Análise em Variedades - Aula 21

Explore advanced mathematical concepts in this graduate-level lecture on Analysis on Manifolds, presented in Portuguese by Professor Vinicius Ramos at the Instituto de Matemática Pura e Aplicada (IMPA). Building upon prerequisites in Rn Analysis, delve into differentiable manifolds, manifolds with boundary, and orientable manifolds. Master the partition of unity and discover its application in Whitney's embedding theorem for compact manifolds. Examine tangent and cotangent bundles, differential applications, and regular values. Study alternating forms, differential forms, and exterior derivatives before progressing to surface integrals and Stokes' theorem. Investigate De Rham cohomology, the Mayer-Vietoris sequence, and homotopy invariance. Conclude with vector fields as sections and derivations, tensors, and their applications. Draw from renowned texts by Lima, Spivak, and Tu to enhance understanding of these advanced mathematical concepts.