| 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) Understand and relate basic concepts and results concerning numerical series, improper integrals, and series of functions;
ii) Formulate and solve problems related to numerical series, improper integrals, and series of functions;
iii) Understand and relate basic concepts and results concerning limits, continuity, and derivatives of vector-valued functions of several real variables;
iv) Formulate and solve problems related to limits, continuity, and derivatives of vector-valued functions of several real variables;
v) Analyze and understand mathematical proofs, particularly in the context of vector calculus;
vi) Communicate, in writing and orally, using mathematical language.
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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 |
Assessment during the teaching-learning process will consist of two written tests, each graded out of 10 points and, eventually, an oral exam. A student will pass if the sum of their test grades is greater than or equal to 9.5 points. Whenever the final grade is greater than or equal to 16.5 points, an oral exam will take place to confirm the grade; in that case, a minimum grade of 16 points is guaranteed from the outset. If an oral defense is required and the student does not attend, their final grade will be 16 points. To be eligible to take the exam, the student must have attended at least one class. Students with special status are subject to their own rules, as defined by the Academic Regulations.
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Language |
Portuguese. Tutorial support is available in English.
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