Contents:2.10 Partitions of unity2.11 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 coordinates4.6.2 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 variation6.4 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.
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.