The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

by Mark Levi
     
 

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ISBN-10: 0691140200

ISBN-13: 9780691140209

Pub. Date: 07/06/2009

Publisher: Princeton University Press

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

Overview

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.

Product Details

ISBN-13:
9780691140209
Publisher:
Princeton University Press
Publication date:
07/06/2009
Edition description:
New Edition
Pages:
196
Product dimensions:
6.40(w) x 9.30(h) x 0.90(d)

Table of Contents

Contents

1 Introduction....................1
1.1 Math versus Physics....................1
1.2 What This Book Is About....................2
1.3 A Physical versus a Mathematical Solution: An Example....................6
1.4 Acknowledgments....................8
2 The Pythagorean Theorem....................9
2.1 Introduction....................9
2.2 The "Fish Tank" Proof of the Pythagorean Theorem....................9
2.3 Converting a Physical Argument into a Rigorous Proof....................12
2.4 The Fundamental Theorem of Calculus....................14
2.5 The Determinant by Sweeping....................15
2.6 The Pythagorean Theorem by Rotation....................16
2.7 Still Water Runs Deep....................17
2.8 A Three-Dimensional Pythagorean Theorem....................19
2.9 A Surprising Equilibrium....................21
2.10 Pythagorean Theorem by Springs....................22
2.11 More Geometry with Springs....................23
2.12 A Kinetic Energy Proof: Pythagoras on Ice....................24
2.13 Pythagoras and Einstein?....................25
3 Minima and Maxima....................27
3.1 The Optical Property of Ellipses....................28
3.2 More about the Optical Property....................31
3.3 Linear Regression (The Best Fit) via Springs....................31
3.4 The Polygon of Least Area....................34
3.5 The Pyramid of Least Volume....................36
3.6 A Theorem on Centroids....................39
3.7 An Isoperimetric Problem....................40
3.8 The Cheapest Can....................44
3.9 The Cheapest Pot....................47
3.10 The Best Spot in a Drive-In Theater....................48
3.11 The Inscribed Angle....................51
3.12 Fermat's Principle and Snell's Law....................52
3.13 Saving a Drowning Victim by Fermat's Principle....................57
3.14 The Least Sum of Squares to a Point....................59
3.15 Why Does a Triangle Balance on the Point of Intersection of the Medians?....................60
3.16 The Least Sum of Distances to Four Points in Space....................61
3.17 Shortest Distance to the Sides of an Angle....................63
3.18 The Shortest Segment through a Point....................64
3.19 Maneuvering a Ladder....................65
3.20 The Most Capacious Paper Cup....................67
3.21 Minimal-Perimeter Triangles....................69
3.22 An Ellipse in the Corner....................72
3.23 Problems....................74
4 Inequalities by Electric Shorting....................76
4.1 Introduction....................76
4.2 The Arithmetic Mean Is Greater than the Geometric Mean by Throwing a Switch....................78
4.3 Arithmetic Mean Harmonic Mean for n Numbers....................80
4.4 Does Any Short Decrease Resistance?....................81
4.5 Problems....................83
5 Center of Mass: Proofs and Solutions....................84
5.1 Introduction....................84
5.2 Center of Mass of a Semicircle by Conservation of Energy....................85
5.3 Center of Mass of a Half-Disk (Half-Pizza)....................87
5.4 Center of Mass of a Hanging Chain....................88
5.5 Pappus's Centroid Theorems....................89
5.6 Ceva's Theorem....................92
5.7 Three Applications of Ceva's Theorem....................94
5.8 Problems....................96
6 Geometry and Motion....................99
6.1 Area between the Tracks of a Bike....................99
6.2 An Equal-Volumes Theorem....................101
6.3 How Much Gold Is in a Wedding Ring?....................102
6.4 The Fastest Descent....................104
6.5 Finding d/dt sin t and d/dt cos t by Rotation....................106
6.6 Problems....................108
7 Computing Integrals Using Mechanics....................109
7.1 Computing [[integral].sup.1.sub.0] x dx/[square root of 1-[x.sup.2]] by Lifting a Weight....................109
7.2 Computing [[integral].sup.x.sub.0] sin tdt with a Pendulum....................111
7.3 A Fluid Proof of Green's Theorem....................112
8 The Euler-Lagrange Equation via Stretched Springs....................115
8.1 Some Background on the Euler-Lagrange Equation....................115
8.2 A Mechanical Interpretation of the Euler-Lagrange Equation....................117
8.3 A Derivation of the Euler-Lagrange Equation....................118
8.4 Energy Conservation by Sliding a Spring....................119
9 Lenses, Telescopes, and Hamiltonian Mechanics....................120
9.1 Area-Preserving Mappings of the Plane: Examples....................121
9.2 Mechanics and Maps....................121
9.3 A (Literally!) Hand-Waving "Proof" of Area Preservation....................123
9.4 The Generating Function....................124
9.5 A Table of Analogies between Mechanics and Analysis....................125
9.6 "The Uncertainty Principle"....................126
9.7 Area Preservation in Optics....................126
9.8 Telescopes and Area Preservation....................129
9.9 Problems....................131
10 A Bicycle Wheel and the Gauss-Bonnet Theorem....................133
10.1 Introduction....................133
10.2 The Dual-Cones Theorem....................135
10.3 The Gauss-Bonnet Formula Formulation and Background....................138
10.4 The Gauss-Bonnet Formula by Mechanics....................142
10.5 A Bicycle Wheel and the Dual Cones....................143
10.6 The Area of a Country....................146
11 Complex Variables Made Simple(r)....................148
11.1 Introduction....................148
11.2 How a Complex Number Could Have Been Invented....................149
11.3 Functions as Ideal Fluid Flows....................150
11.4 A Physical Meaning of the Complex Integral....................153
11.5 The Cauchy Integral Formula via Fluid Flow....................154
11.6 Heat Flow and Analytic Functions....................156
11.7 Riemann Mapping by Heat Flow....................157
11.8 Euler's Sum via Fluid Flow....................159
Appendix. Physical Background....................161
A.1 Springs....................161
A.2 Soap Films....................162
A.3 Compressed Gas....................164
A.4 Vacuum....................165
A.5 Torque....................165
A.6 The Equilibrium of a Rigid Body....................166
A.7 Angular Momentum....................167
A.8 The Center of Mass....................169
A.9 The Moment of Inertia....................170
A.10 Current....................172
A.11 Voltage....................172
A.12 Kirchhoff's Laws....................173
A.13 Resistance and Ohm's Law....................174
A.14 Resistors in Parallel....................174
A.15 Resistors in Series....................175
A.16 Power Dissipated in a Resistor....................176
A.17 Capacitors and Capacitance....................176
A.18 The Inductance: Inertia of the Current....................177
A.19 An Electrical-Plumbing Analogy....................179
A.20 Problems....................181
Bibliography....................183
Index....................185

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