may assume that the center of the sphere S is of the form x_0=(0,0,z), if you wish; also when you use the Kirchhoff formula. Also look at the hints at the end of the problem).

for 1/23: Section 9.2: 9(a), 16, Section 9.3: 1, 5, 9

for 1/30: Section 9.1: 2 (and check that the singuarities are on a characteristic surface), Section 9.4: 1, 2, and see below

for 2/6: Section 9.4: 4, 5 (hint: Show that the H_k satisfy the recursion relation H_{k+1}=2xH_k-H_k'; use this to show by induction on k that H_k satisfies Hermite's equation (16) in the book with \lambda = 2k+1). Section 10.1: 2, 4(b),(c)

Midterm: The material of the midterm will go until including the last homework assignment. Moreover, the following problems below may also be relevant.

relevant for midterm: Section 9.4: 6 (you could also use the recursion formula)

for 2/20: Section 10.2: 1, 2, 3, 4, Section 10.5: 3, 4,

for 2/27: Section 10.6: 2, 5 (see formulas (5) and (6) in Section 10.6; the inner product here is given by the integral from -1 to 1), 6 (see Example 2 in Section 10.3), Section 10.3: 4, 6 (I will say more about the last two problems on Monday; I will also try to put some relevant notes on the web some time this weekend).

for 3/6: 10.7: 2(see below), 3(a)(b), 4 (hint: show for k>2\ell that a_k > [\beta/(k+1) ] a_{k-1})

Comments and hints to Problem 10.7.2: You can find the definition of L_x, L_y, L_z on the bottom of page 294. You have to show that (L_xL_y-L_yL_x)(f)=iL_z(f) for any sufficiently often differentiable function f=f(x,y,z). You need not do it for the other two identities, it is more or less the same calculation. This exercise is important for the following reason: The operators L_x, L_y and L_z satisfy the relations of the three-dimensional Lie algebra so_3. This is a mathematical structure closely related to the matrix group SO(3) of orthogonal 3 x 3 matrices with determinant equal to 1. It can be explicitly checked that applying one of these operators to a function Y^m_l, we again get a linear combination of functions Y^m_l, with m possibly varying (see formulas (9),(10) on page 295). One can deduce from this that rotation of the coordinate system changes a given solution Y^m_l (with m fixed) to a linear combination of solutions Y^m_l, with all of these functions having the same l=\ell.

for 3/13: Section 11.1: 1, 5 (hint: Use Green's first identity to prove a formula similar to (1) in the notes for Chapter 11.1 - it will contain an additional term here), Section 11.2: 6, 7. Section 11.6: 3

Final: The same rules apply as for he midterm: you can use your book and a cheat sheet. The problems will be similar to homework problems.

office hours for exam week: Tu:2-4