Mathematical Tools for Real-World Applications: A Gentle Introduction for Students and Practitioners
306
Mathematical Tools for Real-World Applications: A Gentle Introduction for Students and Practitioners
306Paperback
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Overview
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.
Product Details
| ISBN-13: | 9780262543965 |
|---|---|
| Publisher: | MIT Press |
| Publication date: | 08/02/2022 |
| Pages: | 306 |
| Sales rank: | 1,007,954 |
| Product dimensions: | 7.00(w) x 9.00(h) x 0.68(d) |
About the Author
Table of Contents
List of Figures xv
List of Tables xix
Preface xxi
How to Read This Book xxv
1 Units 1
1.1 Using Dimensional Analysis to Solve Problems 3
1.2 The Two Hikers Problem 5
1.3 The Circle and Line Problem 5
1.4 Satellite Coverage 7
1.5 The Cubic Formula 8
1.6 Summary 10
Exercises 11
2 Limiting Cases 19
2.1 The Product of Two Linear Expressions 21
2.2 The Two Hikers Problem 22
2.3 The Riverboat Problem 23
2.4 The Quadratic Equation 25
2.5 The Intersections between a Circle and a Straight Line 27
2.6 The Sum of Two Ratios 30
2.7 The Sum of Two Scaled Ratios 34
2.8 The Sura or Difference of Two Radicals 36
2.9 A Circle Inscribed in a Right Triangle 38
2.10 Draining a Pool 40
2.11 The Sum of an Unknown and Its Reciprocal 44
2.12 Designing Satellite Coverage 46
2.13 Two Circles Inscribed in an Angle 52
2.14 The Intersections between a Circle and a Parabola 53
2.15 Linear Regression 55
2.16 Summary 58
Exercises 59
3 Symmetry 73
3.1 Symmetry in Mathematical Problems 75
3.2 The Product of Two Linear Expressions 79
3.3 The Intersections between a Circle and a Straight Line 80
3.4 A Circle Inscribed in a Right Triangle 85
3.5 Blending Syrups 86
3.6 Draining a Pool 88
3.7 The Sum of an Unknown and Its Reciprocal 90
3.8 Designing Satellite Coverage 91
3.9 Two Circles Inscribed in an Angle 93
3.10 The Sum or Difference of Two Radicals 94
3.11 Symmetric Polynomials 97
3.12 Symmetry in the Quadratic Equation 99
3.13 Linear Regression 102
3.14 Summary 105
Exercises 107
4 Scaling 121
4.1 Allometric Scaling 123
4.2 The Hierarchy of Scaling Behaviors 124
4.3 Scaling and Polynomial Long Division 128
4.4 The Pythagorean Theorem 129
4.5 Olbers's Paradox 130
4.6 A Rope Wrapped around a Pole 132
4.7 Linear Regression 135
4.8 Summary 137
Exercises 139
5 Order of Magnitude Estimates 151
5.1 How Good Should an Estimate Be? 153
5.2 How to Make Order of Magnitude Estimates 154
5.3 Mortgage Payments 156
5.4 Designing a Parachute 158
5.5 Accuracy of a Pendulum Clock 160
5.6 Sizing the Power for a Car Engine 162
5.7 Summary 164
Exercises 166
6 Successive Approximations 175
6.1 Achilles and the Tortoise 177
6.2 How MSA Works 181
6.3 When It Works and When It Doesn't 183
6.4 The Product of Two Linear Expressions 185
6.5 The Quadratic Equation 188
6.6 Archimedes's Spiral 191
6.7 Designing Satellite Coverage 193
6.8 The Intersections between a Circle and a Parabola 197
6.9 Summary 204
Exercises 206
7 Tying It All Together: The Probability of Catastrophic Events 215
7.1 Helpful Concepts from Probability Theory 215
7.2 Generalized Pareto Distribution 218
7.3 Units 220
7.4 Limiting Cases 220
7.5 Symmetry 221
7.6 Scaling 222
7.7 Order of Magnitude Estimates 223
7.8 Successive Approximations 226
7.9 Summary 227
8 Tying It All Together: Tracking a GPS Satellite 229
8.1 Problem Setup 229
8.2 Units 232
8.3 Limiting Cases 232
8.4 Symmetry and Invariance 234
8.5 Scaling 235
8.6 Order of Magnitude Estimates 236
8.7 Successive Approximations 236
8.8 Summary 239
A Problems and Solutions 241
A.1 Two Hikers on a Trail 241
A.2 A Riverboat 242
A.3 The Intersections between a Circle and a Straight Line 243
A.4 The Intersections between a Circle and an Ellipse 244
A.5 The Intersections between a Circle and a Hyperbola 246
A.6 The Intersections between a Circle and a Parabola 247
A.7 The Product of Two Linear Expressions 248
A.8 The Sum of an Unknown and Its Reciprocal 249
A.9 The Difference of an Unknown and Its Reciprocal 249
A.10 The Sum of Two Ratios 250
A.11 The Sum of Two Scaled Ratios 250
A.12 The Difference of Two Ratios 251
A.13 The Sum of Trigonometric Functions, 1st Version 252
A.14 The Sum of Trigonometric Functions, 2nd Version 253
A.15 The Ratio of Cosines 253
A.16 Blending Two Syrups 254
A.17 Blending Three Syrups 255
A.18 Draining a Pool Using Two Pumps 255
A.19 Draining a Pool Using Three Pumps 255
A.20 The Sum of Two Radicals 256
A.21 The Difference of Two Radicals 256
A.22 The Sum of Two Rational Functions 257
A.23 The Difference of Two Rational Functions 258
A.24 Designing Satellite Coverage 259
A.25 Detecting a Vessel by Two Radars 260
A.26 Two Circles Inscribed in an Angle 261
A.27 A Circle Inscribed in a Right Triangle 262
A.28 A Rectangle Inscribed in a Right Triangle 263
A.29 The Cubic Formula 264
A.30 A Spherical Cap 265
A.31 Mortgage Payments 265
A.32 The Kalman Filter 267
A.33 Linear Regression 268
Further Reading 271
Bibliography 273
Index 275
Index of Problems 279