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Contents
1 Logic 1
1:1 Propositions and Truth Values 2
1:2 Sentential Connectives 3
1:3 Truth Tables for Compound Propositions5
1:4 Tautologies and Contradictions 6
1:5 Deductions 7
1:6 Deduction and Induction in Empirical Science 8
1:7 Information Content of a Theory 10
1:8 The Limits of Logic 11
2 Set Theory 13
2:1 Elements and sets 13
2:2 Operations on Sets 14
2:3 Relations between Sets 17
2:4 Connections between Proposition Logic and Set Theory 19
2:5 Product Sets and Relations 21
2:6 Ordering Relations 22
2:7 Scientific Concept Types 24
3 The Number System 25
3:1 Natural Numbers and Rational Numbers 25
3:2 The Continuum of Real Numbers 27
4 Variables and Functions 31
4:1 Variables and constants 31
4:2 Functions and Maps 31
4:3 Function Value and Argument Value 34
4:4 Functional Equations 35
4:5 Domains and Ranges 35
4:6 Explicit and Implicit Functions 36
4:7 Solving Algebraic Equations 36
5 Coordinate Systems and Elementary Functions 41
5:1 Cartesian Coordinates 41
5:2 Polar Coordinates 42
4:3 Coordinate Transformations 43
5:4 Trigonometric Functions 45
5:5 Power Functions 48
5:6 Polynomials 53
5:7 Exponential and Logarithmic Functions 53
6 Calculus: One Arguments 59
6:1 Limits and Continuity 59
6:2 First Derivatives 60
6:3 Derivatives in Practice 64
6:4 General Differentiation Rules 66
6:5 Higher Derivatives 69
6:6 Power Series Expansions 71
6:7 Minima and Maxima 72
6:8 Differentials 77
7 Calculus: Two Arguments 81
7:1 Functions of two arguments 81
7:2 Level Curves and Implicit Functions 83
7:3 Partial Derivatives 85
7:4 Differentials 88
7:5 Minima, Maxima and Saddle Points 90
7:6 Differentiating Implicit Functions 95
7:7 The Elasticity of Substitution 98
7:8 Homogenous Functions 105
7:9 Minima and Maxima with Linear Constraints 111
8 Integration 121
8:1 A First Definition 121
8:2 Indefinite Integral; Arbitrary Constants 122
8:3 Definite Integral; Geometric Interpretation 123
8:4 Integration Techniques 126
8:5 Integration by Parts 127
8:6 Variable Substitution 128
8:7 Multiple Integrals 130
9 Calculus of Variations and Optimal Control 135
9:1 Euler's Equation 136
9:2 A Simpler Case 140
9:3 Variable Endpoints and Transversality 142
9:4 Optimal Control 147
10 Differential Equations 153
10:1 First Order Equations 154
10:2 Second Order Equations 158
10:3 Complex Roots 163
11 Linear Equation Systems and Determinants 177
11:1 Indices 177
11:2 The Summation Sign 178
11:3 Matrix and Determinant 179
11:4 Minor and Co-factor 180
11:5 Expansion of a Determinant 180
11:6 Expansion with "Alien" Co-factors 184
11:7 Cramer's Rule for Linear Equation Systems 185
11:8 Existence of a Unique Solution 188
12 Matrices 193
12:1 Types of Matrices 193
12:2 Unit Matrix and Zero Matrix 194
12:3 Transposition of Matrices 195
12:4 Vectors 196
12:5 Matrix Algebra 196
12:6 Matrix Addition 196
12:7 Matrix Multiplication 197
12:6 Linear Equation Systems 201
12:9 Inverse Matrix 202
12:10 Solution of Linear Equation Systems through Matrix Inversion 206
12:11 Eigenvalue Problems 209
13 Linear and Nonlinear Programming 215
13:4 The Kuhn-Tucker Theorem 215
13:2 Linear Programming 226
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