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  • © 2016

Calculus and Analysis in Euclidean Space

Authors:

  • Concentrates on n-dimensional Euclidean space

  • Uses multivariable calculus to teach mathematics as a blend of reasoning, computing, and problem-solving, doing justice to the structure, the details, and the scope of the ideas

  • Contains figures, formulas, and words to guide the reader to do mathematics resourcefully by marshaling the skills of geometric intuition, algebraic manipulation, and incisive use of natural language

  • Includes supplementary material: sn.pub/extras

  • Request lecturer material: sn.pub/lecturer-material

Part of the book series: Undergraduate Texts in Mathematics (UTM)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiii
  2. Results from One-Variable Calculus

    • Jerry Shurman
    Pages 1-20
  3. Multivariable Differential Calculus

    1. Front Matter

      Pages 21-21
    2. Euclidean Space

      • Jerry Shurman
      Pages 23-58
    3. Linear Mappings and Their Matrices

      • Jerry Shurman
      Pages 59-130
    4. The Derivative

      • Jerry Shurman
      Pages 131-197
    5. Inverse and Implicit Functions

      • Jerry Shurman
      Pages 199-250
  4. Multivariable Integral Calculus

    1. Front Matter

      Pages 251-251
    2. Integration

      • Jerry Shurman
      Pages 253-346
    3. Approximation by Smooth Functions

      • Jerry Shurman
      Pages 347-373
    4. Parametrized Curves

      • Jerry Shurman
      Pages 375-408
    5. Integration of Differential Forms

      • Jerry Shurman
      Pages 409-500
  5. Back Matter

    Pages 501-505

About this book

The graceful role of analysis in underpinning calculus is often lost to their separation in the curriculum.  This book entwines the two subjects, providing a conceptual approach to multivariable calculus closely supported by the structure and reasoning of analysis.  The setting is Euclidean space, with the material on differentiation culminating in the inverse and implicit function theorems, and the material on integration culminating in the general fundamental theorem of integral calculus.  More in-depth than most calculus books but less technical than a typical analysis introduction, Calculus and Analysis in Euclidean Space offers a rich blend of content to students outside the traditional mathematics major, while also providing transitional preparation for those who will continue on in the subject.

The writing in this book aims to convey the intent of ideas early in discussion. The narrative proceeds through figures, formulas, and text, guiding the reader to do mathematics resourcefully by marshaling the skills of

  • geometric intuition (the visual cortex being quickly instinctive)
  • algebraic manipulation (symbol-patterns being precise and robust)
  • incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).

Thinking in these ways renders mathematics coherent, inevitable, and fluid.

The prerequisite is single-variable calculus, including familiarity with the foundational theorems and some experience with proofs.

Keywords

  • derivative
  • implicit functions
  • integration of differential forms
  • linear mappings
  • multivariable differentiable calculus
  • multivariable integral calculus
  • parameterized curves
  • textbook adoption calculus
  • textbook adoption analysis
  • single-variable calculus
  • Euclidean space
  • inverse functions
  • approximation smooth functions
  • parametrized curves

Reviews

“Shurman (mathematics, Reed College) has succeeded in presenting a text that encompasses multivariable calculus, advanced calculus, and an introduction to point-set topology. In short, this book covers aspects one should know about the elementary analysis, geometry, and topology of Euclidean space. … Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty.” (J. T. Zerger, Choice, Vol. 54 (11), July, 2017)


“The author’s writing style is clear and easy to follow, but, more than that, it is exceptionally well-motivated and contains some useful pedagogical ideas. In addition, throughout the book, the author notes issues that are likely to cause trouble to beginning students, and takes the time and effort to single them out and discuss them thoroughly. There are lots of exercises, many of them quite illuminating. … It is highly recommended.” (Mark Hunacek, MAA Reviews, maa.org, March, 2017)

“This book contains a clear and well-planned lecture discussing the most important issues of differential and integral calculus. … The big advantage of this book are nice, transparent and often colourful drawings illustrating some considerations. A nice complement to mathematical statements are explanations and comments.” (Ryszard Pawlak, zbMATH 1357.26002, 2017)


Authors and Affiliations

  • Reed College Dept. Mathematics, Portland, USA

    Jerry Shurman

About the author

Jerry Shurman is professor of mathematics at Reed College, Portland, OR, USA. He is also the co-author of GTM 228 "A First Course in Modular Forms."

Bibliographic Information

  • Book Title: Calculus and Analysis in Euclidean Space

  • Authors: Jerry Shurman

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-49314-5

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing AG 2016

  • License: CC BY

  • Hardcover ISBN: 978-3-319-49312-1Published: 16 December 2016

  • Softcover ISBN: 978-3-319-84130-4Published: 05 July 2018

  • eBook ISBN: 978-3-319-49314-5Published: 26 November 2016

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: XIII, 505

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Analysis

Buy it now

Buying options

eBook USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 19.99 USD 69.99
Discount applied Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access