Problem 3. Two Math Problems: Algebra and Differential Equations Save the Day

Chapters in this book

1. Frontmatter i
2. Contents ix
3. Preface xiii
4. PART I. THE PROBLEMS
5. Problem 1. A Military Question: Catapult Warfare 3
6. Problem 2. A Seemingly Impossible Question: A Shocking Snow Conundrum 4
7. Problem 3. Two Math Problems: Algebra and Differential Equations Save the Day 6
8. Problem 4. An Escape Problem: Dodge the Truck 8
9. Problem 5. The Catapult Again: Where Dead Cows Can’t Go! 9
10. Problem 6. Another Math Problem: This One Requires Calculus 10
11. Problem 7. If Theory Fails: Monte Carlo Simulation 11
12. Problem 8. Monte Carlo and Theory: The Drunkard’s One-Dimensional Random Walk 17
13. Problem 9. More Monte Carlo: A Two-Dimensional Random Walk in Paris 19
14. Problem 10. Flying with (and against) the Wind: Math for the Modern Traveler 21
15. Problem 11. A Combinatorial Problem with Physics Implications: Particles, Energy Levels, and Pauli Exclusion 22
16. Problem 12. Mathematical Analysis: By Physical Reasoning 29
17. Problem 13. When an Integral Blows Up: Can a Physical Quantity Really Be Infinite? 36
18. Problem 14. Is This Easier Than Falling Off a Log? Well, Maybe Not 39
19. Problem 15. When the Computer Fails: When Every Day Is a Birthday 47
20. Problem 16. When Intuition Fails: Sometimes What Feels Right, Just Isn’t 55
21. Problem 17. Computer Simulation of the Physics of NASTYGLASS: Is This Serious? . . . Maybe 60
22. Problem 18. The Falling-Raindrop, Variable-Mass Problem: Falling Slower Than Gravity 72
23. Problem 19. Beyond the Quadratic: A Cubic Equation and Discontinuous Behavior in a Physical System 81
24. Problem 20. Another Cubic Equation: This One Inspired by Jules Verne 93
25. Problem 21. Beyond the Cubic: Quartic Equations, Crossed Ladders, Undersea Rocket Launches, and Quintic Equations 103
26. Problem 22. Escaping an Atomic Explosion: Why the Enola Gay Survived 114
27. Problem 23. “Impossible’’ Math Made Easy: Gauss’s Congruence Arithmetic 122
28. Problem 24. Wizard Math: Fourier’s Series, Dirac’s Impulse, and Euler’s Zeta Function 126
29. Problem 25. The Euclidean Algorithm: The Zeta Function and Computer Science 137
30. Problem 26. One Last Quadratic: Heaviside Locates an Underwater Fish Bite! 147
31. PART II. THE SOLUTIONS
32. Einleitung 157
33. Appendix 1. MATLAB, Primes, Irrationals, and Continued Fractions 225
34. Appendix 2. A Derivation of Brouncker’s Continued Fraction for 4/π 247
35. Appendix 3. Landen’s Calculus Solution to the Depressed Cubic Equation 251
36. Appendix 4. Solution to Lord Rayleigh’s Rotating-Ring Problem of 1876 261
37. Acknowledgments 270
38. Index 273
39. Also by Paul J. Nahin 281