As the title suggests, this course will survey a variety of methods useful
for applications of mathematics to the physical sciences. Particular
topics will include:
real and complex power
series, vector algebra, partial derivatives and multiple integrals
in polar and spherical
coordinates as well as rectangular coordinates (e.g. expressing
gradients, divergences,
and curls in spherical coordinates), Stokes and Divergence Theorems, the
Legendre function, Fourier Series and its applications to solving partial
differential equation systems (heat equation, wave equation) with nice
boundary conditions, and a little bit on the calculus of variations.
Some of the 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, 411-12, 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, 2:00-3:00, Sever 102
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 requirement. 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
Textbook: Mathematical Methods in the Physical Sciences, Second Edition, Mary L.Boas,John Wiley & Sons, 1983.
Topic Outline: 1.
We'll begin with 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.
2. Chapter 3 is basically a review of the Math 233 approach to vectors.
We won't go over this
in class--instead will dwell on the physical assumptions underlying use
of rectangular coordinate systems, what happens when coordinates are changed,
why n-dimensional volumes are always related to determinants of n x n matrices,
and a brief discussion of how vector techniques with finite or infinite
dimensional vector spaces come up in every phase of modern physics:
standing waves and traveling waves, statistical mechanics and other uses
of statistics, quantum mechanics, special and general relativity, etc.
3. Chapter 4 is a fairly standard treatment of partial derivatives.
We'll rapidly review
some things mentioned in 233, then dwell on the matrix formulation of the
chain rule and techniques for rapidly computing things like the divergence
of a vector field in spherical coordinates. In brief, a very common
problem for physics students is inadequate understanding of what the chain
rule "really" says and how to use it for quick calculations. As with
Chapter 2, the lectures will try to fill in the things which Boas doesn't
mention and indicate how they come up frequently in thermodynamics.
4. We'll spend essentially no time on Chapter 5 since it's just a review
of 233, but will instead move quickly into the later part of Chapter 6,
covering the material on Stokes' Theorem and the Divergence Theorem to
which 233 normally doesn't get.
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.
5. Chapter 7 (Fourier series) will be new to virtually everyone.
It uses complex exponentials, some easy integral results, and a "ton"
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.
6. 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 Washington University courses, cheating on exams will be
taken very seriously
with evidence supporting a cheating allegation forwarded to the Arts
and Sciences Integrity
Committee for adjudication. When the Committee concludes that a
student cheated on an
exam, it normally directs the instructor to assign the student a
failing grade for the
course.
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.