Multivariable Calculus and Differential Geometry

Description

This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics and physics. The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem. A thorough introduction to differential geometry Covers all important concepts and theorems Many examples including applications to physics Preface 1 Euclidean Space 1.1 Vector spaces 1.2 Linear transformations 1.3 Determinants 1.4 Euclidean spaces 1.5 Subspaces of Euclidean space 1.6 Determinants as volume 1.7 Elementary topology of Euclidean spaces 1.8 Sequences 1.9 Limits and continuity 1.10 Exercises 2 Differentiation 2.1 The derivative 2.2 Basic properties of the derivative 2.3 Differentiation of integrals 2.4 Curves 2.5 The inverse and implicit function theorems 2.6 The spectral theorem and scalar products 2.7 Taylor polynomials and extreme values 2.8 Vector fields 2.9 Lie brackets 2.10 Partitions of unity 2.11 Exercises 3 Manifolds 3.1 Submanifolds of Euclidean space 3.2 Differentiablemaps on manifolds 3.3 Vector fields on manifolds 3.4 Lie groups 3.5 The tangent bundle 3.6 Covariant differentiation 3.7 Geodesics 3.8 The second fundamental tensor 3.9 Curvature 3.10 Sectional curvature 3.11 Isometries 3.12 Exercises 4 Integration on Euclidean space 4.1 The integral of a function over a box 4.2 Integrability and discontinuities 4.3 Fubini’s theorem 4.4 Sard’s theorem 4.5 The change of variables theorem 4.6 Cylindrical and spherical coordinates 4.6.1 Cylindrical coordinates 4.6.2 Spherical coordinates 4.7 Some applications 4.7.1 Mass 4.7.2 Center ofmass 4.7.3 Moment of inertia 4.8 Exercises 5 Differential Forms 5.1 Tensors and tensor fields 5.2 Alternating tensors and forms 5.3 Differential forms 5.4 Integration on manifolds 5.5 Manifolds with boundary 5.6 Stokes’ theorem 5.7 Classical versions of Stokes’ theorem 5.7.1 An application: the polar planimeter 5.8 Closed forms and exact forms 5.9 Exercises 6 Manifolds as metric spaces 6.1 Extremal properties of geodesics 6.2 Jacobi fields 6.3 The length function of a variation 6.4 The index formof a geodesic 6.5 The distance function 6.6 The Hopf-Rinow theorem 6.7 Curvature comparison 6.8 Exercises 7 Hypersurfaces 7.1 Hypersurfaces and orientation 7.2 The Gaussmap 7.3 Curvature of hypersurfaces 7.4 The fundamental theorem for hypersurfaces 7.5 Curvature in local coordinates 7.6 Convexity and curvature 7.7 Ruled surfaces 7.8 Surfaces of revolution 7.9 Exercises Appendix A Appendix B Index