Base Knowledge
Differential and integral calculus of real functions of a real variable.
Teaching Methodologies
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.
Learning Results
Acquisition of mathematics’ knowledge essential for understanding the subjects taught in the remaining curricular units of the degree, namely knowledge of ordinary differential equations, numerical series and power series, real functions of several real variables and their derivatives and multiple integrals. Ability to apply this knowledge to solve problems. Development of critical thinking and reasoning skills.
Program
1. Differential Equations – Introduction and motivation. First order ordinary differential equations: first order linear equation, Bernoulli’s equation, separable variable equation, zero degree homogeneous equation.
2. Differential and Integral Calculus in IR^n – Conics and quadric surfaces. Brief notes on topology in R^n. Real functions of several real variables and their derivatives: domain, contour and graph of a two-variables function, limit and continuity, partial derivatives and higher order partial derivatives, differentiability, directional derivative and gradient vector, maxima, minima and saddle points, extrema with constraints, Lagrange’s multipliers method. Multiple integrals: double integral (definition, properties, geometric interpretation, calculation of the double integral in Cartesian and polar coordinates, examples of application), triple integral (definition, properties, calculation of the triple integral in Cartesian, cylindrical and spherical coordinates, examples of application).
3. 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 test, 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.
Curricular Unit Teachers
Grading Methods
- - Exame - 100.0%
Internship(s)
NAO
Bibliography
Cardoso, J. (2024). Apontamentos de apoio às aulas de Análise Matemática II. ISEC.
Rodrigues, R. (2023). Notas teóricas e exercícios de Análise Matemática. ISEC.
Grilo, T. (2022). Apontamentos de apoio às aulas de Matemática II. ISEC.
Guidorizzi, H.L. (2019). Um Curso de Cálculo, vol. 1, vol. 2, vol.3, vol. 4. Livros Técnicos e Científicos.
Larson, R., Hostetler, R.P., & Edwards, B.H. (2006). Cálculo, vol. 1, vol. 2. McGraw-Hill.
Pires, G.E. (2016). Cálculo Diferencial e Integral em Rn. IST – Coleção Ensino da Ciência e Tecnologia.