Multivariate Analysis

Basic information


45 hours


Analysis over R


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.

Teaching Plan



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•    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.


•    Thomson, B.S., Bruckner, J.B. and Bruckner, A.M. Elementary real analysis. 2008.
•    Lages, Elon. Real analysis. Volume 2. University Mathematical Collection, IMPA, 1989.
•    Campos Ferreira, J. Introduction to Analysis over Rn. Instituto Superior Técnico de Lisboa. 2004
•    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.