# Differential Geometry of Curves and Surfaces

• Kristopher Tapp
Textbook

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

1. Front Matter
Pages i-viii
2. Kristopher Tapp
Pages 1-60
3. Kristopher Tapp
Pages 61-111
4. Kristopher Tapp
Pages 113-191
5. Kristopher Tapp
Pages 193-245
6. Kristopher Tapp
Pages 247-318
7. Kristopher Tapp
Pages 319-344
8. Back Matter
Pages 345-366

### Introduction

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

#### Authors and affiliations

• Kristopher Tapp
• 1
1. 1.Department of MathematicsSaint Joseph’s UniversityPhiladelphiaUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-39799-3
• Copyright Information Springer International Publishing Switzerland 2016
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-39798-6
• Online ISBN 978-3-319-39799-3
• Series Print ISSN 0172-6056
• Series Online ISSN 2197-5604
• Buy this book on publisher's site
Industry Sectors
Aerospace
Oil, Gas & Geosciences