Mathematical Analysis II

Year
1
Academic year
2023-2024
Code
01018968
Subject Area
Mathematics
Language of Instruction
Portuguese
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Compulsory
Level
1st Cycle Studies

Recommended Prerequisites

Mathematical Analysis I and Linear Algebra and Analytic Geometry.

Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasized. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

Learning Outcomes

The student who successfully completes this course will be able to:
1. Detect non-continuous real functions of two variables at a given point;
2. Compute the directions of greatest growth of a real function of two variables;
3. Solve a constrained extrema problem;
4. Compute areas and volumes, as well as the center of mass of a solid (with arbitrary density function), using double and triple integrals;
5. Compute areas and lengths of curves in space using line and surface integrals;
6. Solve problems involving the connections between the types of integrals studied (double, triple, line and surface integrals);
7. Solve problems involving applications of integration to mathematical modelling.

Work Placement(s)

No

Syllabus

I. Real functions of several variables
I.1 Limits and continuity
I.2 Partial derivatives
I.3 Differentiability
I.4 Chain rule
I.5 Directional derivatives and gradient vector
I.6 Vector valued functions and Jacobian matrix
I.7 Extrema of functions. Lagrange multipliers

II Integral calculus in R2 and R3
II.1 Double integrals and applications
II.2 Triple integrals and applications
II.3 Change of variables
II.4 Line integrals. Green's Theorem
II.5 Surface integrals. Stoke's and divergence theorems.

Head Lecturer(s)

Joana Margarida Mavigné de Andrade Alves de Sousa Nunes da Costa

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
2 or more midterm exams: 100.0%

Bibliography

[1] James Stewart, Cálculo, volume II, tradução da 8ª edição norte-americana, Cengage Learning, 2017.
[2] Ana d'Azevedo Breda e Joana Nunes da Costa, Cálculo com Funções de Várias Variáveis, McGraw-Hill, Lisboa,  1996.
[3] Edwin "Jed" Herman e Gilbert Strang (entre outros), Calculus, volume 3, OpenStax, 2018. (Disponível online em: https://openstax.org/details/books/calculus-volume-3)
[4] Gilbert Strang, Calculus,  Wellesley-Cambridge Press, 1991. (Disponível online em: https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/)
[5] Jerrold E. Marsden e Anthony Tromba, Vector Calculus, 5ª edição, W.H. Freeman, 2003.
[6] Elon Lages Lima, Curso de Análise, volume 2, 11ª edição, Projecto Euclides, IMPA, 2004.