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

This comprehensive textbook simplifies the discussion of differential geometry to an accessible level by introducing Clifford algebra. It includes chapter-by-chapter exercises, and provides a rare undergraduate-level approach to the subject matter.


Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

Includes chapter-by-chapter exercises Provides a rare undergraduate-level approach to the subject matter Promotes the application of Clifford algebra to differential geometry Presents a significant portion of general relativity without reference to such Newtonian terms as "force", "momentum", or "energy" Only up-to-date title of its kind Includes supplementary material: sn.pub/extras

Klappentext
Differential geometry is the study of curvature and calculus of curves and surfaces. Because of an historical accident, the Geometric Algebra devised by William Kingdom Clifford (18451879) has been overlooked in favor of the more complicated and less powerful formalism of differential forms and tangent vectors to deal with differential geometry. Fortuitously a student who has completed an undergraduate course in linear algebra is better prepared to deal with the intricacies of Clifford algebra than with the formalism currently used. Clifford algebra enables one to demonstrate a close relation between curvature and certain rotations. This is an advantage both conceptually and computationallyparticularly in higher dimensions.Key features and topics include:* a unique undergraduate-level approach to differential geometry;* brief biographies of historically relevant mathematicians and physicists;* some aspects of special and general relativity accessible to undergraduates with no knowledge of Newtonian physics;* chapter-by-chapter exercises.The textbook will also serve as a useful classroom resource primarily for undergraduates as well as beginning-level graduate students; researchers in the algebra and physics communities may also find the book useful as a self-study guide.

Inhalt
Preface.- Introduction.- Clifford Algebra in Euclidean 3-Space.- Clifford Algebra in Minkowski 4-Space.- Clifford Algebra in Flat n-Space.- Curved Spaces.- The Gauss-Bonnet Formula.- Non-Euclidean (Hyperbolic) Geometry.- Some Extrinsic Geometry in E^n.- Ruled Surfaces Continued.- Lines of Curvature.- Minimal Surfaces.- Some General Relativity.- Matrix Representation of a Clifford Algebra.- Construction of Coordinate Dirac Matrices.- A Few Terms of the Taylor's Series for the Urd-Copernican Model for the Outer Planets.- A Few Terms of the Taylor's Series for Kepler's Orbits.- References.- Index.

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