| 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 with fixed-sign terms
1.3 Dirichlet's criterion, Leibniz's criterion, and the integral test
1.4 Simple convergence and absolute convergence
1.5 Sequences of functions: pointwise convergence and uniform convergence
1.6 Series of functions
1.7 Power series
1.8 Definition of exponential, logarithmic, sine, and cosine functions
1.9 Taylor series
2. Functions of Several Variables
2.1 Algebraic and topological structure of Rn
2.2 Functions from Rn to R?
2.3 Limits
2.4 Continuity
3. Differential Calculus
3.1 Partial derivatives and directional derivatives
3.2 Differentiability: the derivative as a linear transformation
3.3 Derivative of a composite function
3.4 Inverse function theorem and implicit function theorem
3.5 Higher-order partial derivatives and Taylor’s formula
3.6 Simple extrema
3.7 Constrained extrema and the Lagrange multipliers method
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 will present the concepts and state the results, demonstrating many of them. The theory will also be illustrated with examples and applications. Students are encouraged to interact with the professor and solve exercises and problems.
Knowledge assessment during the teaching-learning process will consist of two written tests, each graded out of 10 points. The final teaching-learning grade will be the rounded sum of the written tests, provided it is less than or equal to 16 points. If, after rounding, the grade exceeds 16 points, the student must take an oral exam, and in this case, the final grade will be determined by the jury of the respective exam, with a minimum grade of 16 points. If a student required to take the oral exam does not attend, their final grade will be set at 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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