The Mathematical Mechanic: Using Physical Reasoning to Solve Problems

Cover
Contents
Preface
1. Introduction
	1.1 Math versus Physics
	1.2 What This Book Is About
	1.3 A Physical versus a Mathematical Solution: An Example
	1.4 Acknowledgments
2. The Pythagorean Theorem
	2.1 Introduction
	2.2 The “Fish Tank” Proof of the Pythagorean Theorem
	2.3 Converting a Physical Argument into a Rigorous Proof
	2.4 The Fundamental Theorem of Calculus
	2.5 The Determinant by Sweeping
	2.6 The Pythagorean Theorem by Rotation
	2.7 Still Water Runs Deep
	2.8 A Three-Dimensional Pythagorean Theorem
	2.9 A Surprising Equilibrium
	2.10 Pythagorean Theorem by Springs
	2.11 More Geometry with Springs
	2.12 A Kinetic Energy Proof: Pythagoras on Ice
	2.13 Pythagoras and Einstein?
3. Minima and Maxima
	3.1 The Optical Property of Ellipses
	3.2 More about the Optical Property
	3.3 Linear Regression (The Best Fit) via Springs
	3.4 The Polygon of Least Area
	3.5 The Pyramid of Least Volume
	3.6 A Theorem on Centroids
	3.7 An Isoperimetric Problem
	3.8 The Cheapest Can
	3.9 The Cheapest Pot
	3.10 The Best Spot in a Drive-In Theater
	3.11 The Inscribed Angle
	3.12 Fermat’s Principle and Snell’s Law
	3.13 Saving a Drowning Victim by Fermat’s Principle
	3.14 The Least Sum of Squares to a Point
	3.15 Why Does a Triangle Balance on the Point of Intersection of the Medians?
	3.16 The Least Sum of Distances to Four Points in Space
	3.17 Shortest Distance to the Sides of an Angle
	3.18 The Shortest Segment through a Point
	3.19 Maneuvering a Ladder
	3.20 The Most Capacious Paper Cup
	3.21 Minimal-Perimeter Triangles
	3.22 An Ellipse in the Corner
	3.23 Problems
4. Inequalities by Electric Shorting
	4.1 Introduction
	4.2 The Arithmetic Mean Is Greater than the Geometric Mean by Throwing a Switch
	4.3 Arithmetic Mean ≥ Harmonic Mean for n Numbers
	4.4 Does Any Short Decrease Resistance?
	4.5 Problems
5. Center of Mass: Proofs and Solutions
	5.1 Introduction
	5.2 Center of Mass of a Semicircle by Conservation of Energy
	5.3 Center of Mass of a Half-Disk (Half-Pizza
	5.4 Center of Mass of a Hanging Chain
	5.5 Pappus’s Centroid Theorems
	5.6 Ceva’s Theorem
	5.7 Three Applications of Ceva’s Theorem
	5.8 Problems
6. Geometry and Motion
	6.1 Area between the Tracks of a Bike
	6.2 An Equal-Volumes Theorem
	6.3 How Much Gold Is in a Wedding Ring?
	6.4 The Fastest Descent
	6.5 Finding dt sin t and dt cos t by Rotation
	6.6 Problems
7. Computing Integrals Using Mechanics
	7.1 Computing by Lifting a Weight
	7.2 Computing sin tdt with a Pendulum
	7.3 A Fluid Proof of Green’s Theorem
8. The Euler-Lagrange Equation via Stretched Springs
	8.1 Some Background on the Euler-Lagrange Equation
	8.2 A Mechanical Interpretation of the Euler-Lagrange Equation
	8.3 A Derivation of the Euler-Lagrange Equation
	8.4 Energy Conservation by Sliding a Spring
9. Lenses, Telescopes, and Hamiltonian Mechanics
	9.1 Area-Preserving Mappings of the Plane: Examples
	9.2 Mechanics and Maps
	9.3 A (Literally!) Hand-Waving “Proof” of Area Preservation
	9.4 The Generating Function
	9.5 A Table of Analogies between Mechanics and
 Analysis
	9.6 “The Uncertainty Principle”
	9.7 Area Preservation in Optics
	9.8 Telescopes and Area Preservation
	9.9 Problems
10. A Bicycle Wheel and the Gauss-Bonnet Theorem
	10.1 Introduction
	10.2 The Dual-Cones Theorem
	10.3 The Gauss-Bonnet Formula Formulation and Background
	10.4 The Gauss-Bonnet Formula by Mechanics
	10.5 A Bicycle Wheel and the Dual Cones
	10.6 The Area of a Country
11. Complex Variables Made Simple(r)
	11.1 Introduction
	11.2 How a Complex Number Could Have Been Invented
	11.3 Functions as Ideal Fluid Flows
	11.4 A Physical Meaning of the Complex Integral
	11.5 The Cauchy Integral Formula via Fluid Flow
	11.6 Heat Flow and Analytic Functions
	11.7 Riemann Mapping by Heat Flow
	11.8 Euler’s Sum via Fluid Flow
Appendix. Physical Background
	A.1 Springs
	A.2 Soap Films
	A.3 Compressed Gas
	A.4 Vacuum
	A.5 Torque
	A.6 The Equilibrium of a Rigid Body
	A.7 Angular Momentum
	A.8 The Center of Mass
	A.9 The Moment of Inertia
	A.10 Current
	A.11 Voltage
	A.12 Kirchhoff’s Laws
	A.13 Resistance and Ohm’s Law
	A.14 Resistors in Parallel
	A.15 Resistors in Series
	A.16 Power Dissipated in a Resistor
	A.17 Capacitors and Capacitance
	A.18 The Inductance: Inertia of the Current
	A.19 An Electrical-Plumbing Analogy
	A.20 Problems
Bibliography
Index