(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 16147, 407]*) (*NotebookOutlinePosition[ 46375, 1419]*) (* CellTagsIndexPosition[ 46281, 1412]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Preface", "Title"], Cell[TextData[{ "To the student:\n\t", StyleBox["Up to this point in your career you have been asked to use \ mathematics to solve rather elementary problems in the physical sciences. \ However, when you graduate and become a working scientist or engineer you \ will often be confronted with complex real-world problems. Understanding the \ material in this book is a first step toward developing the mathematical \ tools that you will need to solve such problems. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[StyleBox["\tMuch of the work detailed in the following chapters \ requires standard pencil-and-paper (i.e. analytical) methods. These methods \ include solution techniques for the partial differential equations of \ mathematical physics such as Poisson's equation, the wave equation, and Schr\ \[ODoubleDot]dinger's equation, Fourier series and transforms, and elementary \ probability theory and statistical methods. These methods are taught from the \ standpoint of a working scientist, not a mathematician. This means that in \ many cases, important theorems will be stated, not proved (although the ideas \ behind the proofs will usually be discussed). Physical intuition will be \ called upon more often than mathematical rigor. ", FontSlant->"Plain"]], "Text", FontSlant->"Italic"], Cell[TextData[StyleBox["\tMastery of analytical techniques has always been \ and probably always will be of fundamental importance to a student's \ scientific education. However, of increasing importance in today's world are \ numerical methods. The numerical methods taught in this book will allow you \ to solve problems that cannot be solved analytically, and will also allow you \ to inspect the solutions to your problems using plots, animations, and even \ sounds, gaining intuition that is sometimes difficult to extract from dry \ algebra. ", FontSlant->"Plain"]], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tIn an attempt to present these numerical methods in the most \ straightforward manner possible, this book employs the software package ", FontSlant->"Plain"], "Mathematica", StyleBox[". There are many other computational environments that we could \ have used instead - for example, software packages such as ", FontSlant->"Plain"], "Matlab", StyleBox[" or ", FontSlant->"Plain"], "Maple", StyleBox[" have similar graphical and numerical capabilities to ", FontSlant->"Plain"], "Mathematica", StyleBox[". Once the principles of one such package are learned, it is \ relatively easy to master the other packages. I chose ", FontSlant->"Plain"], "Mathematica", StyleBox[" for this book because, in my opinion, it is the most flexible \ and sophisticated of such packages. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tAnother approach to learning numerical methods might be to \ write your own programs from scratch, using a language such as C or Fortran. \ This is an excellent way to learn the elements of numerical analysis, and \ eventually in your scientific careers you will probably be required to \ program in one or another of these languages. However, ", FontSlant->"Plain"], "Mathematica", StyleBox[" provides us with a computational environment where it is much \ easier to quickly learn the ", FontSlant->"Plain"], "ideas", StyleBox[" behind the various numerical methods, without the additional \ baggage of learning an operating system, mathematical and graphical \ libraries, or the complexities of the computer language itself. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tAn important feature of ", FontSlant->"Plain"], "Mathematica", StyleBox[" is its ability to perform ", FontSlant->"Plain"], "analytical", StyleBox[" calculations, such as the analytical solution of linear and \ nonlinear equations, integrals and derivatives, and Fourier transforms. You \ will find that these features can help free you from the tedium of performing \ complicated algebra by hand, just as your calculator has freed you from \ having to do long division. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tHowever, as with everything else in life, using ", FontSlant->"Plain"], "Mathematica ", StyleBox["presents us with certain trade-offs. For instance, in part \ because it has been developed to provide a straightforward interface to the \ user, ", FontSlant->"Plain"], "Mathematica", StyleBox[" is not suited for truly large-scale computations such as large \ molecular dynamics simulations with 1000 particles or more, or inversions of \ 100,000 by 100,000 matrices, for example. Such applications require a \ stripped-down precompiled code, running on a mainframe computer. \ Nevertheless, for the sort of introductory numerical problems covered in this \ book, the speed of ", FontSlant->"Plain"], "Mathematica", StyleBox[" on a PC platform is more than sufficient. Once these numerical \ techniques have been learned using ", FontSlant->"Plain"], "Mathematica", StyleBox[", it should be relatively easy to transfer your new skills to a \ mainframe computing environment.", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tI should note here that this limitation does not affect the \ usefulness of ", FontSlant->"Plain"], "Mathematica ", StyleBox[" in the solution of the sort of small to intermediate scale \ problems that working scientists often confront from day to day. In my own \ experience, hardly a day goes by when I do not fire up ", FontSlant->"Plain"], "Mathematica", StyleBox[" to evaluate an integral or plot a function. For more than a \ decade now I have found this program to be truly useful, and I hope and \ expect that you will as well. (No, I am not receiving any kickbacks from \ Stephen Wolfram!) ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tThere is another limitation to ", FontSlant->"Plain"], "Mathematica", StyleBox[". You will find that although ", FontSlant->"Plain"], "Mathematica", StyleBox[" knows a lot of tricks, it is still a dumb program, in the sense \ that it requires precise input from the user. A missing bracket or semicolon \ often will result in long paroxysms of error statements, and less often will \ result in a dangerous lack of error messages and a subsequent incorrect \ answer. It is still true for this (or for any other software) package, that \ garbage in = garbage out. Science fiction movies involving intelligent \ computers aside, this aphorism will probably hold for the foreseeable future. \ This means that, at least at first, you will spend a good fraction of your \ time cursing the computer screen. My advice is to get used to it -- this is \ a process that you will go through over and over again as you use computers \ in your career. I guarantee that you will find it very satisfying when, after \ a long debugging session, you finally get the output you wanted. Eventually, \ with practice, you will become ", FontSlant->"Plain"], "Mathematica masters", StyleBox[".", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\tI developed this book from course notes for two junior-level \ classes in mathematical methods that I have taught at UCSD for several years. \ The book is oriented toward students in the physical sciences and in \ engineering, at either the advanced undergraduate (junior or senior) or \ graduate level. It assumes an understanding of introductory calculus and \ ordinary differential equations. Chapters 1-8 also require a basic working \ knowledge of ", FontSlant->"Plain"], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[". Chapter 9, included only in electronic form on the CD that \ accompanies this book, presents an introduction to the software's \ capabilities. I recommend that ", FontSlant->"Plain"], StyleBox["Mathematica", FontSlant->"Italic"], StyleBox[" novices read this chapter first, and do the exercises.", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell[TextData[StyleBox["\t Some of the material in the book is rather \ advanced, and will be of more interest to graduate students or professionals. \ This material can obviously be skipped when the book is used in an \ undergraduate course. In order to reduce printing costs, four advanced topics \ appear only in the electronic chapters on the CD: Section 5.3 on wave action; \ Section 6.3 on numerically-determined eigenmodes; Section 7.3 on the \ particle-in-cell method; and Section 8.3 on the Rosenbluth-Teller-Metropolis \ Monte Carlo method. These extra sections are highlighted in red in the \ electronic version. ", FontSlant->"Plain"]], "Text", FontSlant->"Italic"], Cell[TextData[{ StyleBox["\t Aside from these differences, the text and equations in the \ electronic and printed versions are, ", FontSlant->"Plain"], "in theory", StyleBox[", identical. However, I take sole responsibility for any \ inadvertent discrepancies, as the good people at Wiley were not involved in \ typesetting the electronic textbook. ", FontSlant->"Plain"] }], "Text", FontSlant->"Italic"], Cell["\<\ \tThe electronic version of this book has several features that are \ not available in printed textbooks: \ \>", "Text"], Cell[TextData[{ "\t(1) ", StyleBox["Hyperlinks.", FontWeight->"Bold", FontSlant->"Italic"], " There are hyperlinks in the text that can be used to view material from \ the web. For example, a brief biographical sketch of Sir Isaac Newton can be \ found by clicking ", ButtonBox["here", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.\ html"], None}, ButtonStyle->"Hyperlink"], ". Also, when the text refers to an equation, the equation number itself is \ a hyperlink that will take you to that equation. For example, Eq. ", ButtonBox["(2.1.3)", ButtonData:>{"Chapter2.nb", "(2.1.3)"}, ButtonStyle->"Hyperlink"], " is a general expression for a trigonometric Fourier series. Furthermore, \ all items in the ", ButtonBox["index", ButtonData:>{"Index.nb", None}, ButtonStyle->"Hyperlink"], " and ", ButtonBox["contents", ButtonData:>{"tableofcontents.nb", None}, ButtonStyle->"Hyperlink"], " are linked to the corresponding material in the book. (For these features \ to work properly, all chapters must be located in the same directory on your \ computer.) You can return to the original reference using the ", StyleBox["Go Back", FontFamily->"Courier", FontWeight->"Bold"], " command, located in the main menu under ", StyleBox["Find", FontFamily->"Courier", FontWeight->"Bold"], ". " }], "Text", CellTags->"i:1"], Cell[TextData[{ "\t(2) ", StyleBox["Mathematica Code", FontWeight->"Bold", FontSlant->"Italic"], ". Certain portions of the book are ", StyleBox["Mathematica ", FontSlant->"Italic"], "calculations that you can use to graph functions, solve differential \ equations, etc. These calculations can be modified at the readers pleasure, \ and run ", StyleBox["in situ", FontSlant->"Italic"], ". " }], "Text"], Cell[TextData[{ "\t(3) ", StyleBox["Animations and Interactive 3D Renderings.", FontWeight->"Bold", FontSlant->"Italic"], " Some of the displayed figures are interactive three-dimensional \ renderings of curves or surfaces, which can be viewed from different angles \ using the mouse. An example is Fig. ", ButtonBox["1.13", ButtonData:>{"Chapter1.nb", "Fig. 1.13"}, ButtonStyle->"Hyperlink"], ", the strange attractor for the Lorenz system. Also, some of the other \ figures are actually animations. Creating animations and interactive 3D plots \ is covered in Sections ", ButtonBox["9.6.7", ButtonData:>{"Chapter9.nb", "c:33"}, ButtonStyle->"Hyperlink"], " and ", ButtonBox["9.6.6", ButtonData:>{"Chapter9.nb", "c:32"}, ButtonStyle->"Hyperlink"], ", respectively." }], "Text"], Cell[TextData[{ "\t(4) ", StyleBox["Searchable text", FontWeight->"Bold", FontSlant->"Italic"], ". Using the commands in the ", StyleBox["Find", FontFamily->"Courier", FontWeight->"Bold"], " menu, you can search through the text for words or phrases. " }], "Text"], Cell[TextData[{ "\t Equations or text may sometimes be typeset in a font that is too small \ to be read easily at the current magnification. You can increase (or \ decrease) the magnification of the notebook under the ", StyleBox["Format", FontFamily->"Courier", FontWeight->"Bold"], " entry of the main menu (choose ", StyleBox["Magnification", FontFamily->"Courier", FontWeight->"Bold"], "), or by choosing a magnification setting from the small window at the \ bottom left side of the notebook. " }], "Text", CellTags->"i:2"], Cell[TextData[{ "\tA number of individuals made important contributions to this project: \ Professor Tom O'Neil, who originally suggested that the electronic version \ should be written in ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook format; Professor C. Fred Driscoll, who invented some of the \ problems on sound and hearing; Jo Ann Christina, who helped with the proof \ reading and indexing; and Dr. Jay Albert, who actually waded through the \ entire manuscript, found many errors and typos, and helped clear up fuzzy \ thinking in several places. Finally, to the many students who have passed \ through my computational physics classes here at UCSD: You have been \ subjected to two experiments -- a ", StyleBox["Mathematica", FontSlant->"Italic"], "-based course that combines analytical and computational methods; and a \ book that allows the reader to interactively explore variations in the \ examples. Although you were beset by many vicissitudes (crashing computers, \ balky code, debugging sessions stretching into the wee hours) your \ interest, energy, and good humor were unflagging (for the most part!) and a \ constant source of inspiration. 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Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{10, 4}, {0, 8}}, CellHorizontalScrolling->True, Hyphenation->False, LanguageCategory->"Formula", FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{2, Inherited}, {Inherited, Inherited}}, FontSize->9.5] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperlink Styles", "Subsection"], Cell["\<\ The cells below define styles useful for making hypertext \ ButtonBoxes. 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