MATH 4111 Advanced Calculus/Intro. to Analysis
This 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 below), want a broader exposure, and are willing to work hard to
acquire it.
Time and Place: Tuesday, Thursday,
Instructor: Edward N. Wilson
Office: Cupples I, Room 18 (in the basement)
Tentative Office Hours: MW
Office Tel: 935-6729 (has voice-mail)
E-mail: enwilson@math.wustl.edu
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 based math course,…)
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 about as well as any book I know 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 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 150 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.
Topic Outline:
I.
Set theory and logic. We’ll go over the set notations and
definitions in Chapter 1 but will go on to quickly discuss the controversies
underlying them. In brief, set theory
and logic are two halves of the same subject (often called symbolic
logic). Sets (intuitively, a set is a “well
defined collection” or “basket” containing things called the
elements of the set) are the objects of discourse not just in all of mathematics
but in every form of critical thinking. Which collections are sets and which
are not depends on which principles of logic (axioms for set theory and logic)
are adopted plus willingness to assume that these principles are not inherently
contradictory. The launch pad for mathematics is the Axiom of Infinity: without it, the collection of natural numbers
and other infinite collections aren’t sets. Using the Axiom of Infinity,
one can quickly create set theory models for the natural numbers and the
rational numbers, thereby establishing a theoretical basis for grade school
mathematics. When, in addition to the
Axiom of Infinity, the Axiom of Choice is adopted, controversy erupts. Without the Axiom of Choice, the real numbers
don’t exist nor does calculus in the way it’s commonly
presented. With the Axiom of Choice,
there are many theorems claiming the existence of things which can never be
found by even the largest and fastest computer imaginable; some people argue
that such theorems ought to be “disallowed” and that the only
worthwhile theorems are those which translate quickly into computer algorithms
to find predicted quantities. As Math
4111 proceeds, we’ll see many theorems of the “can’t find it,
pie in the sky” type proved using the Axiom of Choice, then later see how,
by adding extra hypotheses, algorithms may be devised by which good computer
approximations of the predicted quantities can be found in nice cases.
II.
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, simultaneously 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 notions of a lim inf
and lim sup; we’ll use these notions
frequently.
III.
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 we’ll devote much of next semester 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.
IV.
Differential Calculus (Chapters 5,8,and
9). Here we’ll quickly review
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. After going over the basic
results, we’ll prove one of the most famous theorems in all of
mathematics: the Inverse/Implicit
function theorem for differentiable functions from R^(n+m) to R^m. This theorem is at the heart of every
assertion that a certain set of equations has a unique solution; as we’ll see,
the main tool in the proof is an easy metric space result which gives rise to
computer algorithms for finding solutions.
The same metric space result proves another famous theorem and again
does so in the form of a computer algorithm:
Picard’s Theorem on the existence and
uniqueness of solutions of systems of ordinary differential equations with
prescribed initial values. So, once
again, the main tools for proving fundamental calculus theorems come from
metric space theory.
V.
Integral Calculus (Chapter 6) We’ll do just enough of the theory of
Riemann integrals to prove the existence of Riemann integrals for continuous
functions on closed, bounded rectangles (the proof not surprisingly being yet
another application of metric space results) and the easy subsequent proof of
the Fundamental Theorem of Calculus. The
reason we won’t delve into Riemann integrability
of non-continuous functions or any of the Chapter 10 material on Riemann
integrals is that all of this is a “museum piece” superceded 100 years ago by the theory of Lebesgue integrals, the topic of Math 4121.
VI.
Interchange of Limit Theorems, Infinite Series, and
Power Series (Chapter 7). Like the
material in Chapters 5 and 6, nothing here should strike you as new; you’ve
already heard discussions of all of the main ideas. What you probably haven’t seen are the
proofs justifying various techniques; like everything else, they rely on
metric space ideas. If extra time is
available, we’ll dip into some other topics, e.g.,use of approximate convolution identities to prove Weierstrass’s
Theorem on approximating continuous functions on closed, bounded intervals by
polynomials and to prove basic results on Fourier series.
Exams/Homework: There will be an in-class exam at the end of September, either an in-class exam or a take-home exam in early November, and an in-class 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 = .24E1 + .24E2 + .32FinE +.20HW
Thus, each of the
mid-semester exams will be 24% of the final average, the final exam 32%, 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, work hard on homework, and show improvement 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
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.
The worst form of cheating on homework
is to go one of the Internet sites where you can pose questions and copy off
verbatim the posted solution without acknowledgment. This is flagrant academic dishonesty,
tantamount to plagiarism; when there is clear evidence that it occurred, the
evidence will be forwarded to the Arts and Sciences Integrity Committee with a
recommendation for a stern penalty.
Penalties can be avoided by acknowledging that the solution came from an
internet expert, but the grader will be asked to give only a small amount of
credit for such solutions in whose devising the student played no role.
A lesser form of cheating on homework is
for two or more students in the class to work together on a problem or problems
and to submit essentially the same solution(s) without acknowledging the help
of others. 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 this request 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.