Mathematics II

Teaching Methodologies

The lectures are intended to provide all the mathematical concepts described in the syllabus with examples where the concepts are applied. In the practical classes exercises are solved in order to discuss the topics presented in the lectures. The software Mathematica, Wolfram Alpha and Moodle are used for further treatment of the subjects studied. There is a final exam (100%).

Learning Results

Knowledge and interpretation of the concepts of ordinary differential equation, numerical series and power series, real function of several variables, directional derivative and multiple integral. Ability to apply this knowledge in solving problems. Development of critical thinking and reasoning skills.

Program

1. Introduction to ordinary differential equations: First-order differential equations – First-order linear differential equation, Bernoulli equation, separable equation and homogeneous equation. 2. Differential and integral calculus in IR^n: (a) Scalar fields – Notions of topology. Differentiable functions. A sufficient condition for differentiability. Directional derivatives and tangent plane. Chain rule. Geometric relation of the directional derivative to the gradient vector. Maxima, minima and saddle points. Extrema with constraints and Lagrange’s multipliers. (b) Integral calculus in Rn: Double integrals. 3. Infinite series: Sequences of real numbers. Infinite series. Geometric series and Telescoping series. Necessary condition for convergence. Series of nonnegative terms. Tests for convergence. 4. Real power series: Radius and interval of convergence. Properties of functions represented by power series. The Taylor series generated by a function. Power series expansions.

Curricular Unit Teachers

Grading Methods

Avaliação Por Exame
  • - Exame - 100.0%

Internship(s)

NAO

Bibliography

Rodrigues, R.C., Notas teóricas e exercícios de análise matemática, DFM, ISEC

Abbott, S., Understanding analysis, Springer, 2002

Guidorizzi , H.L., Um curso de cálculo, vol. 1 and vol. 2., , LTC, 2011

Anton, H., Cálculo – um novo horizonte, vol. 1 and vol. 2, Bookman, 2007

Apostol, T.M., Calculus, vol. I and vol. II, John Wiley, 1969

Azenha, A., Jerónimo, M.A., Cálculo diferencial e integral em R e Rn, McGraw-Hill, 1995

Ferreira, M.F., Equações diferenciais ordinárias, McGrawHill, 1995