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

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.

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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 (1845–1879) 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 computationally—particularly 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.

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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 (1845–1879) 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 computationally—particularly 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.

Content:
Front Matter....Pages i-xvii
Introduction....Pages 1-2
Clifford Algebra in Euclidean 3-Space....Pages 3-25
Clifford Algebra in Minkowski 4-Space....Pages 27-46
Clifford Algebra in Flat n-Space....Pages 47-120
Curved Spaces....Pages 121-179
The Gauss–Bonnet Formula....Pages 181-226
Some Extrinsic Geometry in E n ....Pages 227-298
*Non-Euclidean (Hyperbolic) Geometry....Pages 299-331
*Ruled Surfaces Continued....Pages 333-345
*Lines of Curvature....Pages 347-373
*Minimal Surfaces....Pages 375-394
Some General Relativity....Pages 395-430
Back Matter....Pages 431-465

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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 (1845–1879) 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 computationally—particularly 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.

Content:
Front Matter....Pages i-xvii
Introduction....Pages 1-2
Clifford Algebra in Euclidean 3-Space....Pages 3-25
Clifford Algebra in Minkowski 4-Space....Pages 27-46
Clifford Algebra in Flat n-Space....Pages 47-120
Curved Spaces....Pages 121-179
The Gauss–Bonnet Formula....Pages 181-226
Some Extrinsic Geometry in E n ....Pages 227-298
*Non-Euclidean (Hyperbolic) Geometry....Pages 299-331
*Ruled Surfaces Continued....Pages 333-345
*Lines of Curvature....Pages 347-373
*Minimal Surfaces....Pages 375-394
Some General Relativity....Pages 395-430
Back Matter....Pages 431-465
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