A New Approach to Differential Geometry using Clifford's Geometric Algebra
John. Snygg
Bok Engelsk 2011 · Electronic books.
| Utgitt | Dordrecht : : Springer, , 2011.
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| Omfang | 1 online resource (471 p.)
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| Opplysninger | Description based upon print version of record.. - A New Approach to Differential Geometry Using Clifford's Geometric Algebra; Preface; Acknowledgments; Contents; List of Symbols; Chapter 1 Introduction; Chapter 2 Clifford Algebra in Euclidean 3-Space; 2.1 Reflections, Rotations, and Quaternions in E3; 2.1.1 Using Square Matrices to Represent Vectors; 2.1.2 1-Vectors, 2-Vectors, 3-Vectors, and Clifford Numbers; 2.1.3 Reflection and Rotation Operators; 2.1.4 Quaternions; 2.2 The 4 Periodicity of the Rotation Operator; 2.3 *The Point Groups for the Regular Polyhedrons; 2.4 *Élie Cartan 1869-1951; 2.5 *Suggested Reading. - 4.7.5 *Claudius Ptolemy, Al-Tusı, Al-'Urdı, Ibn al-Shatir, Nicholas Copernicus, Tycho Brahe, Johannes Kepler, and Isaac Newton4.8 *Christopher Columbus and Some Bad Geography; 4.9 *Clifford and Grassmann; 4.9.1 *William Kingdon Clifford 1845-1879; 4.9.2 *Hermann Günther Grassmann 1809-1877; Chapter 5 Curved Spaces; 5.1 Gaussian Curvature (Informal); 5.2 n-Dimensional Curved Surfaces (Spaces); 5.3 The Intrinsic Derivative k; 5.4 Parallel Transport and Geodesics; 5.5 The Riemann Tensor and the Curvature 2-form; 5.6 Fock-Ivanenko Coefficients; 5.6.1 Moving Frames. - 5.6.2 Gauss Curvature via Fock-Ivanenko Coefficients5.6.3 *The Riemann Tensor for Orthonormal Frames; 5.7 *Doing Physics Under Stalin; 5.7.1 *George Gamow 1904-1968; 5.7.2 *Lev Davidovich Landau 1908-1968; 5.7.3 *Matvei Petrovich Bronstein 1906-1938; 5.7.4 *Vladimir Alexandrovich Fock 1898-1974; 5.7.5 *Dmitrii Dmitrievich Ivanenko 1904-1994; Chapter 6 The Gauss-Bonnet Formula; 6.1 The Exterior Derivative and Stokes' Theorem; 6.2 *Curvature via Connection 1-Forms; 6.3 Geodesic Curvature on a 2-dimensional Surface; 6.4 *Huygens' Pendulum Clock and the Cycloid; 6.5 The Gauss-Bonnet Formula. - 6.6 The Interpretation of Curvature 2-Formsas Infinitesimal Rotation Operators6.7 *Euler's Theorem for Convex Polyhedrons; 6.8 *Carl Friedrich Gauss and Bernard Riemann; 6.8.1 *Carl Friedrich Gauss 1777-1855; 6.8.2 *Georg Friedrich Bernhard Riemann 1826-1866; Chapter 7 Some Extrinsic Geometry in En; 7.1 The Frenet Frame; 7.2 *Arbitrary Speed Curves with Formulas; 7.3 Ruled Surfaces and Developable Surfaces; 7.4 *Archimedes' Screw; 7.5 Principal Curvatures; 7.5.1 The Normal and Geodesic Curvature Vectors; 7.5.2 The Weingarten Map or Shape Operator; 7.5.3 Principal Directions and Curvatures. - 7.6 *Leonhard Euler 1707-1783. - Chapter 3 Clifford Algebra in Minkowski 4-Space3.1 A Small Dose of Special Relativity; 3.2 *Albert Einstein 1879-1955; 3.3 *Suggested Reading; Chapter 4 Clifford Algebra in Flat n-Space; 4.1 Clifford Algebra; 4.2 The Scalar Product and Metric Tensor; 4.3 The Exterior Product for p-Vectors; 4.4 Some Useful Formulas; 4.5 Gram-Schmidt Formulas; 4.6 *The Qibla (Kibla) Problem; 4.7 *Mathematics of Arab Speaking Muslims; 4.7.1 *Greek Science and Mathematics in Alexandria; 4.7.2 *Hypatia; 4.7.3 *The Rise of Islam and the House of Wisdom; 4.7.4 *The Impact of Al-Ghazalı. - 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
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| Emner | |
| Sjanger | |
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| ISBN | 9780817682828
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