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Real Analysis II

Code 14762
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.
Teaching Methodologies and Assessment Criteria The classes will be theoretical-practical. The teacher presents the concepts and enunciates the results, demonstrating many of them. It also illustrates the theory with examples and applications. The student is encouraged to participate in classes, interacting with the teacher and sometimes solving exercises and problems.
In the teaching-learning period the evaluation will be periodic and will consist of 2 tests to be carried out on April 15 and June 2. Each test will be quoted to 10 values. A student will pass if the sum of test scores has a rating greater than or equal to 9.5 values. Whenever the final classification is greater than or equal to 16.5 values, an oral test for the defense of grade will take place, with a minimum score of 16 values being guaranteed. If the defense takes place but the student does not attend, he will have a final grade equal to 16 values. If the epidemiological situation does not allow the 1st test to be carried out, the evaluation will consist of a sin
Language Portuguese. Tutorial support is available in English.
Last updated on: 2022-06-16

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