Moscow Lectures
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Smooth Functions and Maps

Authors: Makarov, Boris M., Podkorytov, Anatolii N.

  • Contains a consistent theory of smooth functions
  • Deals with critical values of smooth mappings
  • Uses a new technical approach that allows to clarify some of the technically difficult proofs while maintaining full integrity
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About this Textbook

The book contains a consistent and sufficiently comprehensive theory of smooth functions and maps insofar as it is connected with differential calculus.

The scope of notions includes, among others, Lagrange inequality, Taylor’s formula, finding absolute and relative extrema, theorems on smoothness of the inverse map and on conditions of local invertibility, implicit function theorem, dependence and independence of functions, classification of smooth functions up to diffeomorphism. The concluding chapter deals with a more specific issue of critical values of smooth mappings.

In several chapters, a relatively new technical approach is used that allows the authors to clarify and simplify some of the technically difficult proofs while maintaining full integrity. Besides, the book includes complete proofs of some important results which until now have only been published in scholarly literature or scientific journals (remainder estimates of Taylor’s formula in a nonconvex area (Chapter I, §8), Whitney's extension theorem for smooth function (Chapter I, §11) and some of its corollaries, global diffeomorphism theorem (Chapter II, §5), results on sets of critical values of smooth mappings and the related Whitney example (Chapter IV).

The text features multiple examples illustrating the results obtained and demonstrating their accuracy. Moreover, the book contains over 150 problems and 19 illustrations.

Perusal of the book equips the reader to further explore any literature basing upon multivariable calculus.


About the authors

Boris M. Makarov, Professor of the Saint Petersburg State University, Mathematics and Mechanics Faculty, Department of Mathematical Analysis. In 1996-2006, he was a member of the Editorial Board of the journal Functional Analysis and Its Applications of the Russian Academy of Sciences. In 2014, received the Saint Petersburg State University ‘Pedagogic Excellency’ award. In 2010, co-authored (with Anatolii N. Podkorytov) a book Lectures on Real Analysis published in English by Springer in 2013 under the title of Real Analysis: Measures, Integrals and Applications. Co-authored (with Maria G. Goluzina, Andrei A. Lodkin, and Anatolii N. Podkorytov) a book of problems Problems in Real Analysis published in Russian (two editions: 1992 and 2004), English (AMS, 1992), and French (Cassini, 2010). 

Anatolii N. Podkorytov, Associate Professor of the Saint Petersburg State University, Mathematics and Mechanics Faculty, Department of Mathematical Analysis. Author of a number of published works and frequent deliverer of talks on properties of series and Fourier transform of functions of several variables. In 2014, received the Saint Petersburg State University ‘Pedagogic Excellency’ award. In 2010, co-authored (with Boris M. Makarov) a book Lectures on Real Analysis published in English by Springer in 2013 under the title of Real Analysis: Measures, Integrals and Applications. Co-authored (with Maria G. Goluzina, Andrei A. Lodkin, and Boris M. Makarov) a book of problems Problems in Real Analysis published in Russian (two editions: 1992 and 2004), English (AMS, 1992), and French (Cassini, 2010).


Table of contents (5 chapters)

Table of contents (5 chapters)
  • Differentiable Functions

    Pages 25-125

    Makarov, Boris M. (et al.)

  • Smooth Maps

    Pages 127-161

    Makarov, Boris M. (et al.)

  • The Implicit Map Theorem and Its Applications

    Pages 163-199

    Makarov, Boris M. (et al.)

  • Critical Values of Smooth Maps

    Pages 201-243

    Makarov, Boris M. (et al.)

  • Addenda

    Pages 245-278

    Makarov, Boris M. (et al.)

Buy this book

eBook $44.99
$69.99 (listprice)
price for Mexico
  • ISBN 978-3-030-79438-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $59.99
$89.99 (listprice)
price for Mexico
  • ISBN 978-3-030-79437-8
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Smooth Functions and Maps
Authors
Translated by
Tsilevich, N.
Series Title
Moscow Lectures
Series Volume
7
Copyright
2021
Publisher
Springer International Publishing
Copyright Holder
The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
eBook ISBN
978-3-030-79438-5
DOI
10.1007/978-3-030-79438-5
Hardcover ISBN
978-3-030-79437-8
Series ISSN
2522-0314
Edition Number
1
Number of Pages
XL, 244
Number of Illustrations
19 b/w illustrations
Topics