Mathematical Analysis II E


Domain of the basic techniques required for the solution of ordinary differential equations as well as to the Mathematical Analysis of functions of real variable.

The students should acquire not only calculus capabilities, fundamental to the acquisition of some of the knowledge lectured in other Engineering subjects, but also to develop methods of solid logic reasoning and analysis.

These capabilities ensure autonomy in the analysis and in resolution of new problems to the future engineer, opening the possibility to acquire more complex mathematical tools, possibly needed through is career.

General characterization





Responsible teacher

Jorge Manuel Leocádio André


Weekly - 6

Total - 80

Teaching language



The student must master the mathematical knowledge lectured in the curricular unit Mathematical Analysis I, respecting to the Mathematical Analysis of real functions of real variable, with a particular focus in differential and integral calculus.


H. ANTON,  I. BIVENS, S. DAVIS, Cálculo, volume II, ARTMED editora, 2005

T. APOSTOL, Calculus, volume II, John Wiley & Sons, 1969

F. R. DIAS AGUDO, Análise Real, Livraria Escolar Editora, 1994

E. LAGES LIMA, Curso de Análise volume 2, Projecto Euclides, Publicações IMPA, 2000

C. SARRICO, Cálculo Diferencial e Integral para funções de várias variáveis, Esfera do Caos Editores, 2009

A. A. SÁ, B. LOURO, Cálculo Diferencial em R^n, Uma Introdução, Departamento de Matemática, FCT-UNL

A. A. SÁ, F. OLIVEIRA, PH. DIDIER,  Cálculo Integral em R^n, Teoria e Prática, Departamento de Matemática, FCT-UNL

J. STEWART, Calculus, Brooks/Cole Publishing Company, 2005

Teaching method

Theoretical classes consist in a theoretical exposition illustrated by application examples.

Practical classes consist in the resolution of  application exercises for the methods and results presented in the theoretical classes.

Any questions or doubts will be addressed during the classes, during the weekly sessions specially programmed to attend students or in individual sessions previously scheduled between professors and students.

Evaluation method

Please, contact Professor Jorge Manuel L. André (

Subject matter

1. Ordinary differential equations (ODE)

    1.1 First order ODE: Linear ODE, separable ODE.

    1.2 ODE Models in Exact and Social Sciences.

    1.3 Directions fields. Euler''''''''s Method.

2. Revision of some concepts of Analytical Geometry.

    2.1 Conics.

    2.2 Quadrics.

3. Limits and Continuity in Rn

    3.1 Topological notions in Rn.

    3.2 Vectorial functions and functions of several real variables: Domain, graph, level curves and level surfaces.

    3.3 Limits and continuity of functions of several real variables.

4. Differential calculus in Rn

    4.1 Partial derivatives and Schwarz''''''''s Theorem.

    4.2 Directional derivative. Jacobian matrix, gradient vectors and notion of differentiability. 

    4.3 Differentiability of the compose function. Taylor''''''''s Theorem. Implicit Function''''''''s Theorem and Inverse Function''''''''s Theorem.

    4.4 Relative extreme.  Constrained extreme and Lagrange''''''''s multipliers.

5. Integral calculus in Rn

    5.1 Double integrals. Iterated integrals and Fubini''''''''s Theorem. Variables changes in double integrals. Double integrals in polar coordinates. Applications.

    5.2 Triple integrals. Iterated integrals and Fubini''''''''s Theorem. Variables changes in triple integrals. Triple integrals in cylindrical and spherical coordinates. Applications.


Programs where the course is taught: