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  • © 2016

Differential Geometry of Curves and Surfaces

Authors:

  • Can be used as a textbook in elementary and more advanced courses in differential geometry

  • Focuses on applications of differential geometry, lending simplicity to more difficult and abstract concepts

  • Features full-color text and inserts to distinguish fundamental definitions and theorems

  • Includes supplementary material: sn.pub/extras

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

Buying options

eBook USD 54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-39799-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 69.99
Price excludes VAT (USA)
Hardcover Book USD 64.99
Price excludes VAT (USA)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-viii
  2. Curves

    • Kristopher Tapp
    Pages 1-60
  3. Additional Topics in Curves

    • Kristopher Tapp
    Pages 61-111
  4. Surfaces

    • Kristopher Tapp
    Pages 113-191
  5. The Curvature of a Surface

    • Kristopher Tapp
    Pages 193-245
  6. Geodesics

    • Kristopher Tapp
    Pages 247-318
  7. The Gauss–Bonnet Theorem

    • Kristopher Tapp
    Pages 319-344
  8. Back Matter

    Pages 345-366

About this book

This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.

Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.

Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.

In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. 

Keywords

  • Differential Geometry
  • Gauss Bonnet Theoreom
  • conformal functions
  • curves
  • surfaces
  • Geodesics
  • Rigid Motions

Reviews

“This is the first textbook on mathematics that I see printed in color. … This book is not a usual textbook, but a very well written introduction to differential geometry, and the colors really help the reader in understanding the figures and navigating through the text. … this book will surely serve very well for students who want to learn differential geometry from the ground up no matter what their main learning goal is.” (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 84 (1-2), 2018)


“This book is perfect for undergraduate students. ... There is also plenty of figures, examples, exercises and applications which make the differential geometry of curves and surfaces so interesting and intuitive. The author uses a rich variety of colours and techniques that help to clarify difficult abstract concepts.” (Teresa Arias-Marco, zbMATH 1375.53001, 2018)

“This is a visually appealing book, replete with many diagrams, lots of them in full color. … the author’s writing style is extremely clear and well-motivated. … this is still the book I would use as a text for a beginning course on this subject. It would not surprise me if it quickly becomes the market leader.” (Mark Hunacek, MAA Reviews, July, 2017) 

Authors and Affiliations

  • Department of Mathematics, Saint Joseph’s University, Philadelphia, USA

    Kristopher Tapp

About the author

Kristopher Tapp is Professor of Mathematics at Saint Joseph's University. He has been awarded two National Science Foundation research grants to support research in differential geometry, and several teaching awards. He is the author of Symmetry: A Mathematical Exploration (Springer, 2012) and over twenty research papers featured in top journals.

Bibliographic Information

  • Book Title: Differential Geometry of Curves and Surfaces

  • Authors: Kristopher Tapp

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-39799-3

  • Publisher: Springer Cham

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

  • Copyright Information: Springer International Publishing Switzerland 2016

  • Hardcover ISBN: 978-3-319-39798-6

  • Softcover ISBN: 978-3-319-81969-3

  • eBook ISBN: 978-3-319-39799-3

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: VIII, 366

  • Number of Illustrations: 186 illustrations in colour

  • Topics: Differential Geometry

Buying options

eBook USD 54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-39799-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 69.99
Price excludes VAT (USA)
Hardcover Book USD 64.99
Price excludes VAT (USA)