Math 218 Notes

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AN INTRODUCTION TO FUNCTIONS OF SEVERAL REAL VARIABLES Robert C. Gunning c Robert C. Gunning ii Introduction I have taught the honors version of calculus of several variables at Princeton University from time to time over a number of years. In the past I used Michael Spivak’s path breaking and highly original Calculus on Manifolds as the course textbook, supplemented by one or another of the available excellent textbooks covering the standard approaches to calculus in several variables and i
  AN INTRODUCTION TO FUNCTIONS OFSEVERAL REAL VARIABLES Robert C. Gunning c  Robert C. Gunning  ii  Introduction I have taught the honors version of calculus of several variables at PrincetonUniversity from time to time over a number of years. In the past I used MichaelSpivak’s path breaking and highly srcinal Calculus on Manifolds as the coursetextbook, supplemented by one or another of the available excellent textbookscovering the standard approaches to calculus in several variables and includinga wide range of examples and problems. As usual though, for anyone who hastaught a course from the same textbook more than a couple of times, there aresome parts of Spivak’s treatment that I would like to expand, and some partsthat I would like to contract or handle differently. During the 2007-2008 aca-demic year Lillian Pierce and I cotaught the course, while she was a graduatestudent here, and reorganized the presentation of the material and the problemassignments; her suggestions were substantial and she played a significant partin working through the revisions that year. The goal was to continue the ratherabstract treatment of the underlying mathematical topics but to supplement itby more emphasis on using the theoretical material to solve problems and tosee some applications. The problems were divided into two categories: a firstgroup of problems testing a basic understanding of the essential topics discussedand their applications, problems that almost all serious students should be ableto solve without too much difficulty; and a second group of problems coveringmore theoretical aspects and tougher calculations, challenging the students butstill not particularly difficult. The temptation to include a third category of optional very challenging problems, to introduce interested students to a widerrange of other theoretical and practical aspects, was frustrated by a resoundinglack of interest on the part of the students. A team of Princeon undergradu-ate mathematics majors consisting of Robert Haraway, Adam Hesterberg, JayHolt and Alex Schiller took careful notes of the lectures that both Lillian and Igave. In the subsequent years I have used these notes when teaching the courseagain, modified each year with revisions and additions using the very helpfulcorrections and suggestions from the students taking the course and the gradu-ate assistants coteaching and grading the course. The resulting notes still followto an extent the pattern and outlook pioneered in Michael Spivak’s book. Theprincipal differences are more emphasis on differentiation and the inverse map-ping and related theorems; a more extensive treatment of orientation, whichcan be a rather confusing concept; an introduction of differential forms rathermore simply and primitively, rather than as duals to differentiations, whichiii  iv INTRODUCTION  seems sufficient when the interest is really on their role in Euclidean spaces; andmore emphasis on the classical interpretations and analytic aspects of differen-tial forms. The course based on these notes covers a good deal of material, soit meets four hours each week, one of which is devoted principally to examplesand illustrative calculations, while the discussion of the problems is left to officehours outside the regular class schedule. It has been possible at least to discussall the topics covered in general term, focusing on the most difficult proofs andleaving it to the students to read the details of some of the proofs in the notes.I would like to express here my sincere thanks to the students who compiledthe lecture notes that were the basis for these notes, Robert Haraway, AdamHesterberg, Jay Holt and Alex Schiller; to Lillian Pierce for her suggestionsand great help in reorganizing the course; to the graduate students who havecotaught the course and done the major share of grading the assignments; andto the students who have taken the course since its reorganization for a numberof corrections and suggestions for greater clarity in the notes. The remainingerrors and confusion are my own responsibility though.Robert C. GunningFine Hall, Princeton University, May 2011
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