From the eBook - Second edition.
Half Title; Title Page; Copyright; Dedication; Contents; Preface; 1 Differential Forms in Rn, I; 1.0 Euclidean spaces, tangent spaces, and tangent vector fields; 1.1 The algebra of differential forms; 1.2 Exterior differentiation; 1.3 The fundamental correspondence; 1.4 The Converse of Poincaré's Lemma, I; 1.5 Exercises; 2 Differential Forms in Rn, II; 2.1 1-Forms; 2.2 k-Forms; 2.3 Orientation and signed volume; 2.4 The converse of Poincaré's Lemma, II; 2.5 Exercises; 3 Push-forwards and Pull-backs in Rn; 3.1 Tangent vectors; 3.2 Points, tangent vectors, and push-forwards.
3.3 Differential forms and pull-backs3.4 Pull-backs, products, and exterior derivatives; 3.5 Smooth homotopies and the Converse of Poincaré's Lemma, III; 3.6 Exercises; 4 Smooth Manifolds; 4.1 The notion of a smooth manifold; 4.2 Tangent vectors and differential forms; 4.3 Further constructions; 4.4 Orientations of manifolds'227intuitive discussion; 4.5 Orientations of manifolds'227careful development; 4.6 Partitions of unity; 4.7 Smooth homotopies and the Converse of Poincaré's Lemma in general; 4.8 Exercises; 5 Vector Bundles and the Global Point of View.
5.1 The definition of a vector bundle5.2 The dual bundle, and related bundles; 5.3 The tangent bundle of a smooth manifold, and related bundles; 5.4 Exercises; 6 Integration of Differential Forms; 6.1 Definite integrals in textmathbbRn; 6.2 Definition of the integral in general; 6.3 The integral of a 0-form over a point; 6.4 The integral of a 1-form over a curve; 6.5 The integral of a 2-form over a surface; 6.6 The integral of a 3-form over a solid body; 6.7 Chains and integration on chains; 6.8 Exercises; 7 The Generalized Stokes's Theorem; 7.1 Statement of the theorem.
7.2 The fundamental theorem of calculus and its analog for line integrals7.3 Cap independence; 7.4 Green's and Stokes's theorems; 7.5 Gauss's theorem; 7.6 Proof of the GST; 7.7 The converse of the GST; 7.8 Exercises; 8 de Rham Cohomology; 8.1 Linear and homological algebra constructions; 8.2 Definition and basic properties; 8.3 Computations of cohomology groups; 8.4 Cohomology with compact supports; 8.5 Exercises; Index; A; B; C; D; E; F; G; H; I; L; M; N; O; P; R; S; T; V; W.
