Mathematics II

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. Final written exam (100%)

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.

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 IRn:
(a) Scalar fields. (b) Integral calculus in Rn.
3. Infinite series: Sequences of real numbers. Infinite series. Definition and properties. Geometric series and
Telescoping series. Necessary condition for convergence. Series of nonnegative terms. Tests for convergence,
the integral test, the root test and the ratio test. Conditional and absolute convergence. Alternating series.
Leibniz’s rule. Rearrangements of series.
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.

NAO