Faculdade de Ciências e Tecnologia

Mathematical Analysis IV A

Code

10980

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

Elvira Júlia Conceição Matias Coimbra

Weekly hours

6

Teaching language

Português

Objectives

The first chapters of the course are devoted to the complementary study of Ordinary Differencial Equations begun in Mathematical analysis III-A. the work of Cauchy, Liouville and others showed the importance of establishing general theorems to guarantee the existence of solutions to certain specific classes of differential equations. One of the chapters of our course is concerned with the proofs of some of these theorems. With respect to multiple integration, our study is intended to familiarize the student with the properties and with the methods of finding the values of doule and triple integrals. Our course also includes line integrals and surface integrals. These kind of integrals are of fundamental importance on both pure and applied mathematics. Finally we present two generalizations of the Green´s theorem, namely the Stokes theorem and the divergence theorem. The divergence theorem is useful in connection with the consideration of solid angles.

Prerequisites

A background in ordinary differential equations and knowledge in functions of several variables ( Mathematical Analysis III-A).

Subject matter

1. Complements on ordinary differential equations- Use of power series to obtain solutions of a linear differential equation of order two. The homogeneous linear equation of second order- singular points. The Bessel equation. The Bessel functions. Systems of linear differential equations. Existence and uniqueness theorems for differential equations.

2. Double integrals. Definition of a double integral. Some properties. Evaluation of a double integral. Geometric interpretation of a double integral as a volume.

3. Line integrals. Green´s theorem in the plane. Some applications of  Green´s theorem. Change of variables in a double integral.

4. Surfaces. Surface area.

5. Integrals depending on a parameter. Leibniz´s rule.

6. Triple integrals. Definition of a triple integral. Some properties. Cylindrical co-ordinates. Spherical co-ordinates. Applications.

7. Scalar fields and vector fields. Gradient, curl, divergence and laplacian. Conservative fields.

8. Surface integrals.  Stokes theorem and the divergence theorem.  Applications of the divergence theorem- solid angle.

Bibliography

Bibliografia :

1. Freitas, A. C. - Análise infinitésimal - Volumes 1 e 2 - Notas de Lições para alunos do 2º ano em engenharias.
2. Marsden - Basic Complex Analysis
3. Taylor A. E; Man, W.R. - Advanced Calculus - John Wiley and sons
4. Taylor A. E; Man, W.R. - Advanced Calculus - John Wiley and sons

Teaching method

Theoretical issues are presented and explained  in the first part of each lecture. These issues are immediatly applied  by solving problems, where the application of the concepts is necessary. The students also solve a few exercices as homework.

Evaluation method

Evaluation is made by three tests along the semester or a final exam. The final classification is the weighted mean of the classification of the tests or, in alternative, the mark obtained in the final exam.


 

First test 1 of  April from 14:30h  to 16:30h;

Second test 15 of May from 13:00h to 15:00h;

Third  test  9 of  June from 15:00 to 17:00h


Courses