Preface 1 Differential Forms in Rn���� 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 Poincar��'s Lemma���� 1.5 Exercises 2 Differential Forms in Rn ,�� 2.1 1-Forms 2.2 k-Forms 2.3 Orientation and signed volume 2.4 The Converse of Poincar��'s Lemma���� 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-backs 3.4 Pull-backs��products��and exterior derivatives 3.5 Smooth homotopies and the Converse of Poincar��'s Lemma���� 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 intuitive discussion 4.5 Orientations of manifolds��careful development 4.6 Partitions of unity 4.7 Smooth homotopies and the Converse of Poincar��'s Lemma in general 4.8 Exercises 5 Vector Bundles and the Global point of View 5.1 The definition of a vector bundle 5.2 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 Rn 6.2 Definition of the integral in general 6.3 The integral of a O-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 ca1culus and its analog for line integrals 7.3 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 if cohomology groups 8.4 Cohomology with compact supports 8.5 Exercises Index