Over the course of 20 years, my father gave a large number of lectures as a professor of mathematics at the University of Augsburg. Derived from his lecture notes was a notable textbook on "Gewöhnliche Differenzialgleichungen" ("Ordinary Differential Equations") that was published by Elsevier and has firmly established itself as a standard reference in German-speaking countries. I was able to assure myself of the high quality of this book during exam preparations for my mathematics minor, which formed part of my physics master degree examinations in 2006. The subject area of this exam included Ordinary Differential Equations, even though I did not attend any lecture on this subject! Within a few weeks, I worked through my father's textbook by way of self-study and passed the exam with flying colors.

Less well known, but of equally high quality are my father's undergraduate lecture notes on analysis. For a long time these have not been available in regular bookstores, and instead were distributed only in limited numbers and without ISBN among the students who attended the lectures. Nevertheless, the lecture notes gained popularity beyond the University of Augsburg, and were frequently recommended as supplementary material by lecturers at other universities. Therefore, two years ago my family and I decided to publish our father's lectures notes on Analysis I and Analysis II for regular bookstores. The two lectures were combined to one volume, which made sense due to the structure of this textbook. Unlike the traditional approach to first cover the one-dimensional analysis, then the multi-dimensional one and finally the abstract one, this book starts out with the ultimately desired generality. This avoids repetitions and allows for a compact presentation. Of course, the one- and multi-dimensional special cases are not ignored. This is achieved thanks to 170 detailed examples that are explained and solved in full within the main text.

Below you will find the cover, the table of contents as well as a few reading samples from the book. Have fun! We welcome inquiries, corrections and suggestions - just write an e-mail to *aulbach.mathematik-AT-gmail.com*!

### Cover

Front |
Back |

### Reading samples as PDF

### Contents

*(translated from German)*

**1 Basics**

1.1 Logic

1.2 Sets

1.3 Relations

1.4 Functions

1.5 Fields

1.6 The real numbers

1.7 The complex numbers

1.8 Euclidean, normed and metric spaces

**2 Sequences**

2.1 Convergence

2.2 Calculating with sequences

2.3 Bounded sequences

2.4 Cauchy sequences

2.5 Function sequences

2.6 Series

2.7 Convergence tests

2.8 Function series

**3 Continuity**

3.1 Topology

3.2 Limits of functions

3.3 Continuous functions

3.4 Exponential, logarithmic and power functions

3.5 Trigonometric and hyperbolic functions

**4 Differentiation**

4.1 Definition of the derivative

4.2 Partial derivatives

4.3 Mean value theorems

4.4 Inverse function theorem

4.5 Implicit function theorem

4.6 Higher order derivatives

4.7 Taylor's theorem

4.8 Extreme values

4.9 Sequences of differentiable functions

4.10 Antiderivatives

4.11 Differential equations

**5 Integration**

5.1 Definition of the integral

5.2 Criteria of integrability

5.3 Properties of integrable functions

5.4 Fubini's theorem

5.5 Definite and indefinite integrals

5.6 Improper integrals

5.7 Jordan measurable sets

5.8 Volume calculation

5.9 Change of variables theorem

5.10 Line integrals

5.11 Surface integrals

5.12 Green's, Gauss's and Stokes' theorems