Learning and teaching mathematics using simulations: plus 2000 examples from physics
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
De Gruyter
2011
|
| Schriftenreihe: | De Gruyter textbook
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XVII, 238 S. Ill., graph. Darst. |
| ISBN: | 9783110250053 |
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| 240 | 1 | 0 | |a Mathematik mit Simulationen lehren und lernen |
| 245 | 1 | 0 | |a Learning and teaching mathematics using simulations |b plus 2000 examples from physics |c Dieter Röss |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2011 | |
| 300 | |a XVII, 238 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
1 INTRODUCTION 1
1 .1 GOAL AND STRUCTURE OF THE DIGITAL BOOK 1
1.2 DIRECTORIES 2
1.3 USAGE AND TECHNICAL CONVENTIONS 4
1.4 EXAMPLE OF A SIMULATION: THE MOEBIUS BAND 6
2 PHYSICS AND MATHEMATICS 10
2.1 MATHEMATICS AS THE "LANGUAGE OF PHYSICS" 10
2.2 PHYSICS AND CALCULUS 11
3 NUMBERS 1 3
3.1 NATURAL NUMBERS 13
3.2 WHOLE NUMBERS 15
3.3 RATIONAL NUMBERS 17
3.4 IRRATIONAL NUMBERS 17
3.4.1 ALGEBRAIC NUMBERS 18
3.4.2 TRANSCENDENTAL NUMBERS 18
3.4.3 IT AND THE QUADRATURE OF THE CIRCLE, ACCORDING TO ARCHIMEDES . 19
3.5 REAL NUMBERS 22
3.6 COMPLEX NUMBERS 23
3.6.1 REPRESENTATION AS A PAIR OF REAL NUMBERS 23
3.6.2 NORMAL REPRESENTATION WITH THE "IMAGINARY UNIT I" 25 3.6.3 COMPLEX
PLANE 28
3.6.4 REPRESENTATION IN POLAR COORDINATES 29
3.6.5 SIMULATION OF COMPLEX ADDITION AND SUBTRACTION 30
3.6.6 SIMULATION OF COMPLEX MULTIPLICATION AND DIVISION 33 3.7 EXTENSION
OF ARITHMETIC 33
4 SEQUENCES OF NUMBERS AND SERIES 35
4.1 SEQUENCES AND SERIES 35
4.1.1 SEQUENCE AND SERIES OF THE NATURAL NUMBERS 35
4.1.2 GEOMETRIC SERIES 36
4.2 LIMITS 37
4.3 FIBONACCI SEQUENCE 40
4.4 COMPLEX SEQUENCES AND SERIES 41
4.4.1 COMPLEX GEOMETRIC SEQUENCE AND SERIES 42
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010341332
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
4 . 4 .2 C O M P L EX E X P O N E N T I AL S E Q U E N CE A ND E X P O N
E N T I AL S E R I ES . . . 44
4 .5 I N F L U E N CE OF L I M I T ED A C C U R A CY OF M E A S U R E M
E N TS A ND N O N L I N E A R I TY . . 48
4 . 5 .1 N U M B E RS IN M A T H E M A T I CS A ND P H Y S I CS 48
4 . 5 .2 R E AL S E Q U E N CE W I TH N O N L I N E AR C R E A T I ON L
A W: L O G I S T IC S E Q U E N CE 50
4 . 5 .3 C O M P L EX S E Q U E N CE W I TH N O N L I N E AR C R E A T I
ON L A W: F R A C T A LS . . 56
FUNCTIONS AND THEIR INFINITESIMAL PROPERTIES 61
5.1 DEFINITION OF FUNCTIONS 61
5.2 DIFFERENCE QUOTIENT AND DIFFERENTIAL QUOTIENT 62
5.3 DERIVATIVES OF A FEW FUNDAMENTAL FUNCTIONS 63
5.3.1 POWERS AND POLYNOMIALS 63
5.3.2 EXPONENTIAL FUNCTION 65
5.3.3 TRIGONOMETRIC FUNCTIONS 65
5.3.4 RULES FOR THE DIFFERENTIATION OF COMBINED FUNCTIONS 66 5.3.5
DERIVATIVES OF FURTHER FUNDAMENTAL FUNCTIONS 66
5.4 SERIES EXPANSION: THE TAYLOR SERIES 67
5.4.1 COEFFICIENTS OF THE TAYLOR SERIES 67
5.4.2 APPROXIMATION FORMULAS FOR SIMPLE FUNCTIONS 71
5.4.3 DERIVATION OF FORMULAS AND ERRORS BOUNDS FOR NUMERICAL
DIFFERENTIATION 72
5.4.4 INTERACTIVE VISUALIZATION OF TAYLOR EXPANSIONS 73
5.5 GRAPHICAL PRESENTATION OF FUNCTIONS 75
5.5.1 FUNCTIONS OF ONE TO THREE VARIABLES 75
5.5.2 FUNCTIONS OF FOUR VARIABLES: WORLD LINE IN THE THEORY OF
RELATIVITY 78 5.5.3 GENERAL PROPERTIES OF FUNCTIONS Y - F(X) 80
5.5.4 EXOTIC FUNCTIONS 81
5.6 THE LIMITING PROCESS FOR OBTAINING THE DIFFERENTIAL QUOTIENT 82 5.7
DERIVATIVES AND DIFFERENTIAL EQUATIONS 84
5.8 PHASE SPACE DIAGRAMS 85
5.9 ANTIDERIVATIVES 86
5.9.1 DEFINITION OF THE ANTIDERIVATIVE VIA ITS DIFFERENTIAL EQUATION . .
86 5.9.2 DEFINITE INTEGRAL AND INITIAL VALUE 87
5.9.3 INTEGRAL AS LIMIT OF A SUM 88
5.9.4 THE DEFINITION OF THE RIEMANN INTEGRAL 90
5.9.5 LEBESGUE INTEGRAL 92
5.9.6 RULES FOR THE ANALYTICAL INTEGRATION 93
5.9.7 NUMERICAL INTEGRATION METHODS 94
5.9.8 ERROR ESTIMATES FOR NUMERICAL INTEGRATION 96
5.10 SERIES EXPANSION (2): THE FOURIER SERIES 98
5.10.1 TAYLOR SERIES AND FOURIER SERIES 98
5.10.2 DETERMINATION OF THE FOURIER COEFFICIENTS 99
5.10.3 VISUALIZING THE CALCULATION OF COEFFICIENTS AND SPECTRUM . . . .
103
IMAGE 3
CONTENTS XI
5.10.4 EXAMPLES OF FOURIER EXPANSIONS 103
5.10.5 COMPLEX FOURIER SERIES 105
5.10.6 NUMERICAL SOLUTION OF EQUATIONS AND ITERATIVE METHODS . . . 105
6 VISUALIZATION OF FUNCTIONS IN THE SPACE OF REAL NUMBERS 108
6.1 STANDARD FUNCTIONS Y = F(X) 108
6.2 SOME FUNCTIONS Y - F{X) THAT ARE IMPORTANT IN PHYSICS 112
6.3 STANDARD FUNCTIONS OF TWO VARIABLES Z = F(X,Y) 115
6.4 WAVES IN SPACE 119
6.5 PARAMETER REPRESENTATION OF SURFACES: X - F X (P,Q); Y - FY(P,Q)', Z
= F Z (P,Q) 121
6.6 PARAMETER REPRESENTATION OF CURVES AND SPACE PATHS: X = F X (T)', Y
= FY(T);Z = F Z (T) 123
7 VISUALIZATION OF FUNCTIONS IN THE SPACE OF COMPLEX NUMBERS 126 7.1
CONFORMAI MAPPING 126
7.2 VISUALIZATION OF THE COMPLEX POWER FUNCTION 127
7.3 COMPLEX EXPONENTIAL FUNCTION 131
7.4 COMPLEX TRIGONOMETRIC FUNCTIONS: SINE, COSINE, TANGENT 133 7.4.1
COMPLEX SINE 134
7.4.2 COMPLEX COSINE 134
7.4.3 COMPLEX TANGENT 134
7.5 COMPLEX LOGARITHM 136
8 VECTORS 139
8.1 VECTORS AND OPERATORS AS SHORTHAND FOR -TUPLES OF NUMBERS AND
FUNCTIONS 139
8.2 3D-VISUALIZATION OF VECTORS 140
8.3 BASIC OPERATIONS OF VECTOR ALGEBRA 142
8.3.1 MULTIPLICATION BY A CONSTANT 142
8.3.2 ADDITION AND SUBTRACTION OF VECTORS 143
8.3.3 SCALAR PRODUCT, INNER PRODUCT 143
8.3.4 VECTOR PRODUCT, OUTER PRODUCT 144
8.4 VISUALIZATION OF THE BASIC OPERATIONS FOR VECTORS 145
8.5 FIELDS 146
8.5.1 SCALAR FIELDS AND VECTOR FIELDS 146
8.5.2 VISUALIZATION POSSIBILITIES FOR SCALAR AND VECTOR FIELDS 147 8.5.3
BASIC FORMALISM OF VECTOR ANALYSIS 148
8.5.4 POTENTIAL FIELDS OF POINT SOURCES AS 3D SURFACES 150
8.5.5 POTENTIAL FIELDS OF POINT SOURCES AS CONTOUR DIAGRAMS 152 8.5.6
PLANE VECTOR FIELDS 154
8.5.7 3D FIELD DUE TO POINT CHARGES 157
IMAGE 4
XII CONTENTS
8.5.8 3D MOVEMENT OF A POINT CHARGE IN A HOMOGENEOUS ELECTROMAGNETIC
FIELD 157
9 ORDINARY DIFFERENTIAL EQUATIONS 161
9.1 GENERAL CONSIDERATIONS 161
9.2 DIFFERENTIAL EQUATIONS AS GENERATORS OF FUNCTIONS 162
9.3 SOLUTION METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS 169
9.4 NUMERICAL SOLUTION METHODS: INITIAL VALUE PROBLEM 170
9.4.1 EXPLICIT EULER METHOD 172
9.4.2 HEUN METHOD 174
9.4.3 RUNGE-KUTTA METHOD 175
9.4.4 FURTHER DEVELOPMENTS 177
9.5 SIMULATION OF ORDINARY DIFFERENTIAL EQUATIONS 177
9.5.1 COMPARISON OF EULER, HEUN AND RUNGE-KUTTA METHODS . . . 177 9.5.2
FIRST ORDER DIFFERENTIAL EQUATIONS 179
9.5.3 SECOND ORDER DIFFERENTIAL EQUATIONS 183
9.5.4 DIFFERENTIAL EQUATIONS FOR OSCILLATORS AND THE GRAVITY PENDULUM
187 9.5.5 CHARACTER OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS 190 9.5.6
CHAOTIC SOLUTIONS OF COUPLED DIFFERENTIAL EQUATIONS 190
10 PARTIAL DIFFERENTIAL EQUATIONS 196
10.1 SOME IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS IN PHYSICS 196 10.2
SIMULATION OF THE DIFFUSION EQUATION 199
10.3 SIMULATION OF THE SCHROEDINGER EQUATION 200
10.4 SIMULATION OF THE WAVE EQUATION FOR A VIBRATING STRING 201
11 COLLECTION OF PHYSICS SIMULATIONS 204
11.1 SIMULATIONS VIA OSP/EJS PROGRAMS 204
11.2 A SHORT INTRODUCTION TO EJS (EASY JAVA SIMULATION) 206
11.3 PUBLISHED EJS SIMULATIONS 213
11.3.1 ELECTRODYNAMICS 214
11.3.2 FIELDS AND POTENTIALS 214
11.3.3 MATHEMATICS, DIFFERENTIAL EQUATIONS 214
11.3.4 MECHANICS 217
11.3.5 NEWTON 219
11.3.6 OPTICS 219
11.3.7 OSCILLATORS AND PENDULUMS 220
11.3.8 QUANTUM MECHANICS 222
11.3.9 THEORY OF RELATIVITY 223
11.3.10 STATISTICS 223
11.3.11 THERMODYNAMICS 224
11.3.12 WAVES 224
IMAGE 5
CONTENTS XIII
11.3.13 MISCELLANEOUS 225
11.4 OSP SIMULATIONS THAT WERE NOT CREATED WITH EJS 228
11.4.1 LIST OF OSP LAUNCHER PACKAGES 229
11.5 EJS SIMULATIONS PACKAGED AS LAUNCHERS 233
11.6 COSMOLOGICAL SIMULATIONS BY EUGENE BUTIKOV 234
12 CONCLUSION 239 |
| any_adam_object | 1 |
| author | Röß, Dieter 1932- |
| author_GND | (DE-588)142077208 |
| author_facet | Röß, Dieter 1932- |
| author_role | aut |
| author_sort | Röß, Dieter 1932- |
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| building | Verbundindex |
| bvnumber | BV039536331 |
| classification_rvk | SM 600 SM 730 ST 340 |
| ctrlnum | (OCoLC)708357702 (DE-599)DNB1010341332 |
| dewey-full | 510.113 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510.113 |
| dewey-search | 510.113 |
| dewey-sort | 3510.113 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Book |
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| series2 | De Gruyter textbook |
| spelling | Röß, Dieter 1932- Verfasser (DE-588)142077208 aut Mathematik mit Simulationen lehren und lernen Learning and teaching mathematics using simulations plus 2000 examples from physics Dieter Röss Berlin [u.a.] De Gruyter 2011 XVII, 238 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter textbook Lehrbuch (DE-588)4123623-3 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Java Programmiersprache (DE-588)4401313-9 gnd rswk-swf Computersimulation (DE-588)4148259-1 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf (DE-588)4144384-6 Beispielsammlung gnd-content Analysis (DE-588)4001865-9 s Computersimulation (DE-588)4148259-1 s Java Programmiersprache (DE-588)4401313-9 s Lehrbuch (DE-588)4123623-3 s DE-604 Physik (DE-588)4045956-1 s X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3678596&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024388504&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Röß, Dieter 1932- Learning and teaching mathematics using simulations plus 2000 examples from physics Lehrbuch (DE-588)4123623-3 gnd Physik (DE-588)4045956-1 gnd Java Programmiersprache (DE-588)4401313-9 gnd Computersimulation (DE-588)4148259-1 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4123623-3 (DE-588)4045956-1 (DE-588)4401313-9 (DE-588)4148259-1 (DE-588)4001865-9 (DE-588)4144384-6 |
| title | Learning and teaching mathematics using simulations plus 2000 examples from physics |
| title_alt | Mathematik mit Simulationen lehren und lernen |
| title_auth | Learning and teaching mathematics using simulations plus 2000 examples from physics |
| title_exact_search | Learning and teaching mathematics using simulations plus 2000 examples from physics |
| title_full | Learning and teaching mathematics using simulations plus 2000 examples from physics Dieter Röss |
| title_fullStr | Learning and teaching mathematics using simulations plus 2000 examples from physics Dieter Röss |
| title_full_unstemmed | Learning and teaching mathematics using simulations plus 2000 examples from physics Dieter Röss |
| title_short | Learning and teaching mathematics using simulations |
| title_sort | learning and teaching mathematics using simulations plus 2000 examples from physics |
| title_sub | plus 2000 examples from physics |
| topic | Lehrbuch (DE-588)4123623-3 gnd Physik (DE-588)4045956-1 gnd Java Programmiersprache (DE-588)4401313-9 gnd Computersimulation (DE-588)4148259-1 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Lehrbuch Physik Java Programmiersprache Computersimulation Analysis Beispielsammlung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3678596&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024388504&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT roßdieter mathematikmitsimulationenlehrenundlernen AT roßdieter learningandteachingmathematicsusingsimulationsplus2000examplesfromphysics |