Mathematical Tools for Real-World Applications. A Gentle Introduction for Students and Practitioners

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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