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Category: Mathematics edition: Authors: Mark Levi serie: ISBN : 9780691140209, 0691140200 publisher: PUP publish year: 2009 pages: 197 language: English ebook format : DJVU (It will be converted to PDF, EPUB OR AZW3 if requested by the user) file size: 2 MB

price : $14.45 17 With 15% OFF

Cover)......Page 1

Contents......Page 6

1.1 Math versus Physics......Page 12

1.2 What This Book Is About......Page 13

1.3 A Physical versus a Mathematical Solution: An Example......Page 17

1.4 Acknowledgments......Page 19

2.2 The 'Fish Tank' Proof of the Pythagorean Theorem......Page 20

2.3 Converting a Physical Argument into a Rigorous Proof......Page 23

2.4 The Fundamental Theorem of Calculus......Page 25

2.5 The Determinant by Sweeping......Page 26

2.6 The Pythagorean Theorem by Rotation......Page 27

2.7 Still Water Runs Deep......Page 28

2.8 A Three-Dimensional Pythagorean Theorem......Page 30

2.9 A Surprising Equilibrium......Page 32

2.10 Pythagorean Theorem by Springs......Page 33

2.11 More Geometry with Springs......Page 34

2.12 A Kinetic Energy Proof: Pythagoras on Ice......Page 35

2.13 Pythagoras and Einstein?......Page 36

3 Minima and Maxima......Page 38

3.1 The Optical Property of Ellipses......Page 39

3.3 Linear Regression (The Best Fit) via Springs......Page 42

3.4 The Polygon of Least Area......Page 45

3.5 The Pyramid of Least Volume......Page 47

3.6 A Theorem on Centroids......Page 50

3.7 An Isoperimetric Problem......Page 51

3.8 The Cheapest Can......Page 55

3.9 The Cheapest Pot ......Page 58

3.10 The Best Spot in a Drive-In Theater......Page 59

3.11 The Inscribed Angle......Page 62

3.12 Fermat's principle and Snell's law......Page 63

3.13 Saving a drowning victim by Fermat's principle......Page 68

3.14 The Least Sum of Squares to a Point......Page 70

3.15 Why Does a Triangle Balance on the Point of Intersection of the Medians?......Page 71

3.16 The Least Sum of Distances to Four Points in Space......Page 72

3.17 Shortest Distance to the Sides of an Angle......Page 74

3.18 The Shortest Segment through a Point......Page 75

3.19 Maneuvering a Ladder......Page 76

3.20 The Most Capacious Paper Cup......Page 78

3.21 Minimal-Perimeter Triangles......Page 80

3.22 An Ellipse in the Corner......Page 83

3.23 Problems......Page 85

4.1 Introduction......Page 87

4.2 The Arithmetic Mean Is Greater than the Geometric Mean by Throwing a Switch......Page 89

4.3 Arithmetic Mean > Harmonic Mean for n numbers......Page 91

4.4 Does Any Short Decrease Resistance?......Page 92

4.5 Problems......Page 94

5.1 Introduction......Page 95

5.2 Center of Mass of a Semicircle by Conservation of Energy......Page 96

5.3 Center of Mass of a Half-Disk (Half-Pizza)......Page 98

5.4 Center of Mass of a Hanging Chain......Page 99

5.5 Pappus's Centroid Theorems......Page 100

5.6 Ceva's Theorem......Page 103

5.7 Three applications of Ceva's theorem......Page 105

5.8 Problems......Page 107

6.1 Area between the Tracks of a Bike......Page 110

6.2 An Equal-Volumes Theorem......Page 112

6.3 How Much Gold Is in a Wedding Ring?......Page 113

6.4 The Fastest Descent ......Page 115

6.5 Finding d dt sin t and d dt cos t by Rotation......Page 117

6.6 Problems......Page 119

7.1 Computing integral by lifting a weight......Page 120

7.2 Computing int x 0 sin tdt with a Pendulum......Page 122

7.3 A fluid proof of Green's theorem......Page 123

8.1 Some Background on the Euler-Lagrange Equation......Page 126

8.2 A Mechanical Interpretation of the Euler-Lagrange Equation......Page 128

8.3 A Derivation of the Euler-Lagrange Equation......Page 129

8.4 Energy Conservation by Sliding a Spring......Page 130

9 Lenses, Telescopes, and Hamiltonian Mechanics......Page 131

9.2 Mechanics and Maps......Page 132

9.3 A (Literally!) Hand-Waving 'Proof' of Area Preservation......Page 134

9.4 The Generating Function......Page 135

9.5 A Table of Analogies between Mechanics and Analysis......Page 136

9.7 Area Preservation in Optics......Page 137

9.8 Telescopes and Area Preservation......Page 140

9.9 Problems......Page 142

10.1 Introduction......Page 144

10.2 The Dual-Cones Theorem......Page 146

10.3 The Gauss-Bonnet Formula Formulation and Background......Page 149

10.4 The Gauss-Bonnet Formula by Mechanics......Page 153

10.5 A Bicycle Wheel and the Dual Cones......Page 154

10.6 The Area of a Country......Page 157

11.1 Introduction......Page 159

11.2 How a Complex Number Could Have Been Invented ......Page 160

11.3 Functions as Ideal Fluid Flows......Page 161

11.4 A Physical Meaning of the Complex Integral......Page 164

11.5 The Cauchy Integral Formula via Fluid Flow......Page 165

11.6 Heat Flow and Analytic Functions......Page 167

11.7 Riemann Mapping by Heat Flow......Page 168

11.8 Euler's Sum via Fluid Flow......Page 170

A.1 Springs......Page 172

A.2 Soap Films......Page 173

A.3 Compressed Gas......Page 175

A.5 Torque......Page 176

A.6 The Equilibrium of a Rigid Body......Page 177

A.7 Angular Momentum......Page 178

A.8 The Center of Mass......Page 180

A.9 The Moment of Inertia......Page 181

A.11 Voltage......Page 183

A.12 Kirchhoff's Laws......Page 184

A.14 Resistors in Parallel......Page 185

A.15 Resistors in Series......Page 186

A.17 Capacitors and Capacitance......Page 187

A.18 The Inductance: Inertia of the Current......Page 188

A.19 An Electrical-Plumbing Analogy......Page 190

A.20 Problems......Page 192

Bibliography......Page 194

Index ......Page 196