#### Objective

#### Content

- Submanifold of a manifold: Immersion, submersion, the rank theorem, immersed parts, sub-tangent space of an immersed part, submanifolds.
- Manifolds: Charts, Atlas, definition of a manifold, induced topology, differentiable map between 2 manifolds, germs.
- Tangent spaces : Tangent vector, the manifold TM, vector fields, derivation on an algebra, bracket, the tangent map, the cotangent manifold T*M, fields of 1-form.
- Differential equations and integral manifolds: System of differential equations, theorem of existence and uniqueness, flows, sub-integral manifolds, Frobenius theorem.
- Differential Forms: Exterior Algebra, differential forms on a open of IRn. Lie derivation, Poincare's Lemma Integration on a manifold, Stokes' Theorem.
- Tensorâ€™s Algebra on a manifold: Tensors of type , tensor fields, alternating tensors, tensor products.