3-2-30-6 Qualitative

0 - 3 Observe 3-2 In Depth 3-2-30 Communications

 Concrete 3-2-30-8

3-2-30-7 * Mathematical Preparation Course - Before Studying Physics

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Introduction

3-2-30-7-0 0 Preface
3-2-30-7-1 1 Measuring
3-2-30-7-2 2 Signs And Numbers
3-2-30-7-3 3 Sequences And Series
3-2-30-7-4 4 Functions
3-2-30-7-5 5 Differentiation
3-2-30-7-6 6 Taylor Series
3-2-30-7-7 7 Integration
3-2-30-7-8 8 Complex Numbers
3-2-30-7-9 9 Vectors

 

http://www.freebookcentre.net/physics-books-download/Mathematical-Preparation-Course-Before-Studying-Physics.html

3-2-30-8.pdf


Communications in Mathematical Physics Publisher Springer Berlin / Heidelberg
ISSN 0010-3616 (Print) 1432-0916 (Online)
Springer Link Date Friday, April 05, 2002
Subject Mathematics and Statistics and Physics and Astronomy


For individual spiritual, physical, intellectual and scientific growth; we need a clear operational language, the accumulation of discoveries and insights of free societies and cultures. We need clear sets of postulates in the social sciences grounded in the physical sciences and using unambiguous operational definitions like in the physical sciences.

Before the time of Galileo, many Natural Philosophers wrote with deep understanding on the qualitative level. But, their work was not clearly understood because they used words like work, energy, impulse, momentum, power etc. in different ways. It was not until the introduction of quantative operational definitions founded on clearly stated postulates that the confusion began to end.

Then the process of quantified scientific investigation and reasoned mathematical modeling gave the foundation for thermodynamics, electricity and magnetism, atomic and nuclear physics and cosmology.

We are aware of the process for creating mathematical, cosmological models. We need to develop models of the Cosmos. In general we need Mathematical models to help show our understanding of space, matter and motion. See Diagram 22, "Need Operational Language (Mathematical Insights).


What is healthy child development, home and culture? What is the truth and what is propaganda? We need clear language, science and education that can separate the true and false statements. See Diagram 22 Operational Language.

 

Introduction


MATHEMATICALPREPARATION COURSE
before studying Physics
Accompanying Booklet to the Online Course:
www.thphys.uni-heidelberg.de/∼hefft/vk1
without Animations, Function Plotter
and Solutions of the Exercises
Klaus Hefft
Institute of Theoretical Physics
University of Heidelberg

March 4, 2017
Contents

3-2-30-7-0 0 Preface
3-2-30-7-1
1 MEASURING:- Measured Value and Measuring Unit 
1.1 The Empirical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Order of Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3-2-30-7-2
2 SIGNS AND NUMBERS
and Their Linkages 13
2.1 Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3-2-30-7-3
3 SEQUENCES AND SERIES
and Their Limits 27
3.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Monotony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
i
3-2-30-7-4
4 FUNCTIONS 39
4.1 The Function as Input-Output Relation or Mapping . . . . . . . . . . . . . 39
4.2 Basic Set of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.4 Functions with Kinks and Cracks . . . . . . . . . . . . . . . . . . . 52
4.3 Nested Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6 Monotony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7 Bi-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.8.1 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.8.2 Cyclometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.8.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3-2-30-7-5
5 DIFFERENTIATION 77
5.1 Differential quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Differential Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 The Technique of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1 Four Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.2 Simple Differentiation Rules: Basic Set of Functions . . . . . . . . . 88
5.5.3 Chain and Inverse Function Rules . . . . . . . . . . . . . . . . . . . 92
5.6 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7 Preview of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 99
ii
3-2-30-7-6
6 TAYLOR SERIES 103
6.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Geometric Series as Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Form and Non-ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Examples from the Basic Set of Functions . . . . . . . . . . . . . . . . . . 107
6.4.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.3 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.4 Further Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Convergence Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6 Accurate Rules for Inaccurate Calculations . . . . . . . . . . . . . . . . . . 113
6.7 Quality of Convergence: the Remainder Term . . . . . . . . . . . . . . . . 116
6.8 Taylor Series around an Arbitrary Point . . . . . . . . . . . . . . . . . . . 117
3-2-30-7-7
7 INTEGRATION 121
7.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 Area under a Function over an Interval . . . . . . . . . . . . . . . . . . . . 123
7.3 Properties of the Riemann Integral . . . . . . . . . . . . . . . . . . . . . . 126
7.3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3.2 Interval Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.4 Mean Value Theorem of the Integral Calculus . . . . . . . . . . . . 129
7.4 Fundamental Theorem of Differential and Integral Calculus . . . . . . . . . 130
7.4.1 Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4.2 Differentiation with Respect to the Upper Border . . . . . . . . . . 131
7.4.3 Integration of a Differential Quotient . . . . . . . . . . . . . . . . . 131
7.4.4 Primitive Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5 The Art of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
iii
7.5.1 Differentiation Table Backwards . . . . . . . . . . . . . . . . . . . . 135
7.5.2 Linear Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.5.3 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.5.4 Partial Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.5.5 Further Integration Tricks . . . . . . . . . . . . . . . . . . . . . . . 143
7.5.6 Integral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.6.1 Infinite Integration Interval . . . . . . . . . . . . . . . . . . . . . . 148
7.6.2 Unbounded Integrand . . . . . . . . . . . . . . . . . . . . . . . . . 150
3-2-30-7-8
8 COMPLEX NUMBERS 155
8.1 Imaginary Unit and Illustrations . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.1.2 Imaginary Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.1.3 Definition of complex numbers . . . . . . . . . . . . . . . . . . . . . 157
8.1.4 Gauss Number Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.1.5 Euler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.1.6 Complex Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2 Calculation Rules of Complex Numbers . . . . . . . . . . . . . . . . . . . . 164
8.2.1 Abelian Group of Addition . . . . . . . . . . . . . . . . . . . . . . . 164
8.2.2 Abelian Group of Multiplication . . . . . . . . . . . . . . . . . . . . 167
8.3 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.3.3 Graphic Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3.4 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3.5 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 180
iv
8.3.6 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . 181
8.3.7 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.3.8 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.3.9 General Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
3-2-30-7-9
9 VECTORS 195
9.1 Three-dimensional Euclidean Space . . . . . . . . . . . . . . . . . . . . . . 195
9.1.1 Three-dimensional Real Space . . . . . . . . . . . . . . . . . . . . . 195
9.1.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.1.3 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
9.1.4 Transformations of the Coordinate System . . . . . . . . . . . . . . 198
9.2 Vectors as Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.1 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
9.2.3 Transformations of the Coordinate Systems . . . . . . . . . . . . . 207
9.3 Addition of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.3.1 Vector Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.3.2 Commutative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.3.3 Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.3.4 Zero-vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.3.5 Negatives and Subtraction . . . . . . . . . . . . . . . . . . . . . . . 224
9.4 Multiplication with Real Numbers, Basis Vectors . . . . . . . . . . . . . . 225
9.4.1 Multiple of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.4.2 Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.4.3 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.4.4 Linear Dependence, Basis Vectors . . . . . . . . . . . . . . . . . . . 227
9.4.5 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.5 Scalar Product and the Kronecker Symbol . . . . . . . . . . . . . . . . . . 230
v
9.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.5.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.5.3 Commutative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.5.4 No Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.5.5 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.5.6 Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.5.7 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.5.8 Kronecker Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
9.5.9 Component Representation . . . . . . . . . . . . . . . . . . . . . . 235
9.5.10 Transverse Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.5.11 No Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.6 Vector Product and the Levi-Civita Symbol . . . . . . . . . . . . . . . . . 239
9.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.6.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.6.3 Anticommutative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.6.4 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.6.5 Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.6.6 With Transverse Parts . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.6.7 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.6.8 Levi-Civita Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.6.9 Component Representation . . . . . . . . . . . . . . . . . . . . . . 248
9.6.10 No Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.6.11 No Associative Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.7 Multiple Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.7.1 Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.7.2 Nested Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.7.3 Scalar Product of Two Vector Products . . . . . . . . . . . . . . . . 258
vi
9.7.4 Vector Product of Two Vector Products . . . . . . . . . . . . . . . 259
9.8 Transformation Properties of the Products . . . . . . . . . . . . . . . . . . 262
9.8.1 Orthonormal Right-handed Bases . . . . . . . . . . . . . . . . . . . 262
9.8.2 Group of the Orthogonal Matrices . . . . . . . . . . . . . . . . . . . 263
9.8.3 Subgroup of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.8.4 Transformation of the Products . . . . . . . . . . . . . . . . . . . . 265
vii


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