Topology and analysis in metric and vector spaces, with emphasis on R^d and the space of continuous functions. Classical theorems on the space of continuous functions: Ascoli-Arzèla, Stone-Weierstrass. Fréchet derivative in vector spaces, its properties and Taylor series. Inverse function and implicit function theorems. Outline of the theory of subvarieties of R^d. Null measure and Sard’s theorem. Multidimensional integrals and the Riemann-Lebesgue theorem.

References:
 J.Munkres, Analysis on Manifolds (Westview Press)
S. Lang, Undergraduate Analysis (Springer).
R. I. Oliveira, lecture notes
W. Rudin, Principles of Mathematical Analysis (Ao Livro Técnico).

* Basic syllabus. The teacher has the autonomy to make any changes.