This chapter introduces the notion of a smooth manifold as a topological space locally modeled on \(\mathbb{R}^n\) and equipped with a compatible differentiable structure, providing the setting in which differential calculus extends beyond Euclidean spaces. It develops the basic geometric objects associated with manifolds: tangent space, smooth maps, and vector fields.
The chapter further studies the algebra of differential forms, including the wedge product and the exterior derivative \(d\), establishing the foundations of exterior calculus on manifolds and preparing the framework for de Rham cohomology and Stokes theorem.
- معلم: Bahayou Amine
These notes present a rigorous yet accessible introduction to the fundamental language of smooth manifolds and to the principal tools of differential geometry. Although an initial study of manifolds may appear dominated by definitions and formal structures, this formalism serves a precise purpose: it recasts the familiar concepts of multivariable calculus in a coordinate-independent framework, ensuring that the underlying geometric notions remain invariant under changes of coordinates.
- معلم: Bahayou Amine