Mathematical Methods in Science
| Date: 06 May 2011, 19:46
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'Mathematics, taught and learned appropriately, improves the mind and implants good habits of thought.' This tenet underlies all of Professor Polya's works on teaching and problem-solving. This book captures some of Polya's excitement and vision. In it he provides enlightenment for all those who have ever wondered how the laws of nature were worked out mathematically. The distinctive feature of the present book is the stress on the history of certain elementary chapters of science; these can be a source of enjoyment and deeper understanding of mathematics even for beginners who have little, or perhaps no, knowledge of physics.
About the Author
George Poyla- received his doctor's degree in Budapest and taught at the Swiss Federal Institute of Technology in Zurich from 1914 to 1940, and at Stanford University from 1942 until his retirement in 1953.
Contents
INTRODUCTION
CHAPTER 1 From the History of Astronomy: Measurement and Successive Approximation
SECTION 1 MEASUREMENT
1 The Tunnel
2 Measuring: Triangulating
3 How Far Away is the Moon?
4 Why Teach Triangulation?
SECTION 2 ASTRONOMICAL MEASUREMENTS
1 Aristarchus of Samos
2 Radius of Earth: Eratosthenes
3 Rival Cosmologies
4 The Orbit of Venus
5 Tycho Brahe and Kepler
6 The Mars Year
7 The Orbit of Mars
8 A Word to the Perceptive Reader
9 Newton's Problem of a Comet's Path
SECTION 3 SUCCESSIVE APPROXIMATION
1 First Application
2 Extraction of Square Roots
SECTION 4 NEWTON'S METHOD OF SUCCESSIVE APPROXIMATION
1 The General Method of Newton
2 Newton's Formula
3
4
5
CHAPTEP 2 From the History of Statics
SECTION 1 STEVINUS AND ARCHIMEDES
1 Inclined Plane
2 Lever
SECTION 2 VECTORS
1 Inclined Plane
2 Pulley
3 Lever
4 Archimedes' Application of his Law of the Lever
5 (-) * (-) = (+)
6 Von Mises' Flight Triangle
CHAPTER 3 From the History of Dynamics
SECTION 1 GALILEO
1 Heavier Bodies Fall Faster?
2 Not "Why?", But "How?"
3 How do Heavy Bodies Fall?
4 Dynamics of the Inclined Plane
5 Conservation of Energy
6 Law of Inertia
7 A Cannon Ball's Trajectory
SECTION 2 NEWTON 105
1 Apples. Cannon Balls, and the Moon 106
2 Never Smoke Without Fire 107
3 That the Planets do Accelerate Towards the Sun lOB
4 What is the Law of Universal Gravitation? III
5 Uniform Circular Motion: Hamilton's Hodograph 112
6 On Newton's Discovery of the Law of Universal Gravitation
7 Scientific Attitude: Verification 117
8 Hindsight and Foresight 122
SECTION 3 THE PENDULUM 125
1 The Dimensions Test 125
2 Simple Pendulum's Time of Swing 128
3 Determination of g by Pendulum Experiment 131
4 The Conical Pendulum 132
SECTION 4 ESCAPE VELOCITY 138
1 Go-Around Velocity 139
2 Apropos Go-Away Velocity 141
3 The Force of Gravity 142
4 That Kepler's Third Law is a Consequence of Newton's Law of Gravitation
5 Planetary Mass 147
6 Go-Away Velocity 148
7 Ratio of Escape and Orbital Velocities 157
CHAPTER 4 Physical Reasoning in Mathematics 159
CHAPTER 5 Differential Equations and Their Use in Science
SECTION 1 FIRST EXAMPLFS
1 Rotating Fluid
2 Galileo: Free Fall
3 Catenary
4 Fall with Friction
SECTION 2 APPROXIMATE FORMULAE: POWER SERIES
Introduction
1 Calculation of ...
2 Fall with Friction Again
3 How Deep is a Well?
4 Pendulum: Small Oscillations
SECTION 3 PHYSICAL ANALOGY
SECTION 4 WHAT IS A DIFFERENTIAL EQUATION?
1 Example
2 Vector Fields
3 Direction Fields
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