Multivariate Analysis
Topology of the Euclidean Space: ball, open set, interior, adherent point, closed set, closure, density, isolated point, accumulation point, compact set, Cantor's Theorem (compact fit), Cauchy sequence, T. de Bolzano- Weierstrass, related set. Rn functions in Rm: limit, continuity, partial derivatives, directional derivatives, differentiable functions, Mean-value theorem, Leibniz's rule, Schwarz's theorem, critical points, second order optimality criteria, Lagrange multipliers, Implicit function theorem . Metric spaces: metric topology, convergence, density, separability, isometries. Complete metric spaces. Compactness, sequential compactness. Continuous applications between metric spaces, fixed-point theorems.
Basic Information
Workload
60 hours
Requirements
Analysis over R
Mandatory:
- Lima, E. L. Curso de Análise. Volume 2. Projeto Euclides, IMPA, 2000.
- Rudin, Walter. Principles of mathematical analysis. Vol. 3. New York: McGraw-hill, 1964.
- Spivak M. Calculus on manifolds: a modern approach to classical theorems of advanced calculus. CRC Press; 2018.
Complementary:
- Thomson, B.S., Bruckner, J.B. and Bruckner, A.M. Elementary real analysis. 2008. ClassicalRealAnalysis.com
- Lages, Elon. Análise real. Volume 2. Coleção Matemática Universitária, IMPA, 1989.
- Campos Ferreira, J. Introdução à Análise em Rn. Instituto Técnico Superior de Lisboa. 2004 https://math.tecnico.ulisboa.pt/textos/iarn.pdf
- Apostol, T. M. Mathematical analysis. Addison-Wesley Reading, 1964.
- Apostol TM. Calculus. Vol. 2, Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability. 1962.