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)    compute the sum of a geometric series or telescopic series;
ii)   decide whether a series is convergent or not;
iii)  expand a real function as a power series;
iv)  compute the Fourier series of a periodic function;
v)   compute the direction of greatest rate of increase of a real function of two variables;
vi)  solve a constrained extrema problem;
vii) compute areas and volumes using double and triple integration, respectively.

Work Placement(s)

No

Syllabus

I.   Series of real numbers and Function Series
I.1 Introduction to series of real numbers, elementary series
I.2 Convergence criteria
I.3 Function series
I.4 Power Series
I.5 Taylor series
I.6 Fourier Series

II.   Real functions of several real variables
II.1 Level curves and level surfaces
II.2 Limits and continuity
II.3 Partial derivatives and directional derivatives
II.4 Gradient vector, tangent plane and linearization
II.5 Chain Rule
II.6 Global and local extrema. Lagrange multipliers

III.   Multiple integrals
III.1 Introduction to multiple integration
III.2 Fubini's Theorem
III.3 Change of coordinates.

Head Lecturer(s)

Jorge Manuel Sentieiro Neves

Assessment Methods

Final Assessment
Exam: 100.0%

Assessment
Frequency: 100.0%

Bibliography

[1] Stewart, J., Cálculo, Volumes I e II, 5ª edição, Pioneira, S. Paulo. (2006)
[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).