I Differential Calculus. - 1. Categories. - 2. Topological vector spaces. - 3. Derivatives and composition of map. - 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 exponential map. - 5. Existence of tubular neighborhoods. - 6. Uniqueness of tubular neighborhoods. - V Differential Forms. - 1. Vector fields differential operators brackets. - 2. Lie derivative. - 3. Exterior derivative. - 4. The canonical 2-form. - 5. The Poincaré lemma. - 6. Contractions and Lie derivative. - 7. Darboux 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 Riemannian Metrics. - 1. Definition and functoriality. - 2. The Hilbert group. - 3. Reduction to the Hilbert group. - 4. Hilbertian tubular neighborhoods. - 5. Non-singular bilinear tensors. - 6. Riemannian metrics and sprays. - 7. The Morse-Palais lemma. - VIII Integration of Differential Forms. - 1. Sets of measure 0. - 2. Change of variables formula. - 3. Orientation. - 4. The measure associated with a differential form. - IX Stokes' Theorem. - 1. Stokes' theorem for a rectangular simplex. - 2. Stokes' theorem on a manifold. - 3. Stokes' theorem with singularities. - 4. The divergence theorem. - 5. Cauchy'stheorem. - 6. The residue theorem. - The Spectral Theorem. - 1 Hilbert space. - 2 Functionals and operators. - 3 Hermitian operators. Language: English