Mathematical Analysis III

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 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.Compute areas and volumes, as well as the center of mass of a solid (with arbitrary density function), using double and triple integrals;

2. Compute areas and lengths of curves in space using line and surface integrals;            

3. Solve problems involving the connections between the types of integrals studied (double, triple, line and surface integrals);

4. Solve problems involving applications of integration to mathematical modelling.

5. Solve higher order linear differential equations;

6. Solve systems of linear differential equations with constant parameters;

7. Compute the Laplace transform and its inverse;

8. Solve a differential equation using the Laplace or Fourier transforms;

9. Solve problems involving applications of differential equations in mathematical modelling.    

Work Placement(s)

No

Syllabus

I. Integral calculus in R2 and R3

I.1 Double integrals and applications

I.2 Triple integrals and applications

I.3 Change of variables

I.4 Line integrals. Green's Theorem

I.5 Surface integrals. Stoke's and divergence theorems

 

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 in solving differencial equations.    

Head Lecturer(s)

Raquel Susana Giraldes Caseiro

Assessment Methods

Continuous assessment
2 or more midterm exams: 100.0%

Final assessment
Course assessment can also be made by exam as an alternative to the midterm exams assessment. : 100.0%

Bibliography

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

[2] Gabriel E. Pires: Cálculo diferencial e integral em Rn. IST Press (Colecção Ensino da Ciência e da Tecnologia), 2012.

[3] Ana d'Azevedo Breda, Joana Nunes da Costa: Cálculo com funções de várias variáveis. McGraw-Hill, Lisboa (1996).

[4] Dennis G. Zill: Equações Diferenciais com aplicações em modelagem. Cengage Learning (tradução da 10ª edição norte-americana), 2016 

[5] Figueiredo, D.; Neves, A.. Equações Diferenciais Aplicadas. Coleção Matemática Universitária, IMPA, R. Janeiro, 2018.

[6] Spiegel, M. –  Análise de Fourier, Colecção Schaum 1977.