# Mathematical Analysis IV B

## Objectives

The program deals with the study of first and second order differential equations, system of differential equations and partial differential equations. Some related topics will be studied in the end.

## General characterization

##### Code

5006

##### Credits

6.0

##### Responsible teacher

Fábio Augusto da Costa Carvalho Chalub

##### Hours

Weekly - 5

Total - 70

##### Teaching language

Português

### Prerequisites

Linear Algebra and Mathematical Analysis I, II, III.

### Bibliography

Notes made available at "Clip" to the students, authored by the course professor.

Other material

Basic textbooks.

Online materal: Vilatte, Jaime, Equações diferenciais e equações de diferenças, FEUP http://villate.org/doc/eqdiferenciais/eqdif_20110426.pdf

Apostol, T.M., Calculus, Volume I and Volume II, Blaidsell Publishing Company.

Howard, Anton, Calculus: A New Horizon, John Wiley and Sons.

Taylor, A.E., Man, W.R., Advanced Calculus, John Wiley and Sons.

Stewart, J. Cálculo, Thomson Learning.

Ferreira, M. A. e Amaral, I, Matemática, Integrais míltiplos, equações diferenciais, Edições Síabo

Extra references:

Topics 7, 8 e 10. Butkov, E. Mathematical Physics.

Topic 9. The Mathematics of Medical Imaging: A Beginner''''s Guide, Timothy G. Feeman, Springer

### Teaching method

Lectures (3hs/week, online), exercise class (1hs in person - 1st time students only; 1hs onle - all students). Homework.

### Evaluation method

**Presence in class of optional.**

Two tests (T1 and T2); it is required minimum grade of 7 in the second test.

If T2=9.5, then MF=9; otherwise MF=(T1+T2)/2.

The final grade consist in the rounding of MF to the nearest integer (n.5 is rounded to n+1).

For students that to not obtain approval in tests ("continuous evaluation") it is possible to do a final exam ("recurso").

## Subject matter

1. First order differential equations. Exact differentials. Integrating factors. Separation of variables. Homogeneous equations. Linear equations. Qualitative theory.

2. Second order differential equatons. Linear equations and Euler equation. Newton''''s second law. The free, damped and forced harmonic oscilator. Variation of constants.

3. Solution in series. Bessel, Lagrange and Hermite functions.

4. Linear equations of higher order.

5. Systems of linear equation with constant coefficients. Differential equations in polar coordinates. Linearization of non-linear systens near equilibria.

6. Partial differential equations. Heat, wae and Laplace equations. Laplacian in spherical coordinates.

8. Laplace transform and its use in differential equations. Dirac delta.

9. Introduction to variational calculus. The law of sinus (Snell-Descartes). Catenary. The principle of minimum action. Euler-Lagrange equation and the lagrangean. Brachistochone curve.

10. Introduction to inverse problems. Radon transform.

## Programs

Programs where the course is taught: