Differential Geometry of Curves and Surfaces

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Beschreibung

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.

Über den Autor

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.

Zusammenfassung

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: [...]

Inhaltsverzeichnis

Introduction.- Curves.- Additional topics in curves.- Surfaces.- The curvature of a surface.- Geodesics.- The Gauss-Bonnet theorem.- Appendix A: The topology of subsets of Rn.- Recommended excursions.- Index.

Details

Erscheinungsjahr:	2016
Fachbereich:	Geometrie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Seiten:	320
Reihe:	Undergraduate Texts in Mathematics
Inhalt:	viii 366 S. 186 farbige Illustr. 366 p. 186 illus. in color.
ISBN-13:	9783319397986
ISBN-10:	3319397982
Sprache:	Englisch
Einband:	Gebunden
Autor:	Tapp, Kristopher
Hersteller:	Springer-Verlag GmbH Springer International Publishing
Abbildungen:	1 schwarz-weiße und 198 farbige Abbildungen, Bibliographie
Maße:	241 x 159 x 22 mm
Von/Mit:	Kristopher Tapp
Erscheinungsdatum:	27.09.2016
Gewicht:	0,822 kg

preigu-id: 107564444