German original

# Analysis II (dt. Analysis II)

 Level, degree of commitment in original study programme Basic module, required module Forms of teaching and learning,workload Lecture (4 SWS), recitation class (2 SWS), Werkstatt (2 SWS), 270 hours (120 h attendance, 150 h private study) Credit points,formal requirements 9 CP Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises. Written test (60-120 min.). Examination type: Oral examination Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for study course B.Sc. Mathematics. Original study programme B.Sc. Mathematik / Mathematik Basismodule Duration,frequency One semester, each winter semester Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Thomas Bauer, Prof. Dr. Pablo Ramacher

## Contents

In addition to the contents of the Analysis I module, which are also relevant for the final oral examination, the following contents are covered in the module:

• Metric spaces: Basic topological concepts, convergence, complete, compact, connected metric spaces, space of continuous functions on a compact set (this subject area can be treated by the lecturer alternatively in Analysis I)
• Differentiation in R^n: total and partial differentiability, gradient, inverse function and implicit function theorem, Taylor formula, local extrema without and with constraints, if possible transformation formula for integrals
• Ordinary differential equations: elementary solution methods, linear differential equation systems, homogeneous and inhomogeneous ordinary differential equations, theorem of Picard-Lindelöf

## Qualification Goals

The general qualification goals correspond to those of Analysis I. Building on these, the students are to

• understand the basic principles of calculus of several variables, understand inasmuch calculus of one variable - as taught in Analysis I - is a special case, and work out the differences,
• understand the linearization of nonlinear problems as a technique of analysis and be able to apply methods of linear algebra in analysis,
• practice the modelling of mathematical / scientific processes on the basis of the theory of differential equations.

Upon completion of the module, students should understand and master analysis as a uniform mathematical subject in its entirety.

## Prerequisites

None. The competences taught in the following module are recommended: Analysis I.

• Forster, O.: Analysis 1 und Analysis 2, Vieweg-Verlag
• Heuser, H.: Lehrbuch der Analysis, Teil 1 und Teil 2, Teubner-Verlag
• Rudin, W.: Analysis, Oldenbourg-Verlag.