This curricular unit consists in an introduction to ordinary differential equations and to the differential and integral calculus of functions of several real variables.

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

The students should acquire not onlycalculus 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 intheanalysis and in resolution of new problems to the future engineer, opening thepossibility to acquiremore complex mathematical tools, possibly neededthrough is career.

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 inRn

3.1 Topological notions inRn.

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 inRn

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 inRn

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.

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

The student must master the mathematical knowledgelectured 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.