Mathematical Analysis III

Year
2
Academic year
2023-2024
Code
01018979
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, Mathematical Analysis II 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 will be 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. Determine the nature of convergence of a series and compute the sum of convergent geometric and telescoping series;
2. Compute the Taylor series expansion of a function;
3. Compute the Fourier series of a periodic function;
4. Solve higher order linear differential equations;
5. Solve a system of linear differential equations with constant coefficients;
6. Compute the Laplace transform and its inverse;
7. Solve a differential equation using the Laplace or Fourier transforms;
8. Solve problems involving applications of differential equations in mathematical modelling.

Work Placement(s)

No

Syllabus

I. Series
I.1 Sequences and number series
I.2 Convergence criteria
I.3 Series of functions
I.4 Convergence of series of functions
I.5 Power series (Taylor polynomial and Taylor series)
I.6 Fourier Series

II. Linear Differential Equations
II.1 Annihilator, reduction of order and variation of parameters methods
II.2 Systems of linear differential equations with constant parameters
II.3 Laplace and Fourier transforms and applications.

Head Lecturer(s)

Amílcar José Pinto Lopes Branquinho

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] Dennis G. Zill, Equações Diferenciais com Aplicações em Modelagem, tradução da 10ª edição norte-americana, Cengage Learning, 2016.
[3] Edwin "Jed" Herman e Gilbert Strang (entre outros), Calculus, volumes 2 e 3, OpenStax, 2018. (Disponíveis online em: https://openstax.org/details/books/calculus-volume-2; 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] Elon Lages Lima, Curso de Análise, volume 2, 11ª edição, Projecto Euclides, IMPA, 2004.