Riemann integral in Rn. Line and surface integrals.
Fubini, Green, divergence, and Stokes' theorems.
Applications to problems in electromagnetism and continuum mechanics.
1. Multiple integrals
2. Line and surface integrals
3. Fundamental theorems of integral calculus in R^n
After concluding this course the student should be able to:
(i) know the definition and the basic properties of the Riemann integral of real functions defined in Rn (linearity, Fubini's theorem, change of integration variables, the Fundamental Theorem);
(ii) know the definition, the basic properties, and be able to compute line integrals on sectionally C¹ paths; (iii) know the definition, basic properties and be able to compute surface integrals on orientable sectionally C¹ surfaces; (iv) know and know how to apply the classical theorems of vector analysis (Green, divergence, and Stokes theorems) to problems in Electromagnetism and Continuum Mechanics.
1. Riemann integral in Rn
2. Line integrals
3. Surface integrals
4. Classical theorems of vector analysis
5. Applications of the classical theorems to Electromagnetism and Continuum Mechanics
Gabriel E. Pires; Cálculo Diferencial e Integral em Rn, Coleção Ensino da Ciência e da Tecnologia, vol. 45, IST Press, Lisboa, 2012
 João Palhoto Matos, Cálculo Diferencial e Integral em Rn - available online - https://cdi2tp.math.tecnico.ulisboa.pt/texto/
 Gabriel Pires e Departamento de Matemática do IST, Exercícios de Cálculo Integral em Rn, Colecção de Apoio ao Ensino, volume d, IST Press, Lisboa, 2007
 B. Demidovich et al.; Problemas e Exercícios de Análise Matemática, McGraw Hill/Mir, Amadora/Moscovo, 1999.
Continuous assessment is privileged: 2 or 3 digital written documents (e-folios) during the semester (40%) and a presence-based final exam (p-folio) in the end of the semester (60%). In due time, students can alternatively choose to perform one final presence-based exam (100%).
Students are assumed to be conversant with the subject matter studied in Linear Algebra I and II, and in Elements of Infinitesimal Analysis I and II.