Definition and examples of differentiable manifolds. Uniqueness of differential structures. Differentiable manifolds with boundary. Smooth applications. Tangent space. The differential of an application. Submersions, immersions, embeddings. Singularities of a smooth application. Constant rank theorem. Critical points, critical values, regular values. Null subsets of a variety. Sard theorem. Partitions of unity.

Multilinear applications, tensor product, exterior product. Definition of a vector bundle. Tangent bundle. Morphisms of vector bundles. Operations on vector bundles.

Vector fields. Autonomous differential equations on a manifold. Flow theorem. Lie bracket of two vector fields.

Integration on manifolds. Differential forms. Orientations and volume forms. Integration of differential forms on oriented manifolds. Exterior differential. Stokes' theorem. Introduction to de Rham cohomology.