Free Calculus
(A Liberation from Concepts and Proofs)
Qun Lin(林群)
2008
英文

Contents
Preface vii
0. Calculus in Terms of Images: 1
0.1. HillBehaviorandSlope . . . . . . . . . . . . . . . . . . . . . . 1
0.2. Hill Height and Slope: Unconstructive Tangent Formula . . . . . 2
0.3. ReviewforFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.4. Hillside Length and Slope: Pythagoras Theorem . . . . . . . . . . 9
0.5. Area andSlope . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
0.6. ExplainingAll ofCalculus in aSingleFigure . . . . . . . . . . . 11
0.7. Calculus andNovels . . . . . . . . . . . . . . . . . . . . . . . . 12
1. Official Calculus: 15
1.0. ACase:Height andSlopes . . . . . . . . . . . . . . . . . . . . . 15
1.1. Translating into Function Language . . . . . . . . . . . . . . . . 17
1.2. Generalized First Inequality . . . . . . . . . . . . . . . . . . . . 30
1.3. Generalized Second Inequality . . . . . . . . . . . . . . . . . . . 31
1.4. Rules ofDifferentiation . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.1. Arithmeticofderivatives . . . . . . . . . . . . . . . . . . 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 ofDerivativesandIntegrals . . . . . . . . . . . . . . . . . 39
1.6. Rules of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.7. ACalculusNet . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.8. Taylor’sSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.9. Euler’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.10. PossibleGeneralizations . . . . . . . . . . . . . . . . . . . . . . 46
2. Differential Equations of First Order 51
2.1. ASimplestDifferentialEquation . . . . . . . . . . . . . . . . . . 51
2.2. Varieties ofSimplestDifferentialEquation. . . . . . . . . . . . . 52
2.2.1. Thetest equation . . . . . . . . . . . . . . . . . . . . . . 52
2.2.2. Lineardifferentialequation . . . . . . . . . . . . . . . . . 54
2.2.3. Separableequation . . . . . . . . . . . . . . . . . . . . . 54
2.3. MoreGeneralEquations . . . . . . . . . . . . . . . . . . . . . . 55
2.4. Tests forEuler’sAlgorithm . . . . . . . . . . . . . . . . . . . . . 55
2.5. GeneralEuler’sAlgorithm . . . . . . . . . . . . . . . . . . . . . 57
3. Differential Equations of Second Order 59
3.1. Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . 59
3.2. EigenvalueProblem. . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3. Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 65
3.4. WeakEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5. FiniteElementSolutionandInterpolation . . . . . . . . . . . . . 68
3.6. Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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. Different Inner Products and Norms . . . . . . . . . . . . . . . . 81
4.6. AbstractCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 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