Mathematics II

Differential and integral calculus of real functions of a real variable.

Presentation and analysis of the subject of the course unit in theoretical classes. Resolution and discussion of exercises with guidance from the teacher in theoretical-practical classes.

Knowledge of mathematics essential for understanding the subjects taught in the remaining curricular units of the degree, namely knowledge of ordinary differential equations, real functions of several real variables and their derivatives, multiple integrals and series. Ability to apply this knowledge to solve problems. Development of critical thinking and reasoning skills.

1. Introduction to the study of ordinary differential equations – Introduction and motivation.  First order differential equations: first order linear equation; Bernoulli’s equation; separable variable equation; zero degree homogeneous equation.

2. Differential and integral calculus in IRⁿ – Conics and quadrics surfaces. Notions of topology in IRⁿ. Real functions of several real variables: domain; contour and graph of a two-variables function; limit and continuity; partial derivatives; gradient vector; differentiable functions; directional derivative; tangent plane and normal straight line; linear approximation; free extrema; extrema with constraints and Lagrange’s multipliers method.  Multiple integrals: double integral (definition, properties and geometric interpretation, calculation in Cartesian and polar coordinates, examples of application); triple integral (definition, properties, calculation in Cartesian, cylindrical and spherical coordinates, examples of applications).

3. Introduction to the study of series – Brief revision about numerical sequences. Numerical series: definition; nature; properties; examples (geometric series, Dirichlet’s series and telescoping series); necessary condition for convergence; series of non-negative terms; comparison tests; integral test, root and ratio test; conditional and absolute convergence; alternating series; Leibniz’s rule. Real power series: definition; radius and interval of convergence; properties; Taylor series expansions.

NAO

CARDOSO, J. (2020). Apontamentos de apoio às aulas de Análise Matemática II. ISEC (moodle)

RODRIGUES, R. (2020). Notas teóricas e exercícios de Análise Matemática. ISEC (moodle)

GUIDORIZZI, H. L. (1987). Um Curso de Cálculo, vol. 1, vol. 2, vol.3, vol. 4. Livros Técnicos e Científicos (available in the Library of ISEC:  3-2-197 (V1); 3-2-198 (V2); 3-2-199 (V3); 3-2-200 (V4); 3-2-361 (V1))

LARSON, R. E., HOSTETLER, R. P., & EDWARDS, B. H. (1998). Cálculo com geometria analítica – Volume 2. Rio de Janeiro: Livros Técnicos e Científicos (available in the Library of ISEC:  3-2-245; 3-2-249).

PIRES, G.E. (2016). Cálculo Diferencial e Integral em Rⁿ. IST – Coleção Ensino da Ciência e Tecnologia (available in the Library of ISEC: 3-2-396)