# A New Approach to Differential Geometry using Clifford's Geometric Algebra

Springer Science & Business Media, Dec 8, 2011 - Mathematics - 465 pages

Differential geometry is the study of the curvature and calculus of curves and surfaces. The conceptual complications introduced by a multitude of spaces and mappings normally required in the study of differential geometry usually postpones the topic to graduate-level courses. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an undergraduate level of differential geometry by introducing Clifford algebra. This presentation is relevant since Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.

Key features include:

· a rare undergraduate-level approach to differential geometry;

· brief biographies of historically relevant mathematicians and physicists;

· significant aspects of general relativity and Riemannian geometry and

· chapter-by-chapter exercises.

This accessible and comprehensive textbook offers a valuable introduction to differential geometry, simplifying the complicated theory by using Clifford algebra. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

This accessible and comprehensive textbook offers a valuable introduction to differential geometry, simplifying the complicated theory by using Clifford algebra. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

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### Contents

 Chapter 1 Introduction 1 Chapter 2 Clifford Algebra in Euclidean 3Space 3 Chapter 3 Clifford Algebra in Minkowski 4Space 27 Chapter 4 Clifford Algebra in Flat nSpace 47 Chapter 5 Curved Spaces 121 Chapter 6 The GaussBonnet Formula 181 Chapter 7 Some Extrinsic Geometry in En 227 Chapter 8 NonEuclidean Hyperbolic Geometry 299
 Chapter 11 Minimal Surfaces 375 Chapter 12 Some General Relativity 395 Chapter A A Matrix Representation of a Clifford Algebra 431 Chapter B Construction of Matrix Representations for Dirac Vectors 436 Chapter C A Few Terms of the Taylors Series for the UrdıCopernican Model for the Outer Planets 441 Chapter D A Few Terms of the Taylors Series for Keplers Orbits 444 References 449 Index 459

 Chapter 9 Ruled Surfaces Continued 333 Chapter 10 Lines of Curvature 347