Mathematics
MATH 414 GMU Frame Indifferent Functions & Free Energy Function Exercises

math 414

George Mason University

MATH

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Need help with my Mathematics question - I’m studying for my class.

Solve 6.19 and explain why 6.71 are frame-indifferent. Please show me the how to solve and don't use any coding to solve.

MATH 414 GMU Frame Indifferent Functions & Free Energy Function Exercises
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MATH 414 GMU Frame Indifferent Functions & Free Energy Function Exercises
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Texts in Applied Mathematics 56 Mark H. Holmes Introduction to the Foundations of Applied Mathematics Second Edition Texts in Applied Mathematics Volume 56 Editors-in-chief A. Bloch, University of Michigan, Ann Arbor, USA C. L. Epstein, University of Pennsylvania, Philadelphia, USA A. Goriely, University of Oxford, Oxford, UK L. Greengard, New York University, New York, USA Series Editors J. Bell, Lawrence Berkeley National Lab, Berkeley, USA R. Kohn, New York University, New York, USA P. Newton, University of Southern California, Los Angeles, USA C. Peskin, New York University, New York, USA R. Pego, Carnegie Mellon University, Pittsburgh, USA L. Ryzhik, Stanford University, Stanford, USA A. Singer, Princeton University, Princeton, USA A. Stevens, Universität Münster, Münster, Germany A. Stuart, University of Warwick, Coventry, UK T. Witelski, Duke University, Durham, USA S. Wright, University of Wisconsin, Madison, USA The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The aim of this series is to provide such textbooks in applied mathematics for the student scientist. Books should be well illustrated and have clear exposition and sound pedagogy. Large number of examples and exercises at varying levels are recommended. TAM publishes textbooks suitable for advanced undergraduate and beginning graduate courses, and complements the Applied Mathematical Sciences (AMS) series, which focuses on advanced textbooks and research-level monographs. More information about this series at http://www.springer.com/series/1214 Mark H. Holmes Introduction to the Foundations of Applied Mathematics Second Edition 123 Mark H. Holmes Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY, USA ISSN 0939-2475 ISSN 2196-9949 (electronic) Texts in Applied Mathematics ISBN 978-3-030-24260-2 ISBN 978-3-030-24261-9 (eBook) https://doi.org/10.1007/978-3-030-24261-9 Mathematics Subject Classification (2010): Primary: 76Axx, 76Bxx, 76Dxx 74Bxx, 74Dxx, 74Hxx, 74Jxx 74Axx, 34D05, 34E05, 34E10, 34E13, 35C06, 35C07, 35F50, 60J60, 60J65 1st edition: © Springer Science+Business Media, LLC 2009 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Colette, Matthew, and Marianna Preface to the Second Edition The principal changes are directed to improving the presentation of the material. This includes rewriting and reorganizing certain sections, adding new examples, and reorganizing and embellishing the exercises. The added examples range from the relatively minor to the more extensive, such as the added material for water waves. This edition also provided an opportunity to update the references. Another reason for this edition concerns the changes in publishing over the last decade. The improvements in digital books and the interest in students for having an ebook version were motivating reasons for working on a new edition. Finally, the one or two typos in the first edition were also corrected, and thanks go to Ash Kapila, Emily Fagerstrom, Jan Medlock, and Kevin DelBene for finding them. Troy, NY, USA March 2019 Mark H. Holmes vii Preface to the First Edition FOAM. This acronym has been used for over 50 years at Rensselaer to designate an upper-division course entitled, Foundations of Applied Mathematics. This course was started by George Handelman in 1956, when he came to Rensselaer from the Carnegie Institute of Technology. His objective was to closely integrate mathematical and physical reasoning, and in the process enable students to obtain a qualitative understanding of the world we live in. FOAM was soon taken over by a young faculty member, Lee Segel. About this time a similar course, Introduction to Applied Mathematics, was introduced by Chia-Ch’iao Lin at the Massachusetts Institute of Technology. Together Lin and Segel, with help from Handelman, produced one of the landmark textbooks in applied mathematics, Mathematics Applied to Deterministic Problems in the Natural Sciences. This was originally published in 1974, and republished in 1988 by the Society for Industrial and Applied Mathematics, in their Classics Series. This textbook comes from the author teaching FOAM over the last few years. In this sense, it is an updated version of the Lin and Segel textbook. The objective is definitely the same, which is the construction, analysis, and interpretation of mathematical models to help us understand the world we live in. However, there are some significant differences. Lin and Segel, like many recent modeling books, is based on a case study format. This means that the mathematical ideas are introduced in the context of a particular application. There are certainly good reasons why this is done, and one is the immediate relevance of the mathematics. There are also disadvantages, and one pointed out by Lin and Segel is the fragmentary nature of the development. However, there is another, more important reason for not following a case studies approach. Science evolves, and this means that the problems of current interest continually change. What does not change as quickly is the approach used to derive the relevant mathematical models, and the methods used to analyze the models. Consequently, this book is written in such a way as to establish the mathematical ideas underlying model development independently of a specific application. This does not mean applications are not considered, they are, and connections with experiment are a staple of this book. ix x Preface to the First Edition The first two chapters establish some of the basic mathematical tools that are needed. The model development starts in Chap. 3, with the study of kinetics. The goal of this chapter is to understand how to model interacting populations. This does not account for the spatial motion of the populations, and this is the objective of Chaps. 4 and 5. What remains is to account for the forces in the system, and this is done in Chap. 6. The last three chapters concern the application to specific problems and the generalization of the material to more geometrically realistic systems. The book, as well as the individual chapters, is written in such a way that the material becomes more sophisticated as you progress. This provides some flexibility in how the book is used, allowing consideration for the breadth and depth of the material covered. The principal objective of this book is the derivation and analysis of mathematical models. Consequently, after deriving a model, it is necessary to have a way to solve the resulting mathematical problem. A few of the methods developed here are standard topics in upper-division applied math courses, and in this sense there is some overlap with the material covered in those courses. Examples are the Fourier and Laplace transforms, and the method of characteristics. On the other hand, other methods that are used here are not standard, and this includes perturbation approximations and similarity solutions. There are also unique methods, not found in traditional textbooks, that rely on both the mathematical and physical characteristics of the problem. The prerequisite for this text is a lower-division course in differential equations. The implication is that you have also taken two or three semesters of calculus, which includes some component of matrix algebra. The one topic from calculus that is absolutely essential is Taylor’s theorem, and for this reason a short summary is included in the appendix. Some of the more sophisticated results from calculus, related to multidimensional integral theorems, are not needed until Chap. 8. To learn mathematics you must work out problems, and for this reason the exercises in the text are important. They vary in their difficulty, and cover most of the topics in the chapter. Some of the answers are available, and can be found at www.holmes.rpi.edu. This web page also contains a typos list. I would like to express my gratitude to the many students who have taken my FOAM course at Rensselaer. They helped me immeasurably in understanding the subject, and provided much-needed encouragement to write this book. It is also a pleasure to acknowledge the suggestions of John Ringland, and his students, who read an early version of the manuscript. Troy, NY, USA March 2009 Mark H. Holmes Contents 1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Maximum Height of a Projectile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Drag on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Toppling Dominoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Pattern Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Similarity Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Similarity Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nondimensionalization and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Weakly Nonlinear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 5 6 14 16 16 20 22 23 25 27 27 31 34 34 2 49 49 53 53 56 60 64 66 69 76 77 77 79 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Regular Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 How to Find a Regular Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Given a Specific Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Given an Algebraic or Transcendental Equation . . . . . . . . . . . 2.2.3 Given an Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Scales and Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Introduction to Singular Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Introduction to Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Examples Involving Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Example 1: Layer at Left End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Example 2: Layer at Right End. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi xii Contents 2.6.3 Example 3: Boundary Layer at Both Ends. . . . . . . . . . . . . . . . . . Multiple Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Regular Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Multiple Scales Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 84 85 88 92 3 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Predator-Prey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Epidemic Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Modeling Using the Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Disease Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Reverse Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 General Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Steady States and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Reaction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Solving the Michaelis-Menten Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Quasi-Steady-State Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Perturbation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Modeling with the QSSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 103 103 104 104 105 107 109 111 111 113 114 115 116 118 119 123 123 124 126 134 134 135 137 143 145 148 151 151 4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Random Walks and Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Calculating w(m, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Large n Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Continuous Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 What Does D Signify?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Solutions of the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Point Source Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 A Step Function Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 165 167 170 172 174 175 178 178 183 2.7 Contents xiii 4.5 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Transformation of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Solving the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Continuum Formulation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Balance Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Fick’s Law of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Random Walks and Diffusion in Higher Dimensions . . . . . . . . . . . . . . . . 4.7.1 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Properties of the Random Forcing . . . . . . . . . ...
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