- give precise definitions of some important concepts from real analysis and work with them, and also, the student is able to give rigorous proof of some important results related to these concepts;
- explain and apply the concepts of convergence/divergence of a series of real numbers and the same for a sequence or series of real-valued functions;
- explain and work with the concepts related to metric spaces, compact and connected sets, limit and continuity of functions on metric spaces;
- explain and work with the concepts related to differentiability of functions on R, and also, apply the related results such as Taylor expansion, implicit and inverse function theorem;
Modelling the physical phenomena in science, technology and in the financial world leads, very often, to differential, integral or difference equations. Typical questions in these situations are: Does a solution exist? Under what conditions? Can we construct the solution? If we cannot construct the solution explicitly, can we approximate it? How accurate is the approximation?|
Within the Analysis-line of a degree programme, one not only learns the fundamental concepts in Real Analysis that are indispensable for the further study, but also sees how the main results can be applied in answering some of the previous questions. Another integral part of the Analysis-line is, of course, proper and rigorous manner of proving the results.
Analysis-II builds further on Analysis-I of Module-02-AM (“Mathematical Proof Techniques”). The topics that will be treated in this module are: convergence of series (of real numbers), convergence of sequence and series of real-valued functions, metric spaces, compact and connected subsets, limits and continuity of functions defined on metric spaces, and differentiability of functions on R together with related concepts such as Taylor expansion, implicit and inverse function theorem.