Six Themes On Variation (Student Mathematical Library, V. 26)

The purpose of this series is to introduce undergraduates to the varied areas of current research in mathematics; and this text, of all in the series, most clearly achieves that goal. There are six essays here written by different authors. The best is the introduction to Morse theory by Forman. This author is gifted with the ability to know when to include and exclude detail so as to not lose the developing intuitive thrust. What this book recognizes, perhaps not explicitly, is that there are different learning styles amoung undergraduates and this text has something to offer in this regard. For example, many undergraduates need to see numbers computed and be connected to a physical experiment in order to grasp and motivate the math. The essay on modeling traffic flow and the one on the inquiry into plucked strings will satisfy those readers. The more legalistic undergrads will appreciate the minimal surfaces essays. Since the entire book seems to have been written by the math faculty at Rice university, it seems that whoever put that department together did a great job of eclectic casting in assembling such a talented group of teachers. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 8 List of Contributors......Page 12 Calculus of Variations: What Does "Variations" Mean?......Page 14 II. The domain is n-dimensional......Page 15 IV. The Euler-Lagrange scenario......Page 16 A. Minimal surfaces. G......Page 19 B. Geodesics.......Page 21 C. Isoperimetric problem. T......Page 22 VI. Important disclaimer......Page 24 How Many Equilibria Are There? An Introduction to Morse Theory......Page 26 I. Dynamical systems......Page 27 II. Topology......Page 35 III. Morse Theory......Page 38 V. The idea of the proof......Page 41 VI. What does it really mean to build a shape from cells?......Page 43 VII. What now?......Page 46 1. Introduction......Page 50 2. Acquiring the data......Page 51 3. A mathematical model......Page 56 4. Solving the wave equation......Page 60 5. The damped wave equation......Page 62 6. Discerning the presence of additionaldamping......Page 66 7. Concluding remarks......Page 68 Bibliography......Page 69 1. The news.......Page 72 3. Proof of the Planar Double Bubble Theorem.......Page 73 4. The standard double bubble. For......Page 77 6. Symmetry Theorem. A......Page 78 7. Monotonicity [12, THM. 3.2].......Page 79 9. Hutchings Structure Theorem [12, THM. 5.1].......Page 80 10. Hutchings component bound. C......Page 81 12. Theorem ([7], [9]). F......Page 83 13. Theorem [11].......Page 85 14. Higher dimensions.......Page 87 Bibliography......Page 89 Minimal Surfaces, Flat Cone Spheres and Moduli Spaces of Staircases......Page 92 1. Minimal surfaces......Page 93 2. Some history......Page 97 3.1. Direct method. W......Page 99 3.2. Intrinsic vs. extrinsic geometry. T......Page 100 3.3. Riemann surfaces. O......Page 101 3.4. Some real differential geometry. W......Page 102 3.5. Digression on complex analysis. It......Page 104 3.6. From real differential geometry to complex analysis.......Page 107 3.7. The Weierstrass representation.......Page 112 3.8. Examples. A......Page 115 3.9. Some restrictions. A......Page 116 4.1. The problem and some history. W......Page 119 4.2. The proof.......Page 123 References......Page 137 1. Introduction: A continuum model for traffic flow......Page 140 2. Some conservation law theory......Page 145 3. An application of the model: The timing of traffic lights......Page 155 4. Extensions and other models......Page 160 Bibliography......Page 164 Back Cover......Page 170 The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics courses or curricula. This small volume should nevertheless give the undergraduate reader a sense of its great character and importance. Interesting functionals, such as area or energy, often give rise to problems whose most natural solution occurs by differentiating a one-parameter family of variations of some function. The critical points of the functional are related to the solutions of the associated Euler-Lagrange equation. These differential equations are at the heart of the calculus of variations. Some of the topics addressed here are Morse theory, wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow. All are readily accessible to advanced undergraduates. This book is derived from a workshop that was sponsored by Rice University. The Calculus Of Variations Is A Subject With A Rich History And With Origins In The Minimization Problems Of Calculus. Although It Is Now At The Core Of Many Modern Mathematical Fields, It Does Not Have A Well-defined Place In Most Undergraduate Mathematics Courses Or Curricula. This Small Volume Should Nevertheless Give The Undergraduate Reader A Sense Of Its Great Character And Importance.--jacket. Calculus Of Variations : What Does Variations Mean? / Frank Jones -- How Many Equilibria Are There? An Introduction To Morse Theory / Robin Forman -- Aye, There's The Rub : An Inquiry Into Why A Plucked String Comes To Rest / Steven J. Cox -- Proof Of The Double Bubble Conjecture / Frank Morgan -- Minimal Surfaces, Flat Cone Spheres And Moduli Spaces Of Staircases / Michael Wolf -- Hold That Light! Modeling Of Traffic Flow By Differential Equations / Barbara Lee Keyfitz. Steven J. Cox ... [et Al.] ; Robert M. Hardt, Editor. Includes Bibliographical References. "The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics courses or curricula. This small volume should nevertheless give the undergraduate reader a sense of its great character and importance."--Page 4 de la couverture