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El. knyga: Mathematical Mechanic: Using Physical Reasoning to Solve Problems

3.65/5 (96 ratings by Goodreads)
  • Formatas: 216 pages
  • Serija: Princeton Science Library
  • Išleidimo metai: 24-Jan-2023
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9780691244174
Kitos knygos pagal šią temą:
Mathematical Mechanic: Using Physical Reasoning to Solve ProblemsDidesnis paveikslėlis
  • Formatas: 216 pages
  • Serija: Princeton Science Library
  • Išleidimo metai: 24-Jan-2023
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9780691244174
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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.

Recenzijos

"A pleasure to read. . . . Newton himself would have been charmed by this book.

"---Steven G. Krantz, UMAP Journal "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 problem-solving 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 non-specialists, especially physicists and engineers. In conclusion--a thoroughly enjoyable and thought-provoking 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 physically-minded students approach mathematics and helping mathematically-minded 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 chock-full 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

Preface ix
1 Introduction
1(8)
1.1 Math versus Physics
1(1)
1.2 What This Book Is About
2(4)
1.3 A Physical versus a Mathematical Solution: An Example
6(2)
1.4 Acknowledgments
8(1)
2 The Pythagorean Theorem
9(18)
2.1 Introduction
9(1)
2.2 The "Fish Tank" Proof of the Pythagorean Theorem
9(3)
2.3 Converting a Physical Argument into a Rigorous Proof
12(2)
2.4 The Fundamental Theorem of Calculus
14(1)
2.5 The Determinant by Sweeping
15(1)
2.6 The Pythagorean Theorem by Rotation
16(1)
2.7 Still Water Runs Deep
17(2)
2.8 A Three-Dimensional Pythagorean Theorem
19(2)
2.9 A Surprising Equilibrium
21(1)
2.10 Pythagorean Theorem by Springs
22(1)
2.11 More Geometry with Springs
23(1)
2.12 A Kinetic Energy Proof: Pythagoras on Ice
24(1)
2.13 Pythagoras and Einstein?
25(2)
3 Minima and Maxima
27(49)
3.1 The Optical Property of Ellipses
28(3)
3.2 More about the Optical Property
31(1)
3.3 Linear Regression (The Best Fit) via Springs
31(3)
3.4 The Polygon of Least Area
34(2)
3.5 The Pyramid of Least Volume
36(3)
3.6 A Theorem on Centroids
39(1)
3.7 An Isoperimetric Problem
40(4)
3.8 The Cheapest Can
44(3)
3.9 The Cheapest Pot
47(1)
3.10 The Best Spot in a Drive-In Theater
48(3)
3.11 The Inscribed Angle
51(1)
3.12 Fermat's Principle and Snell's Law
52(5)
3.13 Saving a Drowning Victim by Fermat's Principle
57(2)
3.14 The Least Sum of Squares to a Point
59(1)
3.15 Why Does a Triangle Balance on the Point of Intersection of the Medians?
60(1)
3.16 The Least Sum of Distances to Four Points in Space
61(2)
3.17 Shortest Distance to the Sides of an Angle
63(1)
3.18 The Shortest Segment through a Point
64(1)
3.19 Maneuvering a Ladder
65(2)
3.20 The Most Capacious Paper Cup
67(2)
3.21 Minimal-Perimeter Triangles
69(3)
3.22 An Ellipse in the Corner
72(2)
3.23 Problems
74(2)
4 Inequalities by Electric Shorting
76(8)
4.1 Introduction
76(2)
4.2 The Arithmetic Mean Is Greater than the Geometric Mean by Throwing a Switch
78(2)
4.3 Arithmetic Mean ≥ Harmonic Mean for n Numbers
80(1)
4.4 Does Any Short Decrease Resistance?
81(2)
4.5 Problems
83(1)
5 Center of Mass: Proofs and Solutions
84(15)
5.1 Introduction
84(1)
5.2 Center of Mass of a Semicircle by Conservation of Energy
85(2)
5.3 Center of Mass of a Half-Disk (Half-Pizza)
87(1)
5.4 Center of Mass of a Hanging Chain
88(1)
5.5 Pappus's Centroid Theorems
89(3)
5.6 Ceva's Theorem
92(2)
5.7 Three Applications of Ceva's Theorem
94(2)
5.8 Problems
96(3)
6 Geometry and Motion
99(10)
6.1 Area between the Tracks of a Bike
99(2)
6.2 An Equal-Volumes Theorem
101(1)
6.3 How Much Gold Is in a Wedding Ring?
102(2)
6.4 The Fastest Descent
104(2)
6.5 Finding d/dt sin t and d/dt cos t by Rotation
106(2)
6.6 Problems
108(1)
7 Computing Integrals Using Mechanics
109(6)
7.1 Computing ∫10 by x dx/1-x2 Lifting a Weight
109(2)
7.2 Computing ∫x0 sin tdt with a Pendulum
111(1)
7.3 A Fluid Proof of Green's Theorem
112(3)
8 The Euler-Lagrange Equation via Stretched Springs
115(5)
8.1 Some Background on the Euler-Lagrange Equation
115(2)
8.2 A Mechanical Interpretation of the Euler-Lagrange Equation
117(1)
8.3 A Derivation of the Euler-Lagrange Equation
118(1)
8.4 Energy Conservation by Sliding a Spring
119(1)
9 Lenses, Telescopes, and Hamiltonian Mechanics
120(13)
9.1 Area-Preserving Mappings of the Plane: Examples
121(1)
9.2 Mechanics and Maps
121(2)
9.3 A (Literally!) Hand-Waving "Proofv of Area Preservation
123(1)
9.4 The Generating Function
124(1)
9.5 A Table of Analogies between Mechanics and Analysis
125(1)
9.6 "The Uncertainty Principle"
126(1)
9.7 Area Preservation in Optics
126(3)
9.8 Telescopes and Area Preservation
129(2)
9.9 Problems
131(2)
10 A Bicycle Wheel and the Gauss-Bonnet Theorem
133(15)
10.1 Introduction
133(2)
10.2 The Dual-Cones Theorem
135(3)
10.3 The Gauss-Bonnet Formula Formulation and Background
138(4)
10.4 The Gauss-Bonnet Formula by Mechanics
142(1)
10.5 A Bicycle Wheel and the Dual Cones
143(3)
10.6 The Area of a Country
146(2)
11 Complex Variables Made Simple(r)
148(13)
11.1 Introduction
148(1)
11.2 How a Complex Number Could Have Been Invented
149(1)
11.3 Functions as Ideal Fluid Flows
150(3)
11.4 A Physical Meaning of the Complex Integral
153(1)
11.5 The Cauchy Integral Formula via Fluid Flow
154(2)
11.6 Heat Flow and Analytic Functions
156(1)
11.7 Riemann Mapping by Heat Flow
157(2)
11.8 Euler's Sum via Fluid Flow
159(2)
Appendix. Physical Background
161(22)
A.1 Springs
161(1)
A.2 Soap Films
162(2)
A.3 Compressed Gas
164(1)
A.4 Vacuum
165(1)
A.5 Torque
165(1)
A.6 The Equilibrium of a Rigid Body
166(1)
A.7 Angular Momentum
167(2)
A.8 The Center of Mass
169(1)
A.9 The Moment of Inertia
170(2)
A.10 Current
172(1)
A.11 Voltage
172(1)
A.12 Kirchhoff's Laws
173(1)
A.13 Resistance and Ohm's Law
174(1)
A.14 Resistors in Parallel
174(1)
A.15 Resistors in Series
175(1)
A.16 Power Dissipated in a Resistor
176(1)
A.17 Capacitors and Capacitance
176(1)
A.18 The Inductance: Inertia of the Current
177(2)
A.19 An Electrical-Plumbing Analogy
179(2)
A.20 Problems
181(2)
Bibliography 183(2)
Index 185
Mark Levi is professor of mathematics at Pennsylvania State University and the author of Why Cats Land on Their Feet (Princeton).
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