Techniques for applying mathematical concepts in the real world: six rarely taught but crucial tools for analysis, research, and problem-solving.
Many young graduates leave school with a solid knowledge of mathematical concepts but struggle to apply these concepts in practice. Real scientific and engineering problems are different from those found in textbooks: they are messier, take longer to solve, and standard solution recipes might not apply. This book fills the gap between what is taught in the typical college curriculum and what a practicing engineer or scientist needs to know. It presents six powerful tools for analysis, research, and problem-solving in the real world: dimensional analysis, limiting cases, symmetry, scaling, making order of magnitude estimates, and the method of successive approximations.
The book does not focus on formulaic manipulations of equations, but emphasizes analysis and explores connections between the equations and the application. Each chapter introduces a set of ideas and techniques and then shows how these techniques apply to a series of problems. (Knowledge of algebra and trigonometry, but not calculus, is required.) The final two chapters tie all six techniques together and apply them to two real-world problems: computing the probability of a rare, catastrophic event, and tracking a satellite with a GPS receiver. Readers will learn how to analyze, dissect, and gain insight into the results by using all the techniques presented in earlier chapters—and discover how analysis tools work on problems not concocted for a textbook. The appendix provides solutions to many of the problems found throughout the book.
Alexandr Draganov was born and raised in Kyiv, Ukraine; in light of the current war in Ukraine he will donate 100% of his royalties for the first year to support medical and humanitarian efforts there.
Author(s): Alexandr Draganov
Edition: 1
Publisher: The MIT Press
Year: 2022
Language: English
Commentary: Publisher's PDF
Pages: 306
City: Cambridge, MA
Tags: Problem Solving; Mathematics
Contents
List of Figures
List of Tables
Preface
How to Read This Book
1. Units
1.1 Using Dimensional Analysis to Solve Problems
1.2 The Two Hikers Problem
1.3 The Circle and Line Problem
1.4 Satellite Coverage
1.5 The Cubic Formula
1.6 Summary
Exercises
2. Limiting Cases
2.1 The Product of Two Linear Expressions
2.2 The Two Hikers Problem
2.3 The Riverboat Problem
2.4 The Quadratic Equation
2.5 The Intersections between a Circle and a Straight Line
2.6 The Sum of Two Ratios
2.7 The Sum of Two Scaled Ratios
2.8 The Sum or Difference of Two Radicals
2.9 A Circle Inscribed in a Right Triangle
2.10 Draining a Pool
2.11 The Sum of an Unknown and Its Reciprocal
2.12 Designing Satellite Coverage
2.13 Two Circles Inscribed in an Angle
2.14 The Intersections between a Circle and a Parabola
2.15 Linear Regression
2.16 Summary
Exercises
3. Symmetry
3.1 Symmetry in Mathematical Problems
3.2 The Product of Two Linear Expressions
3.3 The Intersections between a Circle and a Straight Line
3.4 A Circle Inscribed in a Right Triangle
3.5 Blending Syrups
3.6 Draining a Pool
3.7 The Sum of an Unknown and Its Reciprocal
3.8 Designing Satellite Coverage
3.9 Two Circles Inscribed in an Angle
3.10 The Sum or Difference of Two Radicals
3.11 Symmetric Polynomials
3.12 Symmetry in the Quadratic Equation
3.13 Linear Regression
3.14 Summary
Exercises
4. Scaling
4.1 Allometric Scaling
4.2 The Hierarchy of Scaling Behaviors
4.3 Scaling and Polynomial Long Division
4.4 The Pythagorean Theorem
4.5 Olbers’s Paradox
4.6 A Rope Wrapped around a Pole
4.7 Linear Regression
4.8 Summary
Exercises
5. Order of Magnitude Estimates
5.1 How Good Should an Estimate Be?
5.2 How to Make Order of Magnitude Estimates
5.3 Mortgage Payments
5.4 Designing a Parachute
5.5 Accuracy of a Pendulum Clock
5.6 Sizing the Power for a Car Engine
5.7 Summary
Exercises
6. Successive Approximations
6.1 Achilles and the Tortoise
6.2 How MSA Works
6.3 When It Works and When It Doesn’t
6.4 The Product of Two Linear Expressions
6.5 The Quadratic Equation
6.6 Archimedes’s Spiral
6.7 Designing Satellite Coverage
6.8 The Intersections between a Circle and a Parabola
6.9 Summary
Exercises
7. Tying It All Together: The Probability of Catastrophic Events
7.1 Helpful Concepts from Probability Theory
7.2 Generalized Pareto Distribution
7.3 Units
7.4 Limiting Cases
7.5 Symmetry
7.6 Scaling
7.7 Order of Magnitude Estimates
7.8 Successive Approximations
7.9 Summary
8. Tying It All Together: Tracking a GPS Satellite
8.1 Problem Setup
8.2 Units
8.3 Limiting Cases
8.4 Symmetry and Invariance
8.5 Scaling
8.6 Order of Magnitude Estimates
8.7 Successive Approximations
8.8 Summary
A. Problems and Solutions
A.1 Two Hikers on a Trail
A.2 A Riverboat
A.3 The Intersections between a Circle and a Straight Line
A.4 The Intersections between a Circle and an Ellipse
A.5 The Intersections between a Circle and a Hyperbola
A.6 The Intersections between a Circle and a Parabola
A.7 The Product of Two Linear Expressions
A.8 The Sum of an Unknown and Its Reciprocal
A.9 The Difference of an Unknown and Its Reciprocal
A.10 The Sum of Two Ratios
A.11 The Sum of Two Scaled Ratios
A.12 The Difference of Two Ratios
A.13 The Sum of Trigonometric Functions, 1st Version
A.14 The Sum of Trigonometric Functions, 2nd Version
A.15 The Ratio of Cosines
A.16 Blending Two Syrups
A.17 Blending Three Syrups
A.18 Draining a Pool Using Two Pumps
A.19 Draining a Pool Using Three Pumps
A.20 The Sum of Two Radicals
A.21 The Difference of Two Radicals
A.22 The Sum of Two Rational Functions
A.23 The Difference of Two Rational Functions
A.24 Designing Satellite Coverage
A.25 Detecting a Vessel by Two Radars
A.26 Two Circles Inscribed in an Angle
A.27 A Circle Inscribed in a Right Triangle
A.28 A Rectangle Inscribed in a Right Triangle
A.29 The Cubic Formula
A.30 A Spherical Cap
A.31 Mortgage Payments
A.32 The Kalman Filter
A.33 Linear Regression
Further Reading
Bibliography
Index
Index of Problems
