{"product_id":"semi-riemannian-geometry-isbn-9781119517535","title":"Semi-Riemannian Geometry","description":"\u003cp\u003e\u003cb\u003eAn introduction to semi-Riemannian geometry as a foundation for general relativity \u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eSemi-Riemannian Geometry: The Mathematical Language of General Relativity\u003c\/i\u003e is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell’s equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eI Preliminaries 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Vector Spaces 5\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Vector Spaces 5\u003c\/p\u003e \u003cp\u003e1.2 Dual Spaces 17\u003c\/p\u003e \u003cp\u003e1.3 Pullback of Covectors 19\u003c\/p\u003e \u003cp\u003e1.4 Annihilators 20\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Matrices and Determinants 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Matrices 23\u003c\/p\u003e \u003cp\u003e2.2 Matrix Representations 27\u003c\/p\u003e \u003cp\u003e2.3 Rank of Matrices 32\u003c\/p\u003e \u003cp\u003e2.4 Determinant of Matrices 33\u003c\/p\u003e \u003cp\u003e2.5 Trace and Determinant of Linear Maps 43\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Bilinear Functions 45\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Bilinear Functions 45\u003c\/p\u003e \u003cp\u003e3.2 Symmetric Bilinear Functions 49\u003c\/p\u003e \u003cp\u003e3.3 Flat Maps and Sharp Maps 51\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Scalar Product Spaces 57\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Scalar Product Spaces 57\u003c\/p\u003e \u003cp\u003e4.2 Orthonormal Bases 62\u003c\/p\u003e \u003cp\u003e4.3 Adjoints 65\u003c\/p\u003e \u003cp\u003e4.4 Linear Isometries 68\u003c\/p\u003e \u003cp\u003e4.5 Dual Scalar Product Spaces 72\u003c\/p\u003e \u003cp\u003e4.6 Inner Product Spaces 75\u003c\/p\u003e \u003cp\u003e4.7 Eigenvalues and Eigenvectors 81\u003c\/p\u003e \u003cp\u003e4.8 Lorentz Vector Spaces 84\u003c\/p\u003e \u003cp\u003e4.9 Time Cones 91\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Tensors on Vector Spaces 97\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Tensors 97\u003c\/p\u003e \u003cp\u003e5.2 Pullback of Covariant Tensors 103\u003c\/p\u003e \u003cp\u003e5.3 Representation of Tensors 104\u003c\/p\u003e \u003cp\u003e5.4 Contraction of Tensors 106\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Tensors on Scalar Product Spaces 113\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Contraction of Tensors 113\u003c\/p\u003e \u003cp\u003e6.2 Flat Maps 114\u003c\/p\u003e \u003cp\u003e6.3 Sharp Maps 119\u003c\/p\u003e \u003cp\u003e6.4 Representation of Tensors 123\u003c\/p\u003e \u003cp\u003e6.5 Metric Contraction of Tensors 127\u003c\/p\u003e \u003cp\u003e6.6 Symmetries of (0, 4)-Tensors 129\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Multicovectors 133\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Multicovectors 133\u003c\/p\u003e \u003cp\u003e7.2 Wedge Products 137\u003c\/p\u003e \u003cp\u003e7.3 Pullback of Multicovectors 144\u003c\/p\u003e \u003cp\u003e7.4 Interior Multiplication 148\u003c\/p\u003e \u003cp\u003e7.5 Multicovector Scalar Product Spaces 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Orientation 155\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Orientation of R\u003ci\u003e\u003csup\u003em \u003c\/sup\u003e\u003c\/i\u003e155\u003c\/p\u003e \u003cp\u003e8.2 Orientation of Vector Spaces 158\u003c\/p\u003e \u003cp\u003e8.3 Orientation of Scalar Product Spaces 163\u003c\/p\u003e \u003cp\u003e8.4 Vector Products 166\u003c\/p\u003e \u003cp\u003e8.5 Hodge Star 178\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Topology 183\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Topology 183\u003c\/p\u003e \u003cp\u003e9.2 Metric Spaces 193\u003c\/p\u003e \u003cp\u003e9.3 Normed Vector Spaces 195\u003c\/p\u003e \u003cp\u003e9.4 Euclidean Topology on R\u003ci\u003e\u003csup\u003em\u003c\/sup\u003e\u003c\/i\u003e 195\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Analysis in R\u003ci\u003e\u003csup\u003em \u003c\/sup\u003e\u003c\/i\u003e199\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Derivatives 199\u003c\/p\u003e \u003cp\u003e10.2 Immersions and Diffeomorphisms 207\u003c\/p\u003e \u003cp\u003e10.3 Euclidean Derivative and Vector Fields 209\u003c\/p\u003e \u003cp\u003e10.4 Lie Bracket 213\u003c\/p\u003e \u003cp\u003e10.5 Integrals 218\u003c\/p\u003e \u003cp\u003e10.6 Vector Calculus 221\u003c\/p\u003e \u003cp\u003e\u003cb\u003eII Curves and Regular Surfaces 223\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Curves and Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 225\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Curves in R\u003csup\u003e3\u003c\/sup\u003e 225\u003c\/p\u003e \u003cp\u003e11.2 Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 226\u003c\/p\u003e \u003cp\u003e11.3 Tangent Planes in R\u003csup\u003e3\u003c\/sup\u003e 237\u003c\/p\u003e \u003cp\u003e11.4 Types of Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 240\u003c\/p\u003e \u003cp\u003e11.5 Functions on Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 246\u003c\/p\u003e \u003cp\u003e11.6 Maps on Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 248\u003c\/p\u003e \u003cp\u003e11.7 Vector Fields along Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e 252\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Curves and Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e255\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Curves in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e256\u003c\/p\u003e \u003cp\u003e12.2 Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e257\u003c\/p\u003e \u003cp\u003e12.3 Induced Euclidean Derivative in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e266\u003c\/p\u003e \u003cp\u003e12.4 Covariant Derivative on Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e274\u003c\/p\u003e \u003cp\u003e12.5 Covariant Derivative on Curves in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e282\u003c\/p\u003e \u003cp\u003e12.6 Lie Bracket in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e285\u003c\/p\u003e \u003cp\u003e12.7 Orientation in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev \u003c\/sub\u003e\u003c\/i\u003e288\u003c\/p\u003e \u003cp\u003e12.8 Gauss Curvature in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev\u003c\/sub\u003e\u003c\/i\u003e 292\u003c\/p\u003e \u003cp\u003e12.9 Riemann Curvature Tensor in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev\u003c\/sub\u003e\u003c\/i\u003e 299\u003c\/p\u003e \u003cp\u003e12.10 Computations for Regular Surfaces in R\u003csup\u003e3\u003c\/sup\u003e\u003ci\u003e\u003csub\u003ev\u003c\/sub\u003e\u003c\/i\u003e 310\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Examples of Regular Surfaces 321\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Plane in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 321\u003c\/p\u003e \u003cp\u003e13.2 Cylinder in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 322\u003c\/p\u003e \u003cp\u003e13.3 Cone in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 323\u003c\/p\u003e \u003cp\u003e13.4 Sphere in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 324\u003c\/p\u003e \u003cp\u003e13.5 Tractoid in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 325\u003c\/p\u003e \u003cp\u003e13.6 Hyperboloid of One Sheet in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 326\u003c\/p\u003e \u003cp\u003e13.7 Hyperboloid of Two Sheets in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 327\u003c\/p\u003e \u003cp\u003e13.8 Torus in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e0\u003c\/sub\u003e 329\u003c\/p\u003e \u003cp\u003e13.9 Pseudosphere in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e1\u003c\/sub\u003e 330\u003c\/p\u003e \u003cp\u003e13.10 Hyperbolic Space in R\u003csup\u003e3\u003c\/sup\u003e\u003csub\u003e1\u003c\/sub\u003e 331\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIII Smooth Manifolds and Semi-Riemannian Manifolds 333\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Smooth Manifolds 337\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Smooth Manifolds 337\u003c\/p\u003e \u003cp\u003e14.2 Functions and Maps 340\u003c\/p\u003e \u003cp\u003e14.3 Tangent Spaces 344\u003c\/p\u003e \u003cp\u003e14.4 Differential of Maps 351\u003c\/p\u003e \u003cp\u003e14.5 Differential of Functions 353\u003c\/p\u003e \u003cp\u003e14.6 Immersions and Diffeomorphisms 357\u003c\/p\u003e \u003cp\u003e14.7 Curves 358\u003c\/p\u003e \u003cp\u003e14.8 Submanifolds 360\u003c\/p\u003e \u003cp\u003e14.9 Parametrized Surfaces 364\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Fields on Smooth Manifolds 367\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Vector Fields 367\u003c\/p\u003e \u003cp\u003e15.2 Representation of Vector Fields 372\u003c\/p\u003e \u003cp\u003e15.3 Lie Bracket 374\u003c\/p\u003e \u003cp\u003e15.4 Covector Fields 376\u003c\/p\u003e \u003cp\u003e15.5 Representation of Covector Fields 379\u003c\/p\u003e \u003cp\u003e15.6 Tensor Fields 382\u003c\/p\u003e \u003cp\u003e15.7 Representation of Tensor Fields 385\u003c\/p\u003e \u003cp\u003e15.8 Differential Forms 387\u003c\/p\u003e \u003cp\u003e15.9 Pushforward and Pullback of Functions 389\u003c\/p\u003e \u003cp\u003e15.10 Pushforward and Pullback of Vector Fields 391\u003c\/p\u003e \u003cp\u003e15.11 Pullback of Covector Fields 393\u003c\/p\u003e \u003cp\u003e15.12 Pullback of Covariant Tensor Fields 398\u003c\/p\u003e \u003cp\u003e15.13 Pullback of Differential Forms 401\u003c\/p\u003e \u003cp\u003e15.14 Contraction of Tensor Fields 405\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Differentiation and Integration on Smooth Manifolds 407\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Exterior Derivatives 407\u003c\/p\u003e \u003cp\u003e16.2 Tensor Derivations 413\u003c\/p\u003e \u003cp\u003e16.3 Form Derivations 417\u003c\/p\u003e \u003cp\u003e16.4 Lie Derivative 419\u003c\/p\u003e \u003cp\u003e16.5 Interior Multiplication 423\u003c\/p\u003e \u003cp\u003e16.6 Orientation 425\u003c\/p\u003e \u003cp\u003e16.7 Integration of Differential Forms 432\u003c\/p\u003e \u003cp\u003e16.8 Line Integrals 435\u003c\/p\u003e \u003cp\u003e16.9 Closed and Exact Covector Fields 437\u003c\/p\u003e \u003cp\u003e16.10 Flows 443\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Smooth Manifolds with Boundary 449\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Smooth Manifolds with Boundary 449\u003c\/p\u003e \u003cp\u003e17.2 Inward-Pointing and Outward-Pointing Vectors 452\u003c\/p\u003e \u003cp\u003e17.3 Orientation of Boundaries 456\u003c\/p\u003e \u003cp\u003e17.4 Stokes's Theorem 459\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Smooth Manifolds with a Connection 463\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Covariant Derivatives 463\u003c\/p\u003e \u003cp\u003e18.2 Christoffel Symbols 466\u003c\/p\u003e \u003cp\u003e18.3 Covariant Derivative on Curves 472\u003c\/p\u003e \u003cp\u003e18.4 Total Covariant Derivatives 476\u003c\/p\u003e \u003cp\u003e18.5 Parallel Translation 479\u003c\/p\u003e \u003cp\u003e18.6 Torsion Tensors 485\u003c\/p\u003e \u003cp\u003e18.7 Curvature Tensors 488\u003c\/p\u003e \u003cp\u003e18.8 Geodesics 497\u003c\/p\u003e \u003cp\u003e18.9 Radial Geodesics and Exponential Maps 502\u003c\/p\u003e \u003cp\u003e18.10 Normal Coordinates 507\u003c\/p\u003e \u003cp\u003e18.11 Jacobi Fields 509\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Semi-Riemannian Manifolds 515\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Semi-Riemannian Manifolds 515\u003c\/p\u003e \u003cp\u003e19.2 Curves 519\u003c\/p\u003e \u003cp\u003e19.3 Fundamental Theorem of Semi-Riemannian Manifolds 519\u003c\/p\u003e \u003cp\u003e19.4 Flat Maps and Sharp Maps 526\u003c\/p\u003e \u003cp\u003e19.5 Representation of Tensor Fields 529\u003c\/p\u003e \u003cp\u003e19.6 Contraction of Tensor Fields 532\u003c\/p\u003e \u003cp\u003e19.7 Isometries 535\u003c\/p\u003e \u003cp\u003e19.8 Riemann Curvature Tensor 539\u003c\/p\u003e \u003cp\u003e19.9 Geodesics 546\u003c\/p\u003e \u003cp\u003e19.10 Volume Forms 550\u003c\/p\u003e \u003cp\u003e19.11 Orientation of Hypersurfaces 551\u003c\/p\u003e \u003cp\u003e19.12 Induced Connections 558\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Differential Operators on Semi-Riemannian Manifolds 561\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Hodge Star 561\u003c\/p\u003e \u003cp\u003e20.2 Codifferential 562\u003c\/p\u003e \u003cp\u003e20.3 Gradient 566\u003c\/p\u003e \u003cp\u003e20.4 Divergence of Vector Fields 568\u003c\/p\u003e \u003cp\u003e20.5 Curl 572\u003c\/p\u003e \u003cp\u003e20.6 Hesse Operator 573\u003c\/p\u003e \u003cp\u003e20.7 Laplace Operator 575\u003c\/p\u003e \u003cp\u003e20.8 Laplace-de Rham Operator 576\u003c\/p\u003e \u003cp\u003e20.9 Divergence of Symmetric 2-Covariant Tensor Fields 577\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 Riemannian Manifolds 579\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Geodesics and Curvature on Riemannian Manifolds 579\u003c\/p\u003e \u003cp\u003e21.2 Classical Vector Calculus Theorems 582\u003c\/p\u003e \u003cp\u003e\u003cb\u003e22 Applications to Physics 587\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Linear Isometries on Lorentz Vector Spaces 587\u003c\/p\u003e \u003cp\u003e22.2 Maxwell's Equations 598\u003c\/p\u003e \u003cp\u003e22.3 Einstein Tensor 603\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIV Appendices 609\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA Notation and Set Theory 611\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eB Abstract Algebra 617\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 Groups 617\u003c\/p\u003e \u003cp\u003eB.2 Permutation Groups 618\u003c\/p\u003e \u003cp\u003eB.3 Rings 623\u003c\/p\u003e \u003cp\u003eB.4 Fields 623\u003c\/p\u003e \u003cp\u003eB.5 Modules 624\u003c\/p\u003e \u003cp\u003eB.6 Vector Spaces 625\u003c\/p\u003e \u003cp\u003eB.7 Lie Algebras 626\u003c\/p\u003e \u003cp\u003eFurther Reading 627\u003c\/p\u003e \u003cp\u003eIndex 629\u003c\/p\u003e  \u003cp\u003e\u003cb\u003eSTEPHEN C. NEWMAN\u003c\/b\u003e is Professor Emeritus at the University of Alberta, Edmonton, Alberta, Canada. He is the author of \u003ci\u003eBiostatistical Methods in Epidemiology\u003c\/i\u003e and \u003ci\u003eA Classical Introduction to Galois Theory\u003c\/i\u003e, both published by Wiley.   \u003c\/p\u003e\u003cp\u003e\u003cb\u003eAn introduction to semi-Riemannian geometry as a foundation for general relativity\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003e\u003ci\u003eSemi-Riemannian Geometry: The Mathematical Language of General Relativity\u003c\/i\u003e is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell's equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47990005891301,"sku":"NP9781119517535","price":118.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119517535.jpg?v=1761786187","url":"https:\/\/k12savings.com\/products\/semi-riemannian-geometry-isbn-9781119517535","provider":"K12savings","version":"1.0","type":"link"}