The paragraph below can
be skipped by those who took Math 411 in Fall '03. It should be read
by all others.
WHAT IS MATH 411-412 ABOUT?
Mathematics 411-412 presents the theory of one and several variable calculus.
It is required for those wishing to complete the so-called "traditional"
math major, a group normally limited to students contemplating going on
to do graduate work in mathematics, physics, mathematical economics, or
some other highly quantitative discipline in which theory plays a large
role. Students who are not traditional math majors are welcome to
take the course provided they have an interest in understanding the theoretical
basis for calculus.
Those interested in building up their prowess with techniques for doing
calculus problems and uninterested in understanding why the techniques
work are STRONGLY ADVISED NOT to take 411-12. While it's true that
occasionally techniques will be introduced in 411-412 which aren't covered
in other calculus courses, these will be few and far between; also, most
of the lecture time will be devoted to the theory of why these methods
work rather than illustrations of the calculations. In brief, there
is virtually nothing in common between the style of earlier calculus courses
(nothing proven, hand-waving justifications, reliance on physical intuition,
working out numerical exercises, developing computational skills) and the
style of 411-12 (nearly everything proven with a fairly high degree of
rigor, at least two thirds of the homework and exam questions theoretical
with only a few numerical problems tossed in for good measure, emphasis
on building up skills at proving theorems and developing mathematical intuition).
Many 411-412 students may have gotten a taste of theory in Math 310 or
a similar course taken elsewhere; others may never have gone past the statements
of unproved theorems in Calculus 1-3. In any event, it will not be
presumed that students have any prior experience in proving theorems or
working out well-reasoned arguments responsive to theoretical homework
questions. To the contrary, one of the main goals of 411-12 is to
provide this kind of experience.
Instructor: Edward
N. Wilson
Office: Cupples I, Room 18 (in the basement)
Office Hours: Tu 11:30-noon, Th 11:30-1 and by appointment
Office Tel: 935-6729 (has voice-mail)
E-mail: enwilson@math.wustl.edu
Textbook: Primer of Modern Analysis, Kennan Smith, 1983 edition.
Pre-requisites: A course like 411 giving a good theoretical foundation for calculus of one one-variable plus a good working knowledge of linear algebra (preferably, a course like 429, but a course like 309 together with a little extra reading will suffice).
Exams: There will be either one or two mid-semester exams plus a final examination. Likely at least one of the exams will be a take-home exam.
Topics in 412:
We'll
begin with standard definitions of differentiability and affine approximations
for functions of several variables. After some preliminaries, we'll
go over in considerable detail the proofs of the Inverse and Implicit Function
Theorems for functions of several variables, then give some illustrations
of the power and utility of these theorems. For example, if interest
warrants, we may spend some time going over the rudiments of the theory
of ordinary differential equations and its link with the theory of curves
and surfaces. Be advised that while we'll need some things from linear
algebra, the chapters in the text on linear algebra will not be covered
in class. Whether or not we take time to lay the groundwork for Stokes'
Theorem in some setting (the proof of the theorem is trivial--just the
Fundamental Theorem of Calculus--but the groundwork is decidedly non-trivial)
will depend on the interests of the class.
After
handling multi-variable differentiation, we will NOT go on to multi-variable
Riemann integration. Instead, we will go through the introduction
to Lebesgue integration given in the text and use it to prove Lebesgue
integral versions of the Fubini theorem (when it's O.K. to reverse the
order of integration), the multi-variable change of variable theorem, and
a few other things about Lebesgue integrals. This will set the stage
for going back to the Lp spaces mentioned in 411 and proving
some basic results in Fourier analysis. As interest warrants, we
may go on to discuss the Fourier transform and prove some of its properties.
Homework: There will be weekly homework assignments to write up and hand in. Usually the homework will be due on Tuesday and the assignment handed out the previous Tuesday. The homework will consist largely of selected exercises from the textbook. Occasionally the instructor will add a supplementary problem or two.
Grading: The exams will count for 70% of the final grade with homework making up the remaining 30%.
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 assignment or not acknowledging the receipt of assistance from others
in completing the exam. 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.
Homework Assignments: Click on the assignment number
below: