Munkres J.R. Analysis on manifolds (1991)

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J.R.Munkres Analysis on manifolds (1991)

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Preface

This book is intended as a text for a course in analysis at the senior of first-year graduate level.

Ayear-long course in real analysis is an essential part of thepreparation of any potential mathematician. For the first half of sucha course, there is substantial agreement as to what the syllabus shouldbe. Standard topics include: sequence and series, the topology ofmetric spaces, and the derivative and the Riemannian integral for thefunctions of a single variable. There are a number of excellent textsfor such a course, including books by Apostol, Rudin, Goldberg, andRoyden, among others.

There is no such universal agreement as towhat the syllabus of the second half of such a course should be. Partof the problem is that there are simply too many topics that belong insuch a course for one to be able to treat them all within the confinesof a single semester, at more than a superficial level.

At M.I.T., we have dealt with the problem by offering two independentsecond-term courses in analysis. One of these deals with the
derivativeand the Riemannian integral for functions of several variables,followed by a treatment of differential forms and a proof of Stokes'theorem for manifolds in euclidean space. The presentbook has resulted from my years of teaching this course. The otherdeals with Lesbesque integral in euclidean space and its applicationsto Fourier analysis.

Prequisites

As indicated, weassume reader has completed a one-term course in analysis that includeda study of metric spaces and of functions of a single variable. We alsoassume the reader has some background in linear algebra, includingvector spaces and linear transformations, matrix algebra, anddeterminants.

The first chapter of the book is devoted toreviewing the basic results from linear algebra and analysis that weshall need. Results that are truly basic are stated without proof, butbut proofs are provided for those that are sometimes omitted in a firstcourse. The student may determine from a perusal of this chapterwhether his or her background is sufficient for the rest of the book.

Howmuch time the instructor will wish to spend on this chapter depend onthe experience and preparation of the students. I usually assignSection 1 and 3 as reading material, and discuss the remainder in class.

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关键词：manifolds Manifold Analysis Munkres Analysi Analysis 流形 Munkres manifolds

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