MATH 308 Mathematics for the Physical Sciences
Spring, 2008
As the
title suggests, this course will survey a variety of methods useful for
applications of mathematics to the physical sciences. The two big topics
are the following: (i)
vector algebra and vector calculus including grad, div, and curl operations in
polar, cylindrical, and spherical coordinates as well as rectangular
coordinates, Stokes’ Theorem, and the Divergence Theorem;
(ii)
real and complex power series, going
on to Fourier Series and its applications to solving partial differential
equation systems (heat equation, wave equation) with nice boundary conditions.
After covering
these topics, we’ll touch a bit on some other topics as time permits.
Possibilities include: properties of the Legendre
function and some of its applications; calculus of variations; Fourier
transforms. Some of the course material will overlap with that of Math
233. The rest will build on 233 and will introduce lots of ideas pursued
in greater depth in other mathematics courses (217, 309, 429, 4111-4121, 416).
All germane
"fundamental theorems" will be carefully stated and illustrated with
proofs given only when the proofs are relatively easy and instructive.
Time and Place: Monday, Wednesday and
Friday,
Instructor: Edward N. Wilson
Office: Cupples I, Room 18 (in the basement)
Office Hours: MWF 1:00-2:00 and by appointment
Office Tel: 935-6729 (has voice-mail)
E-mail: enwilson@math.wustl.edu
Prerequisites: Math 233 is a hard and fast prerequisite. It's not obligatory for 308 students to have taken 217 (Differential Equations) or 309 (Matrix Algebra) or to be concurrently registered in one of these courses but some topics we cover in 308 will be most easily understood by those who have had an exposure to differential equations and here and there we’ll use some matrix algebra tools.
Textbook: Mathematical Methods in the Physical Sciences, Second Edition, Mary L.Boas,John Wiley & Sons, 1983.
Reference Text Especially Recommended for Physics
Majors: Div, Grad, Curl, and all that;
an informal text on vector calculus, (many editions), H. M. Schey, W.W. Norton & Company. This is a famous book written for science
majors; it has some excellent problems and does a much better job than the
textbook of explaining applications of Stokes’ Theorem and the Divergence
Theorem to the theory of electricity and magnetism and the theorem of
hydrodynamics. A fair number of homework
problems will be taken from this book.
One copy is on reserve in the Physics Library; if need be, we can arrange to have a
second copy put on reserve. Amazon has
some used copies selling for roughly $25 plus some new copies selling for
around $33.
Topic Outline: 1. We'll
begin with a fast review of partial derivatives, multiple integrals, and the
Chain Rule (Chapters 4 and 5 with a small amount of Chapter 3). We’ll
also discuss the Implicit function theorem and mention how most of
Thermodynamics revolves around this theorem plus the Chain Rule.
2. Vector calculus (Chapter 6). Those
who are planning on taking Electricity
and Magnetism next fall should bear down hard on this material. It will
be the backbone of the mathematical part of E&M.
3. Chapters 1 and 2 in the text,
quickly reviewing the standard things about real power series, then going on to
complex power series with particular interest in the complex exponential
function and indications why it's essential for doing problems in mechanics and
electricity.
4. Fourier
series (Chapter 7) will be new to virtually everyone. It uses complex
exponentials, some easy integral results, and a multitude of vector ideas
applied to function spaces. We'll give some
applications of Fourier series to a variety of situations. We won't say
much about how Fourier ideas are at the heart of quantum mechanics since
virtually everything about the axioms of quantum mechanics is so "off the
wall" that it needs an entire semester of analyzing experimental data
before anyone is ready to grudgingly admit that there might be something useful
in the theory. But we will give a brief discussion of modern coding
and signal processing methods with an indication of the way in which they
evolved from Fourier series ideas.
5. After
getting through the above material, we'll start "jumping"
around with quick
exposure to the ideas in certain sections of Chapters 9-13. How long we
spend with each of these topics will depend to a considerable extent on student
interest as well as the pragmatic issue of how much time remains in the
semester.
Exams/Homework: There will be two in-class exams during the semester as well a two-hour final exam during finals week. There will usually be a homework assignment to be handed in each week.
Grading: Final averages will be
determined by the following formula:
Final average = .25E1 + .25E2 + .30FinE +.2HW
Thus, each of the
mid-semester exams will be 25% of the final average, the final exam 30%, and
homework 20%. That said, a different formula
giving the final exam higher weight will be used for those who do poorly on
either of the mid-semester exams but improve considerably on the
final. Also, the process of converting final averages to letter grades
won't, from a student's point of view, be any worse than the traditional 90-100
A, 80-90 B,
70-80 C scale but might well be better, i.e., more generous.
Academic
Integrity: As
with all
Cheating on homework consists of either blindly copying off someone else's
solutions or
not acknowledging the receipt of assistance from others in completing the
assignment. It's
not anticipated that students will work in isolation on homework
problems. To the contrary,
discussing problems with others is often a way to avoid frustration and gain
useful insight.
However, all students are expected to write up their own assignments and to indicate in a short note at
the top of the first page the names of any people (other than the instructor) with
whom they discussed the problems or from whom they received some hints. Violation of
these requests will result in an instructor-imposed penalty (e.g., something like half credit
for the assignment) but won't be treated as a "hanging" offense--in particular, won't
be brought to the attention of the Arts and Sciences
Integrity Committee.