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

Learning and Teaching Mathematics Using Simulations (2011)
......Page 1
De Gruyter Textbook......Page 2
ISBN: 9783110250053......Page 5
Preface......Page 6
--> Contents......Page 10
Guide to simulation technique......Page 16
1.1 Goal and structure of the digital book......Page 20
1.2 Directories......Page 21
1.3 Usage and technical conventions......Page 23
1.4 Example of a simulation: The Mobius band......Page 25
2.1 Mathematics as the " Language of physics"......Page 29
2.2 Physics and calculus......Page 30
3.1 Natural numbers......Page 32
3.2 Whole numbers......Page 34
3.4 Irrational numbers......Page 36
3.4.2 Transcendental numbers......Page 37
3.4.3 pi and the quadrature of the circle, according to Archimedes......Page 38
3.5 Real numbers......Page 41
3.6.1 Representation as a pair of real numbers......Page 42
3.6.2 Normal representation with the "imaginary unit i"......Page 44
3.6.3 Complex plane......Page 47
3.6.4 Representation in polar coordinates......Page 48
3.6.5 Simulation of complex addition and subtraction......Page 49
3.7 Extension of arithmetic......Page 52
4.1.1 Sequence and series of the natural numbers......Page 54
4.1.2 Geometric series......Page 55
4.2 Limits......Page 56
4.3 Fibonacci sequence......Page 59
4.4 Complex sequences and series......Page 60
4.4.1 Complex geometric sequence and series......Page 61
4.4.2 Complex exponential sequence and exponential series......Page 63
4.5.1 Numbers in mathematics and physics......Page 67
4.5.2 Real sequence with nonlinear creation law: Logistic sequence......Page 69
4.5.3 Complex sequence with nonlinear creation law: Fractals......Page 75
5.1 Definition of functions......Page 80
5.2 Difference quotient and differential quotient......Page 81
5.3.1 Powers and polynomials......Page 82
5.3.3 Trigonometric functions......Page 84
5.3.5 Derivatives of further fundamental functions......Page 85
5.4.1 Coefficients of the Taylor series......Page 86
5.4.2 Approximation formulas for simple functions......Page 90
5.4.3 Derivation of formulas and errors bounds for numerical differentiation......Page 91
5.4.4 Interactive visualization of Taylor expansions......Page 92
5.5.1 Functions of one to three variables......Page 94
5.5.2 Functions of four variables: World line in the theory of relativity......Page 97
5.5.3 General properties of functions y = f (x)......Page 99
5.5.4 Exotic functions......Page 100
5.6 The limiting process for obtaining the differential quotient......Page 101
5.7 Derivatives and differential equations......Page 103
5.8 Phase space diagrams......Page 104
5.9.1 Definition of the antiderivative via its differential equation......Page 105
5.9.2 Definite integral and initial value......Page 106
5.9.3 Integral as limit of a sum......Page 107
5.9.4 The definition of the Riemann integral......Page 109
5.9.5 Lebesgue integral......Page 111
5.9.6 Rules for the analytical integration......Page 112
5.9.7 Numerical integration methods......Page 113
5.9.8 Error estimates for numerical integration......Page 115
5.10.1 Taylor series and Fourier series......Page 117
5.10.2 Determination of the Fourier coefficients......Page 118
5.10.4 Examples of Fourier expansions......Page 122
5.10.5 Complex Fourier series......Page 124
5.10.6 Numerical solution of equations and iterative methods......Page 125
6.1 Standard functions y = f (x)......Page 127
6.2 Some functions y = f (x) that are important in physics......Page 131
6.3 Standard functions of two variables z = f (x , y)......Page 134
6.4 Waves in space......Page 138
6.5 Parameter representation of surfaces......Page 140
6.6 Parameter representation of curves and space paths......Page 142
7.1 Conformal mapping......Page 145
7.2 Visualization of the complex power function......Page 146
7.3 Complex exponential function......Page 150
7.4 Complex trigonometric functions: sine, cosine, tangent......Page 152
7.4.3 Complex tangent......Page 153
7.5 Complex logarithm......Page 155
8.1 Vectors and operators as shorthand for n-tuples of numbers and functions......Page 158
8.2 3D-visualization of vectors......Page 159
8.3.1 Multiplication by a constant......Page 161
8.3.3 Scalar product, inner product......Page 162
8.3.4 Vector product, outer product......Page 163
8.4 Visualization of the basic operations for vectors......Page 164
8.5.1 Scalarfields and vector fields......Page 165
8.5.2 Visualization possibilities for scalar and vector fields......Page 166
8.5.3 Basic formalism of vector analysis......Page 167
8.5.4 Potential fields of point sources as 3D surfaces......Page 169
8.5.5 Potential fields of point sources as contour diagrams......Page 171
8.5.6 Plane vector fields......Page 173
8.5.8 3D movement of a point charge in a homogeneous electromagnetic field......Page 176
9.1 General considerations......Page 180
9.2 Differential equations as generators of functions......Page 181
9.3 Solution methods for ordinary differential equations......Page 188
9.4 Numerical solution methods: initial value problem......Page 189
9.4.1 Explicit Euler method......Page 191
9.4.2 Heun method......Page 193
9.4.3 Runge-Kutta method......Page 194
9.5.1 Comparison of Euler, Heun and Runge-Kutta methods......Page 196
9.5.2 First order differential equations......Page 198
9.5.3 Second order differential equations......Page 202
9.5.4 Differential equations for oscillators and the gravity pendulum......Page 206
9.5.6 Chaotic solutions of coupled differential equations......Page 209
10.1 Some important partial differential equations in physics......Page 215
10.2 Simulation of the diffusion equation......Page 218
10.3 Simulation of the Schrödinger equation......Page 219
10.4 Simulation of the wave equation for a vibrating string......Page 220
11.1 Simulations via OSP/EJS programs......Page 223
11.2 A short introduction to EJS (Easy Java Simulation)......Page 225
11.3 Published EJS simulations......Page 232
11.3.3 Mathematics, differential equations......Page 233
11.3.4 Mechanics......Page 236
11.3.6 Optics......Page 238
11.3.7 Oscillators and pendulums......Page 239
11.3.8 Quantum mechanics......Page 241
11.3.10 Statistics......Page 242
11.3.12 Waves......Page 243
11.3.13 Miscellaneous......Page 244
11.4 OSP Simulations that were not created with EJS......Page 247
11.4.1 List of OSP launcher packages......Page 248
11.5 EJS simulations packaged as launchers......Page 252
11.6 Cosmological simulations by Eugene Butikov......Page 253
12 Conclusion......Page 258