### 1.1 Goal and Structure of the Digital Book

This digital book is provided in the form of a pdf file, that is connected with the simulation files via hyperlinks. It illustrates selected mathematical methods, that are important for the presentation and the understanding of the interrelated parts of physics.

The foundations of these mathematical methods are introduced by example. The programmability and computational power of the PC is used to visualize these, undertake calculations, change parameters, present connections interactively and present computation processes via interactive simulation and animation in an interesting manner. In addition playfulness is given free range. This presentation is also supposed to make the beauty and aesthetics of mathematics visible.

The material provided in this book allows the user to penetrate mathematical structures and tools in an experimental manner. Especially those topics have been chosen that are difficult to imagine in an abstract manner such as complex numbers, infinite sequences, transitions to the limit, fields, solutions of differential equations and so on.

The individual simulations always contain extensive descriptions and suggestions for experiments. The user can always interactively engage with the simulations, and in many cases pre-programmed functions can be edited or new one can be introduced. After some initial training in the EJS program the user can open all files, change them and develop them further. OSP

With one exception Java programs have been used, that were either created from scratch or taken from the freely available internet projects Open Source Physics (OSP) and Easy Java Simulation (EJS).

EJS Our own mathematical simulations were created with the EJS-program that has been created by Francisco Esquembre.

Due to its graphical user interface this program immensely simplifies the development and modification of simulations very much in comparison to “classical programming” in Java. This program and its documentation are contained in this book, but are also freely available on the internet.

We will however abstain from explaining the mathematical and computational techniques in systematic detail. The in-depth study of the mathematical and numerical methods will be left to the special textbooks. 1.

The pictures contained in the the pdf file mostly show screen shots of the respective simulation. When such a picture with the caption simulation is clicked at for the first time a small context menu appears, that asks as a security measure against viruses, whether this file should be opened. You may confirm this and mark a check box, to avoid this dialog in future. The simulation will in future be started immediately after clicking on the picture.

Where the reader wishes to learn more about the simulation links to internet pages have been inserted next to the text. They often point to Wikipedia-pages, from where further navigation is easy. These links are resolved in the outlined text boxes on the margin. ??Appendix]Simulations

It is advisable to follow the suggestion to install the JRE into $Math content$Programs$Math content$JAVA$Math content$. For newer EJS-simulations with $Math content$-Rendering you can download the Java 3Dprogram from the same page