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.

Math 412 Update: The above paragraphs are copied over from the Math 411 syllabus.  They're still applicable to Math 412.  But specifically, in Math 412, we'll begin with going over some of the topics in Chapters 6 and 7 (sequences and infinite series, uniform convergence, quick look at power series, distinction between conditional and unconditional convergence, interchanging the order of summation in a double series), then embark on integration theory.  All of Chapter 4 will eventually be subsumed under our discussion of Lebesgue measure and general integration theory; we won't take time in class to go over the Chapter 4 material but students will be asked to read through it for background on why the Lebesgue integral is so much better than the Riemann integral.  Notes on Lebesgue measure will be handed out.  Those who want to see more on measures and integrals may wish to consult some of the books mentioned below in the Reference section (or others like them).  After concluding our discussion of integration theory, we'll come back to Folland's book and finish off the rest of Chapter 7 as well as Chapter 8 and, as time permits, a very brief look at the Fourier transform.

Time and Place: Tuesday and Thursday 10:00-11:30 in Cupples I, Room 218

Instructor: Edward N. Wilson
             Office:  Cupples I, Room 18 (in the basement)
             Office Hours:  M-W 11:00-12:30 and by appointment
             Office Tel:    935-6729 (has voice-mail)
             E-mail:        enwilson@math.wustl.edu

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

Reference Books: In Math 412, we'll spend quite a bit of time going over Lebesgue integration.  Folland's book only mentions this topic and gives no details.  In the library, you can find many books covering this topic in considerable depth.  Most of these are graduate textbooks and the difficulty with them is that they have too much material and may take a different approach than the one we'll discuss in class.  There aren't many books written for undergraduates with a chapter giving the highlights of measure and integration theory.  Two that do are mentioned below;  a third is a new book by Charles Pugh.  Among the graduate texts, one of the best is by Folland; I believe the title is Real Analysis.  Other old standbys include Real and Complex Analysis by Walter Rudin (not to be confused with Rudin's undergraduate text) and Real Analysis by Halsey Royden.
       (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.
 

Exams:  There will be two mid-semester exams, one in mid-February and the other in early April.  At least one will likely be a take-home exam. The final exam will also likely be a take-home exam given out the last day of classes in April 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.