Real Analysis III

Code	13908
Year	2
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	N.A.
Learning outcomes	i) To understand, to relate and to apply concepts and basic results of integral calculus; ii) To formulate and to solve problems related to multiple integrals, line integral and surface integrals; iii) To solve problems related to integrals of differential forms; iii) To analyze and understand mathematical proofs; iv) To communicate using mathematical language, written and orally.
Syllabus	1. Multiple integrals 1.1. Construction of the integral and properties 1.2. Null measure sets and integrability 1.3. Fubini’s theorem 1.4. Change of coordinates 1.5. Application to the calculus of physical measures 2. Line integrals 2.1. Parametrization of curves 2.2. Line integrals of scalar and vector fields 2.3. Green’s theorem 3. Surface integrals 3.1. Differentiable surfaces 3.2. Integrals of scalar fields and flows of vector fields 3.3. Stokes’ curl theorem 3.4. Gauss-Ostrogradsky’s divergence theorem 4. Integrals of differential forms 4.1. Differential forms of degree 1 4.2. Line integral of a differential form 4.3. Invariance by homotopy 4.4. Closed and exact forms. Poincaré’s Lemma 4.5. Exterior product and differential forms of degree 2. Exterior differential 4.6. Surface integral of a differential form 4.7. Stokes’ theorem
Main Bibliography	- Dias Agudo, F. R. (1994). Análise Real, vol. I. Escolar Editora. - Edwards, H. M. (2013). Advanced calculus: a differential forms approach. Springer Science & Business Media. - Lages Lima, E. (2015). Curso de Análise, vol. 2. (11ª edição). IMPA. - Lages Lima, E. (2016). Análise Real, vol. 2. (6ª edição). IMPA. - Lages Lima, E. (2016), Análise Real, vol. 3. (4ª edição). IMPA. - Sarrico, C. (2009). Cálculo Diferencial e Integral para Funções de Várias Variáveis. Esfera do Caos.
Language	Portuguese. Tutorial support is available in English.