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; 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:  M, Tu, and Th 1:00-2:00 and by appointment
             Office Tel:    935-6729 (has voice-mail)
             E-mail:        enwilson@math.wustl.edu (or just click on the highlighted name above)

Textbook:  Advanced Calculus, by Gerald Folland, Prentice-Hall, 2002.

Reference Books: The library has many books with titles using the words "Advanced Calculus" or "Introduction to Mathematical Analysis".  Many of these books have been used as texts for 411-412 in previous years and go over much of the same material as Folland's book.  In general, I don't have a very high impression (Folland's book excepted) of recent books--some with publication dates 5-15 years ago have astonishingly large numbers of outright blunders.  Three old standbys I like (for different reasons) are the following:
       (1) Principles of Mathematical Analysis, by Walter Rudin, McGraw-Hill, 1976 (and earlier editions).  Nationwide (and worldwide), Rudin's book has likely been used as a text for courses like 411-412 more than any other book.  The style is dry and "off-putting" to some students.  Also, Chapter 1 not only does things with sets but goes further than necessary with a discussion of cardinality for sets and other fairly abstruse ideas.  But thereafter, the topic selection is superb and Rudin's proofs are generally optimally short and direct without a lot of "beating around the bush". Like Smith's book and many others, Rudin first does essentially "everything" for the real line before going on to R^n and other spaces. He also includes some discussion of metric spaces and Lebesgue integration.
        (2) Primer of Mathematical Analysis, by Kennan Smith, Springer Verlag, 1983.  Smith's book was used as a text last year in 411-12.  His style is witty and sometimes anecdotal.  Part I is a fairly nice treatment of everything for the real line.  Part II begins with metric spaces (including R^n) and contains some results not found in most comparable books.  But the readability of the book goes downhill from early in Part II onward and even in Part I, some of Smith's proofs are more convoluted than necessary.  He also has an irritating habit of leaving the key steps in many proofs as exercises for the reader.  In general, I like Smith's topic selection, like the fact that he does an introduction to Lebesgue integration and includes deep results like the Weierstrass Approximation Theorem,  and gets across the key point that metric spaces are natural and useful
for many applications where R^n is irrelevant.  There is still the downside already mentioned.
          (3) Mathematical Analysis, by Tom Apostol, Addison-Wesley, 1974.  Apostol's book happens to be the one I studied from when I was learning this material.  At the time, I thought it was hard going, but liked it more and more as time went on.  The overlap in style and topics with Rudin's book is large.

Topics:  Over the course of the year, we'll cover essentially all of Folland's textbook plus additional material (especially on metric spaces and Lebesgue integration) provided by the instructor.  The order in which we treat topics is not set in stone.  We'll begin with Chapter 1 of Folland's book in order to get adjusted to his style and notation.  This Chapter includes all of the fundamental theorems about continuous functions.  These theorems and their proofs should be very carefully studied--Exam 1 will ask students to reproduce the proof of at least one of these theorems.  After Chapter 1, we could skip to the end of the book and do the chapters on infinite series and Fourier series along with a brief introduction to integration theory (both the Riemann variety presented in Chapter 4 and the Lebesgue variety only hinted at in the text).  Alternatively, we could proceed more or less in the order in which the book presents things and do Chapter 2 (differentiation theory for functions of several variables) followed by Chapter 3 (implicit and inverse function theorems and their application to surface theory), Chapter 4 (integration theory), and Chapter 5 (various versions of Stokes' Theorem) before getting to infinite series.  Which of these two orders we follow (or some mixture of the two) will be determined largely by student interests.  In brief, the surface theory and Stokes' Theorem topics are highly relevant to modern differential geometry and theoretical physics, while infinite series are relevant to coding theory, data analysis, signal processing, and a variety of other applications.

Exams:  There will be two mid-semester exams, one on or around Sept. 28 and the other on or around Nov. 4--on or around means we could shift these exam times up or back a few days if the indicated times are problematic for a number of students.  As indicated above, each of these in-class exams will ask for the proof of one or more theorems discussed in class.  There will also be questions asking for definions, examples, and counter-examples for the material covered by the exam. Finally, there will be some problems not previously covered in class but often analagous to homework problems. The final exam will very likely be a take-home exam given out in early December and due back in roughly a week.  We will settle on the exact dates when everyone has a firm schedule for exam week. The final exam will consist entirely of problems not covered in class.

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 of a mixture of selected exercises from the textbook and supplementary exercises made up by the instructor.

Grading:       Each of the two mid-semester exams will count 20% toward the final grade, the final exam will count 30%, and the final homework average the last 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 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.

Homework Assignments: Click on the assignment number below: