Contents

Preface
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
  3.4.1 Algebraic Numbers
  3.4.2 Transcendental Numbers
  3.4.3 π and the Quadrature of the Circle according to Archimedes
 3.5 Real Numbers
 3.6 Complex Numbers
  3.6.1 Representation as a Pair of Real Numbers
  3.6.2 Normal representation with the “imaginary unit i”
  3.6.3 Complex Plane
  3.6.4 Representation in Polar Coordinates
  3.6.5 Simulation of Complex Addition and Subtraction
  3.6.6 Simulation of Complex Multiplication and Division.
 3.7 Extension of Arithmetics
4 Sequences of Numbers and Series
 4.1 Sequences and Series
  4.1.1 Sequence and Series of the Natural Numbers
  4.1.2 Geometric Series
 4.2 Limits
 4.3 Fibonacci Sequence
 4.4 Complex Sequences and Series
  4.4.1 Complex geometric Sequence and Series
  4.4.2 Complex Exponential Sequence and Exponential Series
 4.5 Influence of Limited Accuracy of Measurements and Non-linearity
  4.5.1 Numbers in Mathematics and Physics
  4.5.2 Real Sequence with Nonlinear Creation Law; Logistic Sequence
  4.5.3 Complex Sequence with Nonlinear Creation Law; Fractals
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
  5.3.1 Powers and Polynomials
  5.3.2 Exponential Function
  5.3.3 Trigonometric Functions
  5.3.4 Rules for the Differentiation of combined Functions
  5.3.5 Derivatives of further Fundamental Functions
 5.4 Series expansion, Taylor Series
  5.4.1 Coefficients of the Taylor Series
  5.4.2 Approximation Formulas for simple Functions
  5.4.3 Derivation of Formulas and errors bounds for numerical differentiation
  5.4.4 Interactive Visualization of Taylor expansions
 5.5 Graphical Presentation of Functions
  5.5.1 Functions of 1 to 3 Variables
  5.5.2 Functions of four variables: World line in the Theory of Relativity
  5.5.3 General Properties of Functions y = f(x)
  5.5.4 Exotic 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
  5.9.1 Definition of Anti-derivative via its Differential Equation
  5.9.2 Definite Integral and Initial Value
  5.9.3 Integral as Limit of a Sum
  5.9.4 The Definition of the Integral due to Riemann
  5.9.5 Lebesgue Integral
  5.9.6 Rules for the Analytical Integration
  5.9.7 Numerical Integration Methods
  5.9.8 Error Estimate for Numerical Integration
 5.10 Series Expansion (2): the Fourier Series
  5.10.1 Taylor Series and Fourier Series
  5.10.2 Determination of the Fourier coefficients
  5.10.3 Visualizing the Calculation of Coefficients and Spectrum
  5.10.4 Examples of Fourier Expansions
  5.10.5 Complex Fourier Series
  5.10.6 Numerical Solution of Equations and iterative Methods
6 Visualization of Functions in the Space of Real Numbers
 6.1 Standard functions y = f(x)
 6.2 Some Functions y = f(x) that are important in Physics
 6.3 Standard Functions of two variables z = f(x,y)
 6.4 Waves in Space
 6.5 Parameter Representation of Surfaces: x = fx(p,q);y = fy(p,q) z = fz(p,q)
 6.6 Parameter representation of curves, space paths x = fx(t);y = fy(t) z = fz(t)
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
  7.4.1 Complex Sine
  7.4.2 Complex 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
  8.3.1 Multiplication by a Constant
  8.3.2 Addition and Subtraction of Vectors
  8.3.3 Scalar Product, Inner Product
  8.3.4 Vector Product, Outer Product
 8.4 Visualization of the Basic Operations for Vectors
 8.5 Fields
  8.5.1 Scalar Field and Vector Fields
  8.5.2 Visualization Possibilities for Scalar-and Vector fields
  8.5.3 Basic Formalism of Vector Analysis
  8.5.4 Potential Fields of point sources as 3D-Surface
  8.5.5 Potential Fields of Point sources as Contour Diagram
  8.5.6 Plane Vector Fields
  8.5.7 3D-Field due to Point Charges
  8.5.8 3D Movement of a Point Charge in a Homogeneous Electromagnetic Field
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
  9.4.1 Explicit Euler Method
  9.4.2 Heun Method
  9.4.3 Runge-Kutta Method
  9.4.4 Further Developments
 9.5 Simulation of Ordinary Differential Equations
  9.5.1 Comparison of Euler, Heun and Runge-Kutta Methods
  9.5.2 Differential Equations of First Order
  9.5.3 Differential Equations of second order
  9.5.4 Differential Equations for Oscillators and Gravity Pendulum
  9.5.5 Character of Ordinary Linear Differential Equations
  9.5.6 Chaotic Solutions of Coupled 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 Schrödinger 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
  11.3.1 Electrodynamics
  11.3.2 Fields and Potentials
  11.3.3 Mathematics, Differential Equations
  11.3.4 Mechanics
  11.3.5 Newton
  11.3.6 Optics
 11.4 Oscillators and Pendulums
  11.4.1 Quantum Mechanics
  11.4.2 Theory of of Relativity
  11.4.3 Statistics
  11.4.4 Thermodynamics
  11.4.5 Waves
  11.4.6 Miscellaneous
 11.5 OSP Simulations, that were not created with EJS
  11.5.1 List of OSP Launcher Packages
 11.6 EJS-Simulations packaged as Launchers
 11.7 Cosmological Simulations by Eugene Butikov
12 Conclusion