Mathematics: Introduction to Higher Analysis

<p>The course is a compulsory course given in the second semester's studies within the Bachelor's programme in mathematics and is intended to be read in parallel with a course in analysis in several variables. The course can also be given as a stand-alone course or as part of a course package. The overarching goal of the course is for students to develop understanding of central concepts, results and methods of relevance for further studies that involve mathematical analysis. Additionally, the course aims to expose students to research literature in mathematics.</p><p>The course treats:</p> <ul> <li>Properties of the real numbers R: completeness axiom, Cauchy sequences, cardinality of rational and irrational numbers.</li> <li>Topology in Rn: open and closed sets, p-norms, convergence, compactness and the Bolzano-Weierstrass theorem, connected sets.</li> <li>Continuous functions in Rn: intermediate value theorem, min-max theorem, uniform continuity, continuity of inverse functions, implicit function theorem.</li> <li>Convergence of sequences and series of functions: pointwise, absolute and uniform convergence, termwise differentiation and integration, power series.</li> <li>Examples of applications to selected topics relevant to mathematical research at the Centre for mathematical sciences.</li> </ul>