Mathematical Analysis II

Recommended Prerequisites

Mathematical Analysis I and Linear Algebra and Analytic Geometry.

Teaching Methods

The teaching in this course will assume two formats: theoretical and example classes. During a theoretical class teaching will be mostly expository. During an example class teaching will consist 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:

i)     calculate the sum of a geometric or telescopic series;

ii)    decide whether a number series is convergent or not;

iii)   expand a real function as a power series;

iv)   compute the Taylor series of a function;

v)    identify the graph and level curves of a function of R2;

vi)   study the limit of a function of Rn at a point;

vii)  calculate the partial derivatives of a function of Rn;

viii) study the differentiability of a function of Rn at a point;

iv) compute areas and volumes using double and triple integration, respectively.

Work Placement(s)

No

Syllabus

I.   Series

I.1 Sequences

I.2 Elementary number series

I.3 Convergence criteria

I.4 Taylor and power series

 

II.   Calculus in Rn

II.1 Topology and functions of Rn

II.2 Limits and continuity

II.3 Partial derivatives

II.4 Differentiability and applications

II.5 Double integrals and applications

II.6 Triple integrals and applications.

Head Lecturer(s)

Stéphane Louis Clain

Assessment Methods

Continuous assessment
2 or more midterm exams : 100.0%

Final assessment
Exam: 100.0%

Bibliography

[1] James Stewart: Cálculo, Volume II, Cengage Learning (tradução da 8ª edição norte-americana) 2017 

[2] J. Campos Ferreira, Introdução à Análise Matemática, 11ª edição, Fundação Calouste Gulbenkian, Lisboa, 7a. Edição, (2014).

[3] J. Carvalho e Silva, Princípios de Análise Matemática Aplicada,  McGraw-Hill, (2005).

[4] Carlos Sarrico, Análise Matemática, Leitura e exercícios, 6ª edição, Colecção Trajectos Ciência n. 4, Gradiva, (2005).