Subject: Mathematical Analysis II

Scientific Area:



80 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

1 - Stimulation and development of reasoning, formalization, rigour, computational, argumentation, deduction and abstraction skills.
2 - To equip students with the basic theoretical and pratical notions of Mathematical Analysis which will then be used in more advanced courses.
3 - The study of the fundamental concepts in the theory of integration of functions in one variable and its main applications.
4 - Introduction and formalization of the theory of Mathematical Analysis in R^n with the correponding applications to concrete problems.
5 - The generalization to R^n of the notions of limit, continuity and differentiability which were introduced in Mathematical Analysis I.
6 - To familiarize students with prooving results and requiring the knowledge of some them.


1 - Integral calculus in IR. Anti-derivatives Riemann Integral Integrability of piecewise continuous functions Fundamental Theorem of Calculus Integration: by parts; by change of variables; of trigonometric and rational functions Application to the computation of areas Improper Integrals
2 - Algebraic and topological structure of R^n. Norms and Euclidean distance Topological notions Sequences: Boundness; Limits; Cauchy Sequences; Fundamental Theorems; Series
3 - Functions from R^n to R^m: Limits and continuity Limits according to E. Heine and to Cauchy Continuity; Uniform continuity; Hölder and Lipschitz continuity; Continuity and compactness / connectedness Fixed point theorem Sequences and series of functions: Pointwise and Uniform convergence: fundamental results
4 - Differentiation in R^n Partial and Directional Derivatives. Differentiation. Composite Function Differentiation. Properties of differentiable functions Inverse Functions in R^n. Implicit Functions. Inverse Function Theorem and Implicit Function Theorem Taylor's Theorem Critical Points and Constraint Critical Points of a Differentiable Function. Karush-Kuhn-Tucker Theorem


Agudo, D. , 1969 , Lições de Análise Infinitesimal, Vol I e II, , Escolar Editora
Apostol, T. , 1983 , Cálculo, Vol II, 2ª Edição , Ed. Reverté Lda.
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 is done through the realization of two tests (each worth 50% of the final grade) to be solved individually. 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.