• The student is familiar with the notion of smooth manifold and smooth map
• The student understands the notions of tangent space, vector fields, cotangent space and differential forms
• The student understands flows of vector fields
• The student can manage integration theory on smooth manifolds, and knows the general Stokes' Theorem 

A smooth n-dimensional manifold M is a topological space which locally looks like Rⁿ. For example, a sphere in R³ is a 2-dimensional manifold, since locally it can be smoothly parameterized using two coordinates (e.g. the latitude and the longitude). This course consists of the study of this fundamental concept of mathematics and the basic constructions associated with it. Multivariable calculus on Rⁿ extends to any smooth n-dimensional manifold M, and it can be used to understand global properties of M, which leads to a subtle interplay between analysis and topology.

The course should be considered as a natural and necessary preparation for a large number of master courses: algebraic geometry, differential geometry, dynamical systems, global analysis, Lie groups, Riemann surfaces, symplectic geometry.