Conventional calculus is too hard and too complex. Students are forced to learn too many theorems and proofs. In Free Calculus, the author suggests a direct approach to the two fundamental concepts of calculus -- differentiation and integration -- using two inequalities. Regular calculus is condensed into a single concise chapter. This makes the teaching of physics in step with the calculus teaching.
Contents: Calculus in Terms of Images; Official Calculus; Differential Equations of First Order; Differential Equations of Second Order; Free Calculus; Appendix: Calculus of Functional Analysis Becomes Elementary Algebra.
Author(s): Qun Lin
Publisher: World Scientific Publishing Company
Year: 2008
Language: English
Commentary: 166784
Pages: 109
Contents......Page 16
Preface......Page 8
0.1. Hill Behavior and Slope......Page 20
0.2. Hill Height and Slope: Unconstructive Tangent Formula......Page 21
0.3. Review for FT.......Page 24
0.4. Hillside Length and Slope: Pythagoras Theorem......Page 28
0.5. Area and Slope......Page 29
0.6. Explaining All of Calculus in a Single Figure......Page 30
0.7. Calculus and Novels......Page 31
1.0. A Case:Height and Slopes......Page 34
1.1. Translating into Function Language......Page 36
1.2. Generalized First Inequality......Page 49
1.3. Generalized Second Inequality......Page 50
1.4.1. Arithmetic of derivatives......Page 52
1.4.2. Derivatives of rational functions and trigonometric functions......Page 55
1.4.3. Derivatives of composite functions and inverse functions......Page 56
1.5. Tables of Derivatives and Integrals......Page 58
1.6. Rules of Integration......Page 59
1.7. A Calculus Net......Page 61
1.8. Taylor’s Series......Page 63
1.9. Euler’s Formula......Page 64
1.10. Possible Generalizations......Page 65
2.1. A Simplest Differential Equation......Page 70
2.2.1. Thetest equation......Page 71
2.2.3. Separable equation......Page 73
2.4. Tests for Euler’s Algorithm......Page 74
2.5. General Euler’s Algorithm......Page 76
3.1. Initial Value Problems......Page 78
3.2. Eigenvalue Problem......Page 82
3.3. Boundary Value Problem......Page 84
3.4. Weak Equation......Page 85
3.5. Finite Element Solution and Interpolation......Page 87
3.6. Generalization......Page 88
3.7. Summary......Page 91
4.1. Function Spaces, Norms, and Triangle Inequality......Page 92
4.2. Angle and Schwartz’s Inequality......Page 96
4.3. Inner Product......Page 97
4.4. Orthogonality and Projection......Page 98
4.5. Different Inner Products and Norms......Page 100
4.6. Abstract Calculus......Page 101
Appendix......Page 102
2. Derivative Definition Becomes an Elementary Inequality......Page 104
3. Fundamental Theorem Becomes Another Elementary Inequality......Page 105
Acknowledgments......Page 106
References......Page 107
Bibliography......Page 108
