Mathematics 4111
Fall, 2011
Instructor: Edward N. Wilson
Cupples I, Room 18
E-mail: enwilson@math.wustl.edu
Office Tel: 935-6729 (O.K. to leave messages, much better
to use e-mail since voice-mail messages may not be heard for several days)
Office Hours: Tu
Class Meeting Times and Place: Tu Th 10-11:30,
Textbook: Introduction to Analysis, Maxwell
Rosenlicht,
Other
Recommended Texts: There are literally hundreds of textbooks covering the
material in this course. Each has its
pros and cons. Thus, Rosenlicht’s book does metric spaces very well and
there’s much to be said for his approach to the material in Chapters 7-9. But many of his notations are awkward and
non-standard and the fact that he doesn’t use matrix notation and vector space
norms in Chapters 5-9 is a large drawback.
A well-written book which is weak where Rosenlicht is strong and strong where Rosenlicht is weak is Gerald Folland’s Advanced Calculus (published by Addison Wesley in 2002). I also like very much the first 100 pages or
so of Kennan Smith’s Primer of Modern Analysis (published by
Springer in the early 1980’s with some very cheap used paperback copies
available from Amazon) but am not keen on the rest of the book. Worldwide, for over 50 years, Rudin’s Principles of Mathematical Analysis (McGraw-Hill, Second Edition,
1964) has probably been used more often as a text for a course of this kind
than any five other books combined. Rudin’s style is concise and dry:
this has appeal to some and is decidedly unappealing to others.
Prerequisites: Math 233 or its equivalent is a hard
and fast requirement with Math 318 or its equivalent useful but not
essential. Math 309 is very strongly advisable as is some prior exposure
to theoretical ideas in mathematics (e.g.,
outside reading, freshman seminar, Math 310, Math 429 or other theoretically
oriented math course,…)
Overview:
Math 4111 is a course in the theory
of calculus with a radically different style from the 100-300 level calculus
courses. The emphasis is on proving theorems, not on calculation
techniques or applications of calculus to other quantitative
disciplines. The homework and exams will be heavily slanted toward
providing proofs of various statements with very few "routine
calculation" problems. It's not assumed that students have had prior
experience with devising proofs. To the contrary, one of the goals of the
course is to help students develop proof skills. Although the lectures will try
to provide examples illustrating various theoretical ideas, the bulk of class
time will be the proverbial repitition
of definition, theorem statement, proof, next definition, theorem statement,
proof,...
There
are 3 types of students for whom Math 4111 is designed:
(1)
Undergraduates with
good mathematical backgrounds who anticipate going on to do a Ph.D. in a
quantitatively-based discipline;
(2)
Graduate students in a
quantitatively-based discipline who wish to understand the theoretical
underpinnings of every approach to modeling;
(3) Students who have had only a brief prior exposure to mathematical theory (see Prerequisites above), want a broader exposure, and are willing to work hard to acquire it.
Topic Outline:
Sets and set notations. This material is briefly covered in Chapter 1
of the text and covered in more depth in the Notes on Sets and Set Notations
posted on the course web site. We won’t
cover this material in class since much of it is likely familiar to many. Everyone should read carefully Section 1 of
the Notes, scan Section 2 lightly, and read Section 3 carefully.
I.
Real number system. The
discussion of the real number system (more precisely, properties of real number
system models and various set-theoretic constructions of models) in Chapter 2
is a good illustration of Kronecker’s
famous adage: “God created the integers,
everything else is man-made.”
Translating, the Axiom of Infinity is “natural” and it creates the
natural numbers and the integers, simulataneously
establishing their basic properties, but everything else (the real numbers and
their properties, calculus, constructions of other sets using real numbers and
associated theorems) is highly abstract, very artificial, and rooted in the
controversial Axiom of Choice. One of
the important ideas which Rosenlicht
buries in one of his problems sets are the lim
inf and lim sup notions; we’ll go over these notions in class
and use them frequently.
II. Metric
Spaces (Chapters 3 and 4). In this
section, we’ll prove all of the important properties of continuous functions on
compact subsets of a metric space. This
specializes in the case of the metric space R^n to the theorems on which calculus and advanced
calculus are based: existence of maxima
and minima along with uniform continuity for continuous R-valued functions on a
closed, bounded subset of R^n. Why don’t we spare the business of general
metric spaces and just prove these theorems for R^n? There
are two answers to this question. First,
there are lots of interesting metric spaces other than subspaces of R^n and much of 4122 is devoted to studying
some of them in detail. Second, proving
theorems on general metric spaces is easy since all we have to work with is a
distance function satisfying the triangle inequality; with no other clutter around (R^n is very cluttered), proofs
will of necessity just entail repeated use of the triangle inequality. In our examples of metric spaces, we’ll make
heavy use of vector space norms (another very important notion buried by Rosenlicht in a problem section)
and will prove a few key results about norms; these things will be used almost
“daily” for the remainder of Math 4111 and will be used often in Math
4121.
III.
Differential and Integral Calculus (Chapters 5-7). Here we’ll start
with a review of the definitions and properties of ordinary derivatives,
partial derivatives, and differentiable functions from R^n to R^m.
This is where some background in linear algebra is essential. The main tool in establishing basic theorems
about differentiable functions is the Mean Value Theorem which in turn rests on
the existence of maximum values for R-valued continuous functions on a closed,
bounded interval. This is one of the
places where the metric space results pay off in a big way.
Next we’ll go over just the basics of the theory of
Riemann integrals for bounded functions from an n-dimensional rectangle into
R. The nice part of the theory is
integrals of continuous functions; this rests on another metric space
result, the uniform continuity of continuous functions on compact sets. The Fundamental Theorem of Calculus is an
immediate corollary. We’ll resist the
temptation to do a “partial cleanup” of the sloppy treatment of curve and
surface integrals in Calculus III; the best treatment of these integrals
needs tools from Differential Geometry and is best left to course like Math
5041. Also, students should be aware
that the Riemann theory is 100 years out of date for integrals of discontinuous
functions and integrals over non-smooth domains. The Lebesgue theory
of integration (the subject of Math 4122) handles these topics nicely, unlike
the horrendous mess for the Riemann theory presented in Chapter 10; we’ll say
nothing about Chapter 10 in class and only those who have a taste for very bad
jokes should look at it.
We’ll conclude with discussion of various
interchange of limits theorems, including term-by-term integration and
differentiation as well as differentiation under an integral sign.
IV.
Picard’s Theorem
and the Inverse/Implicit Function
Theorem (Chapters 8 and 9) These are two of the most famous and
most heavily used theorems in mathematics.
The heart of the proof for both theorems is an application of yet
another metric space result: the fixed point theorem for metric space contractions.
But getting to the point where the fixed point theorem can be applied takes a
while and much more needs to be done to fully clean up the theorems. In brief, neither of these important theorems
has a quick and easy proof.
V.
Other topics.
As time permits, we’ll delve briefly into such topics as convolutions
and their application to proving approximation theorems, Fourier series, and Fourier integrals.
Homework: There will be weekly homework assignments except for weeks
in which an exam will be given. Homework
assignments will either be given out in class or posted on the course website
(or both).
Most of the
problems will be taken from the textbook.
Homework should either be turned in during class on the due date
(preferred method) or slipped under Prof. Wilson’s office door at some time
before mid-afternoon on the due date. Once the grader has picked up homework
papers, additional papers will NOT BE CAREFULLY GRADED AND WILL RECEIVE LITTLE
OR NO CREDIT. Obviously, it’s much
better to hand in partial solutions on the due date and receive substantial
partial credit than be late will full solutions and receive almost no
credit. Unlike examinations, it’s
permissible for students to talk to each other about homework questions; often
this is the best way to gain insight.
However, each student must write up her/his own homework
assignment. Students are asked to jot
down the names of their homework collaborators at the top of the first page of
their solutions. Those who don’t may
receive a substantial penalty.
Collaborating
with other students on homework is not cheating. However, going to a math site on the web and
asking the site manager to solve one or more homework problems is cheating.
PLEASE
DON’T DO THIS.
Examinations: There will be a total of three examinations in 4111:
II. a take-home
exam to be handed out in early November;
III. a two hour final exam whose date is determined by the
College of Arts and Sciences and is inflexible.
At least
half of Exams I
and III will be short essay questions:
statements of theorems, sketching proofs done in class, easy true/false
questions to be proved or disproved by a counter-example, etc. The
remaining parts of these exams and nearly all of Exam II
will involve
selecting a certain number problems to
be carefully worked out from a list of problems. Most of these problems will require proofs of
various statements. Some may involve tricky computations. Exam problems will often be similar to
homework problems but may, on occasion, be somewhat
more difficult.
Students
who miss an exam for one of the valid reasons permitted by the
Academic Integrity: When there is evidence strongly suggesting
that cheating took place on an exam, the evidence will be forwarded to the Arts
and Sciences Integrity Committtee. If, after a hearing with the instructor and
affected student(s), a majority of the Committee members are convinced that
cheating did occur, the Committee will assess a penalty and inform the
instructor and sudent(s) of its decision. For exam cheating, the most common penalties
are either a failing grade on the exam or a failing grade in the course. In brief, please don’t cheat since cheating cases,
however they turn out, are very painful for all concerned.
Grading Scale: Each of the two in-class
exams will account for 20% of the course grade average, homework will account
for another 20%, and the final exam will account for the remaining 40%. Preliminary letter grades for exams and
course averages will be determined as follows:
A 90-100%
B 75-90%
C 60-75%
F below 60%
In no case
will final grades be lower than indicated by this scaling. However, the instructor reserves the right to
switch to a more lenient scaling if he decides that it is warranted.