A New Approach to Differential Geometry using Cliffords Geometric Algebra 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 Taylors Series for the Urdī Copernican Model for the Outer Planets A Few Terms of the Taylors Series for Keplers Orbits References Index. 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.