Subject: Mathematical Analysis III

Scientific Area:



80 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

O1 - To familiarize the student with the theory of integration in R^n and with its main applications.
O2 - To equip students with the basic theoretical and pratical notions of Fourier Analysis which will then be used in subsequent curricular units.


1 - Integral Calculus in R^n Multiple Integrals: definition, geometric interpretation and properties Change of variables Volumes and areas
2 - Line Integrals Lines in R^n and parametrizations Line integral of scalar functions and of vector fields The Fundamental Theorem of Calculus for Line Integrals Green's Theorem Applications
3 - Surface Integrals Parametrization of elementary surfaces Surface integral of scalar functions and of vector fields Divergence and Curl Divergence Theorem Stokes's Theorem Applications
4 - Introduction to Fourier Analysis Fourier Series: pointwise convergence and uniform convergence Fourier Transform


Agudo, D. , 1969 , Lições de Análise Infinitesimal, Vol I e II, , Escolar Editora
Ávila, G. S. S. , 1979 , Cálculo III-Diferencial e Integral , Livros Técnicos e Científicos Editora S.A.
Campos Ferreira, J. , 2002 , Introdução à Análise em IR^n , IST
Kaplan, W. , 1972 , Cálculo Avançado, Vol II , Editora da Universidade de S. Paulo
Lang, S. , 1980 , Cálculo, Vol II , Livros Técnicos e Científicos Editora
Piskounov N. , 1986 , Cálculo Diferencial e Integral, Vol. I e II , Lopes da Silva Editora
Swokowski, E. W. , 1983 , Cálculo com Geometria Analítica, Vol II , McGraw-Hill
Courant, R., John, F. , 1989 , Introduction to Calculus and Analysis I , Springer
Dieudonné, J. , 1972 , Eléments d'Analyse I , Hermann, Paris
Cartan, H. , 1972 , Cours de Calcul Différentiel , Hermann, Paris
Lang, Serge , 1993 , Real and Functional Analysis , Springer

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Oral and written exposition of the currricular unit's syllabus. Discussion and solving exercises and concrete problems in small groups or individually. The evaluation of the curricular unit will focus on two written tests, each with a weight of 45% of the final grade, and a coursework, worth 10% of the final grade. The coursework will be about solving a theoretical exercise or a theorem, assigned randomly to each student. In this way the student can assess his progress and change his studying strategies if necessary. During the supplementary Exam period students who have not successfully completed the curricular unit can choose between retaking one of the tests or doing an exam which includes all the contents of the curricular unit. The importance of this exam is to test the student's ability to relate the notions and results introduced throughout the curricular unit.