Free Calculus: A Liberation from Concepts and Proofs Book

Preface vii

0 Calculus in Terms of Images: 1

0.1 Hill Behavior and Slope 1

0.2 Hill Height and Slope: Unconstructive Tangent Formula 2

0.3 Review for FT 5

0.4 Hillside Length and Slope: Pythagoras Theorem 9

0.5 Area and Slope 10

0.6 Explaining All of Calculus in a Single Figure 11

0.7 Calculus and Novels 12

1 Official Calculus: 15

1.0 A Case: Height and Slopes 15

1.1 Translating into Function Language 17

1.2 Generalized First Inequality 30

1.3 Generalized Second Inequality 31

1.4 Rules of Differentiation 33

1.4.1 Arithmetic of derivatives 33

1.4.2 Derivatives of rational functions and trigonometric functions 36

1.4.3 Derivatives of composite functions and inverse functions 37

1.5 Tables of Derivatives and Integrals 39

1.6 Rules of Integration 40

1.7 A Calculus Net 42

1.8 Taylor's Series 44

1.9 Euler's Formula 45

1.10 Possible Generalizations 46

2 Differential Equations of First Order 51

2.1 A Simplest Differential Equation 51

2.2 Varieties of Simplest Differential Equation 52

2.2.1 The test equation 52

2.2.2 Linear differential equation 54

2.2.3 Separable equation 54

2.3 More General Equations 55

2.4 Tests for Euler's Algorithm 55

2.5 General Euler's Algorithm 57

3 Differential Equations of Second Order 59

3.1 Initial Value Problems 59

3.2 Eigenvalue Problem 63

3.3 Boundary Value Problem 65

3.4 Weak Equation 66

3.5 Finite Element Solution and Interpolation 68

3.6 Generalization 69

3.7 Summary 72

4 Free Calculus 73

4.1 Function Spaces, Norms, and Triangle Inequality 73

4.2 Angle and Schwartz's Inequality 77

4.3 Inner Product 78

4.4 Orthogonality and Projection 79

4.5 DifferentInner Products and Norms 81

4.6 Abstract Calculus 82

Appendix 83

Calculus of Functional Analysis Becomes Elementary Algebra 85

1 Introduction 85

2 Derivative Definition Becomes an Elementary Inequality 85

3 Fundamental Theorem Becomes Another Elementary Inequality 86

Acknowledgments 87

References 88

Bibliography 89