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More About This Textbook
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 mindbending 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 oneofakind 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 littleknown connections between mathematics and physics and how both endeavors relate to the world around us.
Editorial Reviews
Choice
The Mathematical Mechanic is a pleasant surprise.— E. Kincanon
Mathematics Teacher
The Mathematical Mechanic documents novel ways of viewing physics as a method of understanding mathematics. Levi uses physical arguments as tools to conjecture about mathematical concepts before providing rigorous proofs. . . . The Mathematical Mechanic is an excellent display of creative, interdisciplinary problemsolving strategies. The author has explained complex concepts with simplicity, yet the mathematics is accurate.MAA Reviews
The Mathematical Mechanic reverses the usual interaction of mathematics and physics. . . . Careful study of Levi's book may train readers to think of physical companions to mathematical problems. . . . Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to supplement mathematics courses, helping physicallyminded students approach mathematics and helping mathematicallyminded students appreciate physics.— John D. Cook
SEED Magazine
Mark Levi reverses the old stereotype that math is merely a tool to aid physicists by showing that many questions in mathematics can be easily solved by interpreting them as physical problems. . . . Some sections of the book require readers to brush up on their calculus but Levi's clear explanations, witty footnotes, and fascinating insights make the extra effort painless.The Math Less Traveled
The book is chockfull of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions. . . . I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician.— Boris Yorgey
London Mathematical Society Newsletter
A most interesting book. . . . Many of the ideas in it could be used as motivational or illustrative examples to support the teaching of nonspecialists, especially physicists and engineers. In conclusion—a thoroughly enjoyable and thoughtprovoking read.— Nigel Steele
London Mathematical Society Newsletter
A most interesting book. . . . Many of the ideas in it could be used as motivational or illustrative examples to support the teaching of nonspecialists, especially physicists and engineers. In conclusion—a thoroughly enjoyable and thoughtprovoking read.— Nigel Steele
MAA Reviews
The Mathematical Mechanic reverses the usual interaction of mathematics and physics. . . . Careful study of Levi's book may train readers to think of physical companions to mathematical problems. . . . Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to supplement mathematics courses, helping physicallyminded students approach mathematics and helping mathematicallyminded students appreciate physics.— John D. Cook
SEED Magazine
Mark Levi reverses the old stereotype that math is merely a tool to aid physicists by showing that many questions in mathematics can be easily solved by interpreting them as physical problems. . . . Some sections of the book require readers to brush up on their calculus but Levi's clear explanations, witty footnotes, and fascinating insights make the extra effort painless.Choice
The Mathematical Mechanic is a pleasant surprise.— E. Kincanon
The Math Less Traveled
The book is chockfull of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions. . . . I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician.— Boris Yorgey
Mathematics Teacher
The Mathematical Mechanic documents novel ways of viewing physics as a method of understanding mathematics. Levi uses physical arguments as tools to conjecture about mathematical concepts before providing rigorous proofs. . . . The Mathematical Mechanic is an excellent display of creative, interdisciplinary problemsolving strategies. The author has explained complex concepts with simplicity, yet the mathematics is accurate.UMAP Journal
This is a delightful and unusual book that is a welcome addition to the literature. Certainly, any calculus teacher and many others of us as well will want to have it on the shelf for ready reference. It not only will enhance our teaching experience but will also teach us (the instructors) something in the process.— Steven G. Krantz
Seed Magazine
Mark Levi reverses the old stereotype that math is merely a tool to aid physicists by showing that many questions in mathematics can be easily solved by interpreting them as physical problems. . . . Some sections of the book require readers to brush up on their calculus but Levi's clear explanations, witty footnotes, and fascinating insights make the extra effort painless.London Mathematical Society Newsletter  Nigel Steele
A most interesting book. . . . Many of the ideas in it could be used as motivational or illustrative examples to support the teaching of nonspecialists, especially physicists and engineers. In conclusion—a thoroughly enjoyable and thoughtprovoking read.MAA Reviews  John D. Cook
The Mathematical Mechanic reverses the usual interaction of mathematics and physics. . . . Careful study of Levi's book may train readers to think of physical companions to mathematical problems. . . . Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to supplement mathematics courses, helping physicallyminded students approach mathematics and helping mathematicallyminded students appreciate physics.The Math Less Traveled  Boris Yorgey
The book is chockfull of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions. . . . I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician.Choice  E. Kincanon
The Mathematical Mechanic is a pleasant surprise.UMAP Journal  Steven G. Krantz
This is a delightful and unusual book that is a welcome addition to the literature. Certainly, any calculus teacher and many others of us as well will want to have it on the shelf for ready reference. It not only will enhance our teaching experience but will also teach us (the instructors) something in the process.From the Publisher
One of Amazon.com science editors' Top 10 list for Science, Best for 2009One of Choice's Outstanding Academic Titles for 2009
"The Mathematical Mechanic documents novel ways of viewing physics as a method of understanding mathematics. Levi uses physical arguments as tools to conjecture about mathematical concepts before providing rigorous proofs. . . . The Mathematical Mechanic is an excellent display of creative, interdisciplinary problemsolving strategies. The author has explained complex concepts with simplicity, yet the mathematics is accurate."—Mathematics Teacher
"A most interesting book. . . . Many of the ideas in it could be used as motivational or illustrative examples to support the teaching of nonspecialists, especially physicists and engineers. In conclusion—a thoroughly enjoyable and thoughtprovoking read."—Nigel Steele, London Mathematical Society Newsletter
"The Mathematical Mechanic reverses the usual interaction of mathematics and physics. . . . Careful study of Levi's book may train readers to think of physical companions to mathematical problems. . . . Mathematicians will find The Mathematical Mechanic provides exercise in new ways of thinking. Instructors will find it contains material to supplement mathematics courses, helping physicallyminded students approach mathematics and helping mathematicallyminded students appreciate physics."—John D. Cook, MAA Reviews
"Mark Levi reverses the old stereotype that math is merely a tool to aid physicists by showing that many questions in mathematics can be easily solved by interpreting them as physical problems. . . . Some sections of the book require readers to brush up on their calculus but Levi's clear explanations, witty footnotes, and fascinating insights make the extra effort painless."—SEED Magazine
"The book is chockfull of these seemingly magical physical thought experiments involving bicycle wheels, pistons, springs, soap films, pendulums, and electric circuits, with applications to geometry, maximization and minimization problems, inequalities, optics, integrals, and complex functions. . . . I highly recommend it to anyone who is (even slightly) interested in physics, and appreciates mathematical elegance and cleverness. It would make a great gift for almost anyone, whether a high school student or university professor, armchair physicist or professional mathematician."—Boris Yorgey, The Math Less Traveled
"The Mathematical Mechanic is a pleasant surprise."—E. Kincanon, Choice
"This is a delightful and unusual book that is a welcome addition to the literature. Certainly, any calculus teacher and many others of us as well will want to have it on the shelf for ready reference. It not only will enhance our teaching experience but will also teach us (the instructors) something in the process."—Steven G. Krantz, UMAP Journal
Product Details
Meet the Author
Mark Levi is professor of mathematics at Pennsylvania State University and the author of "Why Cats Land on Their Feet" (Princeton).
Table of Contents
Contents
1 Introduction....................11.1 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 ThreeDimensional 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 DriveIn 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 MinimalPerimeter 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 HalfDisk (HalfPizza)....................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 EqualVolumes 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 EulerLagrange Equation via Stretched Springs....................115
8.1 Some Background on the EulerLagrange Equation....................115
8.2 A Mechanical Interpretation of the EulerLagrange Equation....................117
8.3 A Derivation of the EulerLagrange Equation....................118
8.4 Energy Conservation by Sliding a Spring....................119
9 Lenses, Telescopes, and Hamiltonian Mechanics....................120
9.1 AreaPreserving Mappings of the Plane: Examples....................121
9.2 Mechanics and Maps....................121
9.3 A (Literally!) HandWaving "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 GaussBonnet Theorem....................133
10.1 Introduction....................133
10.2 The DualCones Theorem....................135
10.3 The GaussBonnet Formula Formulation and Background....................138
10.4 The GaussBonnet 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 ElectricalPlumbing Analogy....................179
A.20 Problems....................181
Bibliography....................183
Index....................185