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