Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist.
Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles.
The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.
Author(s): Mark Levi
Edition: 1
Publisher: Princeton University Press
Year: 2009
Language: English
Pages: 196
Tags: Физика;Матметоды и моделирование в физике;
Cover
......Page 1
Contents......Page 6
7.1 Computing ......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
0 √x dx 1−x2 by Lifting a Weight......Page 120
7.2 Computing 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
