| Code |
14762
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| Year |
1
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| Semester |
S2
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| ECTS Credits |
7,5
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| Workload |
TP(75H)
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| Scientific area |
Mathematics
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Entry requirements |
NA
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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
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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.
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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.
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Teaching Methodologies and Assessment Criteria |
The classes will be theoretical-practical. The professor presents the concepts and states the results, demonstrating many of them. Theory is also illustrated with examples and applications. The student is encouraged to interact with the professor and to solve exercises and problems.
The knowledge assessment during the Teaching-Learning process will consist of two written exams, each worth 10 points. The final Teaching-Learning grade will be given by rounding to the units the sum of the written tests, provided it is less than or equal to 16 points. If after rounding the grade is higher than 16 points, the student must take an oral exam, and the final grade, in this case, will be determined by the jury of the respective exam, and it cannot be less than 16 points.
To be admitted to the exam, the student must have attended at least one class.
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Language |
Portuguese. Tutorial support is available in English.
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