Real Analysis II

Code	13903
Year	1
Semester	S2
ECTS Credits	7,5
Workload	TP(75H)
Scientific area	Mathematics
Entry requirements	NA
Learning outcomes	i) To understand and relate concepts and basic results on numerical series; ii) To formulate and solve problems related to numerical series and series of functions; iii) To understand and relate concepts and basic results about limits, continuity and derivative of vector-valued functions; iv) To formulate and to solve problems related to limits, continuity and derivative of vector-valued functions; v) To analyze and to understand mathematical proofs, particularly in the context of vector calculus; vi) To communicate using mathematical language, written and orall
Syllabus	1. Series 1.1. Definition and examples. 1.2. Series of non-negative or non-positive terms: comparison test, d’Alembert test and Cauchy test; 1.3. Alternating series 1.4. Simple and absolute convergence. 1.5. Sequences of functions. Simple and uniform convergence. 1.6. Series of functions. 1.7. Power series. 1.8. Taylor series and analytic functions. 2. Functions of several variables 2.1. Algebraic and topologic structure of IR^n. 2.2. Functions from IR^n to IR^m. 2.3. Limits. 2.4. Continuity. 3. Differential calculus 3.1. Partial derivative and directional derivative. 3.2. Differentiability. The derivative as a linear map. 3.3. Derivative of a composition of functions. 3.4. Inverse function theorem and implicit function theorem. 3.5. Higher order partial derivatives and Taylor formula. 3.6. Extremes. 3.7. Extremes with constraints and method of Lagrange multipliers. 3.8. Applications.
Main Bibliography	- Conway, J. B. (2017). A First Course in Analysis. Cambridge University Press. - Dias Agudo, F. R. (1994). Análise Real, vol. I. (2.ª edição). Escolar Editora. - Ferreira, J. C. (2008). Introdução à Análise Matemática. (9.ª edição). Fundação Calouste Gulbenkian. - Lages Lima, E. (2017). Curso de Análise, vol. 1. (14.ª edição). IMPA. - Lages Lima, E. (2015). Curso de Análise, vol. 2. (11.ª edição). IMPA. - Lages Lima, E. (2017). Análise Real, vol. 1. (12.ª edição). IMPA. - Lages Lima, E. (2016). Análise Real, vol. 2. (6.ª edição). IMPA. - Marsden, J. E., & Tromba, A. J. (2012). Vector calculus. (6th ed.). W H Freeman & Co. - Sarrico, C. (2009). Cálculo Diferencial e Integral para Funções de Várias Variáveis. Esfera do Caos.
Language	Portuguese. Tutorial support is available in English.