Preface

Contents

1 Introduction

1.1 Goal and Structure of the Digital Book

1.2 Directories

1.3 Usage and technical Conventions

1.4 Example of a Simulation: The Moebius band

2 Physics and Mathematics

2.1 Mathematics as “Language of Physics”

2.2 Physics and Calculus

3 Numbers

3.1 Natural Numbers

3.2 Whole Numbers

3.3 Rational Numbers

3.4 Irrational Numbers

3.5 Real Numbers

3.6 Complex Numbers

3.7 Extension of Arithmetics

4 Sequences of Numbers and Series

4.1 Sequences and Series

4.2 Limits

4.3 Fibonacci Sequence

4.4 Complex Sequences and Series

4.5 Influence of Limited Accuracy of Measurements and Non-linearity

5 Functions and their Infinitesimal Properties

5.1 Definition of Functions

5.2 Difference Quotient and Differential Quotient

5.3 Derivatives of a Few Fundamental Functions

5.4 Series expansion, Taylor Series

5.5 Graphical Presentation of Functions

5.6 The Limiting process for Obtaining the Differential Quotient

5.7 Derivative and Differential Equations

5.8 Phase Space Diagrams

5.9 Integral

5.10 Series Expansion (2): the Fourier Series

6 Visualization of Functions in the Space of Real Numbers

6.1 Standard functions $y=f\left(x\right)$

6.2 Some Functions $y=f\left(x\right)$ that are important in Physics

6.3 Standard Functions of two variables $z=f\left(x,y\right)$

6.4 Waves in Space

6.5 Parameter Representation of Surfaces: $x={f}_{x}\left(p,q\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(p,q\right)z={f}_{z}\left(p,q\right)$

6.6 Parameter representation of curves, space paths $x={f}_{x}\left(t\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(t\right)z={f}_{z}\left(t\right)$

7 Visualization of Functions in the Space of complex numbers

7.1 Conformal Mapping

7.2 Visualization of the Complex Power Function

7.3 Complex Exponential Function

7.4 Complex Trigonometric Functions: Sine, Cosine, Tangent

7.5 Complex Logarithm

8 Vectors

8.1 Vectors and Operators as Shorthand for $n$-Tuples of Number and Functions

8.2 3D-Visualization of Vectors

8.3 Basic Operations of Vector Algebra

8.4 Visualization of the Basic Operations for Vectors

8.5 Fields

9 Ordinary Differential Equations

9.1 General Considerations

9.2 Differential equations as Generators of Functions

9.3 Solution Methods for ordinary Differential Equations

9.4 Numerical Solution Methods, Initial Value Problem

9.5 Simulation of Ordinary Differential Equations

10 Partial Differential Equations

10.1 Some Important Partial Differential Equations of Physics

10.2 Simulation of the Diffusion Equation

10.3 Simulation of the Schr�dinger equation

10.4 Simulation of the Wave Equation for a Vibrating String

11 Appendix: Collection of Physics Simulations

11.1 Simulations via OSP/EJS-Program

11.2 A short introduction to EJS (Easy Java Simulation)

11.3 Published EJS simulations

11.4 Oscillators and Pendulums

11.5 OSP Simulations, that were not created with EJS

11.6 EJS-Simulations packaged as Launchers

11.7 Cosmological Simulations by Eugene Butikov

12 Conclusion

Contents

1 Introduction

1.1 Goal and Structure of the Digital Book

1.2 Directories

1.3 Usage and technical Conventions

1.4 Example of a Simulation: The Moebius band

2 Physics and Mathematics

2.1 Mathematics as “Language of Physics”

2.2 Physics and Calculus

3 Numbers

3.1 Natural Numbers

3.2 Whole Numbers

3.3 Rational Numbers

3.4 Irrational Numbers

3.5 Real Numbers

3.6 Complex Numbers

3.7 Extension of Arithmetics

4 Sequences of Numbers and Series

4.1 Sequences and Series

4.2 Limits

4.3 Fibonacci Sequence

4.4 Complex Sequences and Series

4.5 Influence of Limited Accuracy of Measurements and Non-linearity

5 Functions and their Infinitesimal Properties

5.1 Definition of Functions

5.2 Difference Quotient and Differential Quotient

5.3 Derivatives of a Few Fundamental Functions

5.4 Series expansion, Taylor Series

5.5 Graphical Presentation of Functions

5.6 The Limiting process for Obtaining the Differential Quotient

5.7 Derivative and Differential Equations

5.8 Phase Space Diagrams

5.9 Integral

5.10 Series Expansion (2): the Fourier Series

6 Visualization of Functions in the Space of Real Numbers

6.1 Standard functions $y=f\left(x\right)$

6.2 Some Functions $y=f\left(x\right)$ that are important in Physics

6.3 Standard Functions of two variables $z=f\left(x,y\right)$

6.4 Waves in Space

6.5 Parameter Representation of Surfaces: $x={f}_{x}\left(p,q\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(p,q\right)z={f}_{z}\left(p,q\right)$

6.6 Parameter representation of curves, space paths $x={f}_{x}\left(t\right);\phantom{\rule{0em}{0ex}}y={f}_{y}\left(t\right)z={f}_{z}\left(t\right)$

7 Visualization of Functions in the Space of complex numbers

7.1 Conformal Mapping

7.2 Visualization of the Complex Power Function

7.3 Complex Exponential Function

7.4 Complex Trigonometric Functions: Sine, Cosine, Tangent

7.5 Complex Logarithm

8 Vectors

8.1 Vectors and Operators as Shorthand for $n$-Tuples of Number and Functions

8.2 3D-Visualization of Vectors

8.3 Basic Operations of Vector Algebra

8.4 Visualization of the Basic Operations for Vectors

8.5 Fields

9 Ordinary Differential Equations

9.1 General Considerations

9.2 Differential equations as Generators of Functions

9.3 Solution Methods for ordinary Differential Equations

9.4 Numerical Solution Methods, Initial Value Problem

9.5 Simulation of Ordinary Differential Equations

10 Partial Differential Equations

10.1 Some Important Partial Differential Equations of Physics

10.2 Simulation of the Diffusion Equation

10.3 Simulation of the Schr�dinger equation

10.4 Simulation of the Wave Equation for a Vibrating String

11 Appendix: Collection of Physics Simulations

11.1 Simulations via OSP/EJS-Program

11.2 A short introduction to EJS (Easy Java Simulation)

11.3 Published EJS simulations

11.4 Oscillators and Pendulums

11.5 OSP Simulations, that were not created with EJS

11.6 EJS-Simulations packaged as Launchers

11.7 Cosmological Simulations by Eugene Butikov

12 Conclusion