I Differential Calculus.- §1. Categories.- §2. Topological Vector Spaces.- §3. Derivatives and Composition of Maps.- §4. Integration and Taylor's Formula.- §5. The Inverse Mapping Theorem.- II Manifolds.- §1. Atlases, Charts, Morphisms.- §2. Submanifolds, Immersions, Submersions.- §3. Partitions of Unity.- §4. Manifolds with Boundary.- III Vector Bundles.- §1. Definition, Pull Backs.- §2. The Tangent Bundle.- §3. Exact Sequences of Bundles.- §4. Operations on Vector Bundles.- §5. Splitting of Vector Bundles.- IV Vector Fields and Differential Equations.- §1. Existence Theorem for Differential Equations.- §2. Vector Fields, Curves, and Flows.- §3. Sprays.- §4. The Flow of a Spray and the Exponential Map.- §5. Existence of Tubular Neighborhoods.- §6. Uniqueness of Tubular Neighborhoods.- V Operations on Vector Fields and Differential Forms.- §1. Vector Fields, Differential Operators, Brackets.- §2. Lie Derivative.- $3. Exterior Derivative.- §4. The Poincaré Lemma.- §5. Contractions and Lie Derivative.- §6. Vector Fields and 1-Forms Under Self Duality.- §7. The Canonical 2-Form.- §8. Darboux's Theorem.- VI The Theorem of Frobenius.- §1. Statement of the Theorem.- §2. Differential Equations Depending on a Parameter.- §3. Proof of the Theorem.- §4. The Global Formulation.- §5. Lie Groups and Subgroups.- VII Metrics.- §1. Definition and Functoriality.- §2. The Hilbert Group.- §3. Reduction to the Hilbert Group.- §4. Hilbertian Tubular Neighborhoods.- §5. The Morse-Palais Lemma.- §6. The Riemannian Distance.- §7. The Canonical Spray.- VIII Covariant Derivatives and Geodesics.- §1. Basic Properties.- §2. Sprays and Covariant Derivatives.- §3. Derivative Along a Curve and Parallelism.- §4. The Metric Derivative.- §5. More LocalResults on the Exponential Map.- §6. Riemannian Geodesic Length and Completeness.- IX Curvature.- §1. The Riemann Tensor.- §2. Jacobi Lifts.- §3. Application of Jacobi Lifts to dexpx.- §4. The Index Form, Variations, and the Second Variation Formula.- §5. Taylor Expansions.- X Volume Forms.- §1. The Riemannian Volume Form.- §2. Covariant Derivatives.- §3. The Jacobian Determinant of the Exponential Map.- §4. The Hodge Star on Forms.- §5. Hodge Decomposition of Differential Forms.- XI Integration of Differential Forms.- §1. Sets of Measure 0.- §2. Change of Variables Formula.- §3. Orientation.- §4. The Measure Associated with a Differential Form.- XII Stokes' Theorem.- §1. Stokes' Theorem for a Rectangular Simplex.- §2. Stokes' Theorem on a Manifold.- §3. Stokes' Theorem with Singularities.- XIII Applications of Stokes' Theorem.- §1. The Maximal de Rham Cohomology.- §2. Moser's Theorem.- §3. The Divergence Theorem.- §4. The Adjoint of d for Higher Degree Forms.- §5. Cauchy's Theorem.- §6. The Residue Theorem.- Appendix The Spectral Theorem.- §1. Hilbert Space.- §2. Functionals and Operators.- §3. Hermitian Operators.