Learning and Teaching Mathematics using Simulations: Plus 2000 Examples from Physics (De Gruyter Textbook)

Preface\n......Page 6
Contents......Page 10
Guide to simulation technique\n......Page 16
Goal and structure of the digital book......Page 20
Directories......Page 21
Usage and technical conventions......Page 23
Example of a simulation: The Möbius band......Page 25
Mathematics as the ``Language of physics\'\'......Page 29
Physics and calculus......Page 30
Natural numbers......Page 32
Whole numbers......Page 34
Irrational numbers......Page 36
Transcendental numbers......Page 37
 and the quadrature of the circle, according to Archimedes......Page 38
Real numbers......Page 41
Representation as a pair of real numbers......Page 42
Normal representation with the ``imaginary unit i\'\'......Page 44
Complex plane......Page 47
Representation in polar coordinates......Page 48
Simulation of complex addition and subtraction......Page 49
Extension of arithmetic......Page 52
Sequence and series of the natural numbers......Page 54
Geometric series......Page 55
Limits......Page 56
Fibonacci sequence......Page 59
Complex sequences and series......Page 60
Complex geometric sequence and series......Page 61
Complex exponential sequence and exponential series......Page 63
Numbers in mathematics and physics......Page 67
Real sequence with nonlinear creation law: Logistic sequence......Page 69
Complex sequence with nonlinear creation law: Fractals......Page 74
Definition of functions......Page 80
Difference quotient and differential quotient......Page 81
Powers and polynomials......Page 82
Trigonometric functions......Page 84
Derivatives of further fundamental functions......Page 85
Coefficients of the Taylor series......Page 86
Approximation formulas for simple functions......Page 90
Derivation of formulas and errors bounds for numericaldifferentiation......Page 91
Interactive visualization of Taylor expansions......Page 92
Functions of one to three variables......Page 94
Functions of four variables: World line in the theory of relativity......Page 97
General properties of functions y=f(x)......Page 98
The limiting process for obtaining the differential quotient......Page 100
Derivatives and differential equations......Page 103
Phase space diagrams......Page 104
Definition of the antiderivative via its differential equation......Page 105
Definite integral and initial value......Page 106
Integral as limit of a sum......Page 107
The definition of the Riemann integral......Page 109
Lebesgue integral......Page 111
Rules for the analytical integration......Page 112
Numerical integration methods......Page 113
Error estimates for numerical integration......Page 115
Taylor series and Fourier series......Page 117
Determination of the Fourier coefficients......Page 118
Visualizing the calculation of coefficients and spectrum......Page 121
Examples of Fourier expansions......Page 122
Complex Fourier series......Page 124
Numerical solution of equations and iterative methods......Page 125
Standard functions y=f(x)......Page 127
Some functions y=f(x) that are important in physics......Page 131
Standard functions of two variables z=f(x,y)......Page 133
Waves in space......Page 138
Parameter representation of surfaces: x=fx(p,q); y=fy(p,q); z=fz(p,q)......Page 140
Parameter representation of curves and space paths: x=fx(t);y=fy(t); z=fz(t)......Page 142
Conformal mapping......Page 145
Visualization of the complex power function......Page 146
Complex exponential function......Page 150
Complex trigonometric functions: sine, cosine, tangent......Page 152
Complex logarithm......Page 153
Vectors and operators as shorthand for n-tuples of numbers andfunctions......Page 158
3D-visualization of vectors......Page 159
Addition and subtraction of vectors......Page 161
Scalar product, inner product......Page 162
Vector product, outer product......Page 163
Visualization of the basic operations for vectors......Page 164
Scalar fields and vector fields......Page 165
Visualization possibilities for scalar and vector fields......Page 166
Basic formalism of vector analysis......Page 167
Potential fields of point sources as 3D surfaces......Page 169
Potential fields of point sources as contour diagrams......Page 171
Plane vector fields......Page 173
3D movement of a point charge in a homogeneouselectromagnetic field......Page 176
General considerations......Page 180
Differential equations as generators of functions......Page 181
Solution methods for ordinary differential equations......Page 188
Numerical solution methods: initial value problem......Page 189
Explicit Euler method......Page 191
Heun method......Page 193
Runge–Kutta method......Page 194
Comparison of Euler, Heun and Runge–Kutta methods......Page 196
First order differential equations......Page 198
Second order differential equations......Page 202
Differential equations for oscillators and the gravity pendulum......Page 206
Chaotic solutions of coupled differential equations......Page 209
Some important partial differential equations in physics......Page 215
Simulation of the diffusion equation......Page 218
Simulation of the Schrödinger equation......Page 219
Simulation of the wave equation for a vibrating string......Page 220
Simulations via OSP/EJS programs......Page 223
A short introduction to EJS (Easy Java Simulation)......Page 225
Published EJS simulations......Page 232
Mathematics, differential equations......Page 233
Mechanics......Page 236
Optics......Page 238
Oscillators and pendulums......Page 239
Quantum mechanics......Page 241
Statistics......Page 242
Waves......Page 243
Miscellaneous......Page 244
OSP Simulations that were not created with EJS......Page 247
List of OSP launcher packages......Page 248
EJS simulations packaged as launchers......Page 252
Cosmological simulations by Eugene Butikov......Page 253
Conclusion......Page 258