Learn advanced concepts in Analysis on Manifolds in this graduate-level mathematics lecture from IMPA (Instituto de Matemática Pura e Aplicada). Explore differentiable manifolds, manifolds with boundary, and orientable manifolds while delving into topics like partition of unity and Whitney's embedding theorem for compact manifolds. Master tangent and cotangent bundles, differential applications, regular values, alternating forms, differential forms, and exterior derivatives. Study surface integrals, Stokes' theorem, de Rham cohomology, Mayer-Vietoris sequence, and homotopy invariance. Examine vector fields as sections and derivations, tensors, and their applications. Prerequisites include Analysis in Rn. Based on foundational texts by Lima, Spivak, and Tu, this Portuguese-language lecture is part of a comprehensive graduate course series in mathematical analysis.