# Analysis III

Undergraduate Course, *Ruhr University Bochum*, 2019

- Lecturer: Prof. Dr. Abbondandolo
- Language: German
- Credits: 9 CP
- Programs: B.A./B.Sc./M.Ed. Mathematics
- Examination: 100 % Written Exam (240 Minutes) + 10 % Homework

# Course Description

Contents:

- differential and integral calculus of several variables
- Lebesgue integration
- Introduction to the theory of ordinary differential equations
- differential forms and their integration on submanifolds of Euclidean space

# Contents

- Theorem of implicit functions I
- Complete metric spaces
- Banach fixed-point theorem
- Diffeomorphism
- Theorem of implicit functions II
- Submanifolds
- Ordinary Differential Equations
- Linear Differential Equations
- Abstract measure theory
- Lebesgue measure
- Theorems of Tonelli and Fubini
- Transformation theorem
- parameter-dependent integrals
- Convolution
- Approximation of functions
- Length of a curve
- Sector area of flat curves and applications
- Integration on submanifolds of R^n
- First-order differential forms
- Higher-order differential forms
- Differential calculus on submanifolds
- Theorem of Stokes