Complements of Mathematical Analysis

Recommended Prerequisites

Basic knowledge of Real Analysis, Linear Algebra and Analytic Geometry.

Teaching Methods

In theoretical classes, the most relevant concepts and theoretical results are presented, accompanied by illustrative examples of the theory, with simple and motivating applications. In TP classes, students must solve the proposed exercises, having various degrees of difficulty, and will be confronted with problems in the context of Science and Engineering applications. Active participation of students in class discussions, individual and team work, and correct use of  the available office hours should be strongly encouraged by the instructor.

Learning Outcomes

Having a basic knowledge of Real Analysis, acquired  in the curricular units of BSc programs, students learn,  in this curricular unit, concepts and methods of differential and integral calculus of functions defined in Rn. Another goal of the unit is to provide fundamental knowledge about series and series of functions, including Fourier series . The approach used in the presentation of the topics is essentially addressed to applications to Science and Biology.

The main competences to be developed are: ability to formulate and solve problems; knowledge of mathematical results; design, analyze and correctly use mathematical models; ability to work in teams; critical thinking.

Work Placement(s)

No

Syllabus

1. Sequences and infinite series of  real numbers

1.1. Convergence criteria

1.2. Function series: power series; Taylor’s formula and Taylor series; Fourier series

2. Parametric equations of curves and polar coordinates

3. Differential Calculus in Rⁿ

3.1. Limits and continuity

3.2. Partial and direccional derivatives, differentiability

3.3. Chain rule

3.4. Extrema and Lagrange Multipliers

4. Integral Calculus in R² and R³

4.1. Double and triple integrals; applications and change of variable

4.2. Curvilinear integral and Green’s theorem

4.3. Surface integral and Stokes and divergence theorems.

Head Lecturer(s)

Susana Margarida Pereira da Silva Domingues de Moura

Assessment Methods

Continuous assessment
Frequency: 100.0%

Final assessment
Exam: 100.0%

Bibliography

1. J. Stewart, Cálculo, Volumes I e II, 7ª Ed., CENGAGE Learning, 2012.

2. J. Glyn, D. Burley, D. Clements, P. Dyke, J. Searl, N. Steele, Advanced Modern Engineering Mathematics, 4ª Ed., Prentice Hall, 2010.

3. A. Breda e J. Costa, Cálculo com funções de várias variáveis, McGraw-Hill, 1996.

4. M. Spiegel, Análise de Fourier, Coleção Schaum, 1977.

5. J. Carvalho e Silva, Princípios de Análise Matemática Aplicada, McGraw-Hill, Lisboa, 1994.