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Introduction to Mathematical Analysis

  • Textbook
  • © 2013

Overview

Authors:
  1. Igor Kriz

    1. Department of Mathematics, University of Michigan, Ann Arbor, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Aleš Pultr

    1. Department of Applied Mathematics (KAM), Charles University Faculty of Mathematics and Physics, Prague 1, Czech Republic

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • In a single text, the student gets a glimpse of a big picture

  • The treatment of mathematical analysis is rigorous and self-contained

  • Can be used not only as a textbook for a course but also as a reference book for lecturers in analysis or mathematicians and scientists in general ?

  • Includes supplementary material: sn.pub/extras

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xx
    Download chapter PDF

  2. A Rigorous Approach to Advanced Calculus

    1. Front Matter

      Pages 1-1
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    2. Preliminaries

      • Igor Kriz, Aleš Pultr
      Pages 3-31

    3. Metric and Topological Spaces I

      • Igor Kriz, Aleš Pultr
      Pages 33-63

    4. Multivariable Differential Calculus

      • Igor Kriz, Aleš Pultr
      Pages 65-95

    5. Integration I: Multivariable Riemann Integral and Basic Ideas Toward the Lebesgue Integral

      • Igor Kriz, Aleš Pultr
      Pages 97-116

    6. Integration II: Measurable Functions, Measure and the Techniques of Lebesgue Integration

      • Igor Kriz, Aleš Pultr
      Pages 117-143

    7. Systems of Ordinary Differential Equations

      • Igor Kriz, Aleš Pultr
      Pages 145-173

    8. Systems of Linear Differential Equations

      • Igor Kriz, Aleš Pultr
      Pages 175-191

    9. Line Integrals and Green’s Theorem

      • Igor Kriz, Aleš Pultr
      Pages 193-210

  3. Analysis and Geometry

    1. Front Matter

      Pages 211-211
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    2. Metric and Topological Spaces II

      • Igor Kriz, Aleš Pultr
      Pages 213-235

    3. Complex Analysis I: Basic Concepts

      • Igor Kriz, Aleš Pultr
      Pages 237-266

    4. Multilinear Algebra

      • Igor Kriz, Aleš Pultr
      Pages 267-285

    5. Smooth Manifolds, Differential Forms and Stokes’ Theorem

      • Igor Kriz, Aleš Pultr
      Pages 287-310

    6. Complex Analysis II: Further Topics

      • Igor Kriz, Aleš Pultr
      Pages 311-348

    7. Calculus of Variations and the Geodesic Equation

      • Igor Kriz, Aleš Pultr
      Pages 349-366

    8. Tensor Calculus and Riemannian Geometry

      • Igor Kriz, Aleš Pultr
      Pages 367-392

    9. Banach and Hilbert Spaces: Elements of Functional Analysis

      • Igor Kriz, Aleš Pultr
      Pages 393-426

    10. A Few Applications of Hilbert Spaces

      • Igor Kriz, Aleš Pultr
      Pages 427-449
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Keywords

About this book

The book begins at the level of an undergraduate student assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations. After having built on a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis, as understood by a mathematician today.​  

Reviews

From the reviews:

“The book is intended as a second-year course of mathematical analysis for advanced undergraduate students. … The volume is addressed to undergraduate students seriously interested in mathematics and is accessible to students before they start taking graduate classes. Researchers in pure and applied nonlinear analysis will find interesting material in this volume.” (Teodora-Liliana Rădulescu, zbMATH, Vol. 1279, 2014)

“The authors included in their book some topics from topology, calculus of real functions of one and several real variables … elements of functional analysis, as well as some applications. … the present well written book is a valuable addition to the existing ones on similar topics. It can be used by graduate students in mathematics and researchers in mathematics and other areas … . The instructors can recommend the book as a supplementary material for their courses.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Math, Vol. 58 (4), 2013)

Authors and Affiliations

  • Department of Mathematics, University of Michigan, Ann Arbor, USA

    Igor Kriz

  • Department of Applied Mathematics (KAM), Charles University Faculty of Mathematics and Physics, Prague 1, Czech Republic

    Aleš Pultr

About the authors

Igor Kriz studied in Prague, Czech Republic. He has lived in the United States since 1988 and been teaching at the University of Michigan since 1994. His main interests are geometry and algebraic topology. Aleš Pultr studied in Prague and has been teaching at the Charles University since 1961. His main interests are point-free topology, category theory and combinatorics. Both authors have taught numerous courses of mathematical analysis.

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