Differential and Integral Calculus III

Objectives

Master of:
- Resolution of elementary ordinary differential equations; resolution of linear differential equations and systems of linear differential equations.
- Existence, uniqueness and continuous dependence of solutions of ordinary differential equations.
- Gauss and Stokes theorems, general properties of the divergence and curl of vector fields, and applications.
- Resolution of elementary linear partial differential equations of 1st and 2nd order.
- General properties and convergence of Fourier series, Fourier transform and applications.

Program

Ordinary Differential Equations (ODEs): examples of solvable 1st order ODEs, integration factors; existence, uniqueness and continuous dependence of solutions of systems of 1st order ODEs; variation of constants formula; ODEs of order > 1; Laplace transform and applications to ODEs.

Gauss and Stokes Theorems and introduction to Partial Differential Equations (PDEs): surfaces in R^3; surface integrals of scalarand vector fields; Gauss and Stokes Theorems; divergence and curl of vector fields; derivation of the continuity, wave, heat, Laplaceand Poisson differential equations.

PDEs and Fourier series: linear 1st order PDEs; wave, heat, Laplace and Poisson equations; trigonometric Fourier series; solutions of wave, heat, Laplace and Poisson equations, via separation of variables and Fourier series; Fourier transform and applications.

Teaching Methodologies

The teaching methodologies aim to promote learning based on problem solving, reinforcing the practical component, active learning, autonomous work and student accountability. The assessment model incorporates exam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).

Bibliography

* Elementary Differential Equations and Boundary Value Problems, Boyce and Di Prima, 2013, 10th ed Wiley.
* Vector Calculus, Marsden and Tromba, 2012, 6th ed Freeman.
* Análise Complexa e Equações Diferenciais, Luís Barreira, 2019, 4ª ed. IST Press.
* Introdução à Análise Complexa, Séries de Fourier e Equações Diferenciais, Pedro Girão, 2018, 2ª ed. IST Press.
* Métodos de Resolução de Equações Diferenciais e Análise de Fourier com Aplicações, Luís Magalhães, 2013 DM-IST.
* Análise de Fourier e Equações Diferenciais Parciais, Djairo Figueiredo, 2012, 4ª ed IMPA.
* Cálculo Diferencial e Integral em R^n, Gabriel Pires, 2016, 3ª ed. IST Press.
* Integrais em Variedades, Luís T. Magalhães, 1993, 2ª ed. Texto Editora.
* Exercícios de Análise Complexa e Equações Diferenciais, Luís Barreira e Claudia Valls, 2010, 2ª ed. IST Press.
* Exercícios de Cálculo Integral em R^n, Gabriel Pires, 2018, 2ª ed. IST Press.