RUTGERS UNIVERSITY  NEW BRUNSWICK
DEPARTMENT OF MATHEMATICS
MATHEMATICS 423:01 (P. D. E.) HOME PAGE 
FALL 2001
"Mutationem motus proportionalem
esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa
imprimitur." (Newton's Second Law of Motion,
F = d/dt[mv].)
The last section of notes on the Laplace
equation (an additional 14 pp.) is now posted below.
DIRECTORY

Course Materials: (except for the syllabus,
almost all linked names will be .pdf files)

Syllabus

Problem Sheets:

Supplementary Notes (many from Spring 2001see the
footlines):²

Notes on Leibniz's formula (see also Strauss's
A.3) and the little problem at the end of the first lecture last spring.

Notes on discretizing pdes (featuring both the
heat and wave equations, in space dimension 1 only).

The details of deriving the plane Laplacian in polar
coördinates (see also Strauss, pp. 150151).

Notes on the maximum principle for the heat and
Laplace equations (originally from 2/6/2001).
 A new (9/25/01) version of notes on the Cauchy (Poisson) kernel for the Laplace
equation in the upper half plane (and related matters).
 Notes on SturmLiouville orthogonality.
These have now been reëdited for Fall 2001.

Notes on an example.

Notes cleaning up the transportequation
derivation of 9/6/01.
 Notes on an algorithmic approach
to classifying secondorder linear p. d. operators with constant
coefficients.
 Notes giving a proof of the RiemannLebesgue Lemma.
 Notes on the convergence of Fourier
Series, in (presumably) final form.
 Notes, Part I, on the Laplace equation,
including some connections with functions of a complex variable.
These contain some material not (yet) dealt with in class; you may
want to confine your attentions to §§5 through 8.
 Part II of the notes on the Laplace
equation. These include corrections for the original
pp. 1516, whose hypotheses for the Schwarz reflection principle made
it too good to be true.
 Part III, the final installment, of the
notes on the Laplace equation (which seems to get less and less
connected with complexvariable stuff).

Ch. 1: §1.1: 2, 3, 5, 7, 9, 10. §1.2: 1, 3, 5, 6,
8. §1.3: 2, 3, 5, 9. §1.4: 1, 2, 3, 5. §1.5:
1, 2, 4. §1.6: 1(a,b), 3, 5, 6.

Ch. 2: §2.1: 5, 6, 8, 9. §2.2: 1, 2, 5, 6.
§2.3: 1, 2, 3, 6, 8. §2.4: 1,
4, 6, 9, 15, 16, 18, 19. §2.5: 1, 2*.

Ch. 4: §4.1: 1, 2, 4, 5. §4.2: 1, 2, 3. §4.3: 1, 2, 3,
4, 5, 6, 7 (with a [quasi]physical explanation), 9, 11, 17.
 Ch. 5: §5.1: 1, 2, 3 (use a machine!), 4, 5, 9, 10, 11.
§5.2: 3, 4, 5, 6, 9, 13, 14, 15, 16. §5.3: 3, 4, 6, 8, 10,
11, 13, 15. §5.4: 1, 2, 6, 8, 9, 10, 11, 16, 18, 19, 20.
§5.5: 2, 3, 4, 5, 6, 11, 12. §5.6: 1, 2, 3, 6, 7, 13.

Ch. 6: §6.1: 2 (NB: substitute u = v/r, that is, v =
ru [book has a typo]), 4, 5, 6, 10, 11, 12. §6.2: 2, 3.
§6.3: 1, 2, 3 (typo in the trig identity). §6.4: 1, 2, 3,
6, 11, 12(b) (prove the uniquenessyou don't need the Hopf maximum
principle).
¹Albert Einstein® is a trademark of the Hebrew University
of Jerusalem, represented by the Roger Richman Agency, Inc.,
www.alberteinstein.net. However, to the best of my knowledge Isaac
Newton is not a trademark of the Isaac Newton Institute of the
University of Cambridge (UK).
The notes and solutions posted here are class materials. Class members
at Rutgers University may copy them for their personal use. Copyright
rests in Roger D. Nussbaum(³) and
Bertram Walsh(²,³). Duplication of these materials for
commercial purposes is expressly prohibited.
original page last revised 0835 EST 12/11/2001
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