Smooth manifolds. Submanifolds, product manifolds, boundary of a manifold and examples of these. The tangent space, cotangent space and the differential of a smooth map. Embeddings, immersions, submersions and diffeomorphisms. Applications of partitions of unity. Vector bundles, vector fields and flows. Orientability. The Lie derivative and the Lie bracket. Differential forms and operations on them. Integration on manifolds. Applications of the Stokes's Theorem to cohomology.