5 comments about String Theory, Vol. 1 (Cambridge Monographs on Mathematical Physics).

1. The two volumes introduce many important recent developments in string theory not covered in Green, Schwarz & Witten's Superstring text such as D-branes, dualities, etc. However, I found GSW's treatment of basic materials easier to follow because the authors try to explain things intuitively. Although many physical insight in Polchinski's book is sacrificed, it makes up for them through completeness and mathematical rigor. However, I highly recommend that you read Di Francesco et al's conformal field theory book (read chapters 3-7, 10 and 12-13) to get a better feel for stuff like operator state mapping, OPE's, Virasoro algebra, vertex operators, etc. Of course a good course in QFT, GR and some basic familiarity with SUSY, Rep. theory, & some algebraic topology are probably a prereq although Polchinski claims the book is pretty much self-contained. Lastly, the book suffers a little from numerous typos (atleast the 1st edition) but corrections are updated frequently on a ucsb website address.



2. This is an exceptionally well written book of the highest quality. In many ways, it definitely accomplishes what it sets out to do-give graduate level students and professional physicists an advanced string theory book that can prepare them for research. This book is not one that is going to hold your hand. Nonetheless, it is very well written and has a clear and well organized exposition.
   
   Right now, I am writing a string theory book of my own so have been reading and reviewing a lot of books. You will see that most of my reviews are positive. The reason I mention this is that maybe you can take my reviews a little more seriously since I am simultaneously trying to wade through a number of these books.
   
   The notorious "path integral" is of course, a vital component when learning string theory at this level. What I find basically annoying is that nobody, even Polchinski, seems to have found a CLEAR and SIMPLE way to teach people what a path integral is and how to calculate them. I give Polchinski a B+ for his effort in the appendix, but come on people-can't we do better? I have yet to read a description of what a path integral is without 1) Getting a major headache and 2) being able to sit down and calculate them. I think Polchinski or someone else with an inclination to writing textbooks ought to just focus on that-write a small book on path integrals that finally makes this technique accessible to the majority of the human race.
   
   OK so where are you in your string theory program. Just staring out? This is not the book for you. I like Becker, Becker & Schwarz for "beginnners" (I am assuming you have a significant background in math & physics, but maybe you're not a star grad student at Princeton). I also liked the books by Kaku. Once you've gone through Becker et. al. you can tackle this one.
   
   I think just a basic understanding of quantum field theory is all you need, but the stumbling block is going to be the path integral. If you get path integrals, then this book will be a breeze. If you don't, then you're not going to know what the hell he is talking about.



3. This book succeeds in what seems to be the impossible. It actually presents a clear, up to date, and entertaining version of a field that is still very much in a state of active research and is still, after all these years, on quite uncertain ground. By studying this, the reader who thinks intelligently about the material presented will be able to form his/her own opinions on this still somewhat controversial topic and will be able to converse intelligently with others who have opinions on the topic. I know that for me personally, this text opened up beautiful ideas which, to a large extent, are still unexplored. Before I read this book, my gut feelings about the topic were that it was rather dubious at best, but now that I understand (I think) the basic ideas of the field, I feel quite comfortable in it, indeed almost as if it is completely natural. What I think is one of the best things about this book is that it does not assume the pretense that string theory is on firm ground, that everything is quite certain and that string theory HAS to be the final theory of nature in all its glory. I find this attitude EXTREMELY pretensious and annoying. Instead, it simply covers what we know about string theory, and explains in detail just why it is consistent, and why it offers an explanation for what we see in nature. In short, it leaves just enough room for the imagination of an intelligent reader to philosophize as to the meaning of the theory and as to its ultimate place in nature
   As for practical details, it seems to me that the reader should at the very least have a firm understanding of Quantum Field Theory (at least at the level of Weinberg's first volume, see my review on that modern masterpiece), and to a lesser extent of General Relativity, before even attempting to tackle this. I know that I myself, despite the fact that I have read several texts on QFT, had to reread several sizeable chunks of the book to fully digest it.



4. In short, I think volumes I and II of "String Theory" are the best books on string theory available. Presumably any serious student of string theory will study them both. The writing style is clear, physical considerations are at the forefront, the selection of topics is excellent and the treatment is as up-to-date as any I'm aware of.
   
   Volume I covers the bosonic string. Of course this doesn't provide a realistic model for our universe, but understanding it forms the foundation of the study of more realistic string theories.
   
   The first chapter provides the physical motivation for string theory. A brief description of some current unsolved problems in physics, and how string theory may resolve them, is given. Most notably this includes not only providing a quantum theory of gravity, but also providing a grand unified theory. A brief outline of techniques used throughout the book is given. These are covered in more detail as the book develops and include: the Polyakov action (how to get it from the Nambu-Goto form and why it's more useful), the Polyakov action symmetries, string theory as a two-dimensional quantum field theory, string boundary conditions, the string spectra, supersymmetry (worldsheet and spacetime) and the critical dimension. This is an excellent introduction and nicely sets the stage for the rest of the book.
   
   The next chapter presents conformal field theory. It's also an excellent introduction. In particular covering conformal field theory with anticommuting fields. The Virasoro algebra is also derived. He could have covered these conformal field theory concepts as they came up, but I liked having them in one central location early in the book.
   
   Strings take center stage again in the following chapter as the Polyakov path integral is examined in great detail. Among the results are a calculation of the critical dimension and the recovery of general relativity in the low energy limit of string theory. These are just a couple of the interesting results, there is much more in this chapter.
   
   The following chapters quantize the string, calculate the string spectrum, derive the S-matrix, calculate tree level scattering amplitudes and calculate one-loop amplitudes (higher order amplitudes are covered in the final chapter). One of many things that stand out is his discussion of divergences. He describes the difference between infrared and ultraviolet divergences. After showing ultraviolet divergences are absent in string theory he comments on how the mechanisms that remove them is different for open and closed strings. This is just one example of how physical concepts are kept at the forefront.
   
   The chapter on compactification covers more than just the basics such as (D - 4) dimensions must be compactified and this gives rise to some extra gauge fields. Orbifolds are introduced in this chapter. It also covers T-duality, one of the important (and unexpected) symmetries of string theories. D-branes are also introduced (D-branes are covered in more detail in volume II), obviously this is an important concept in string theory. I was happy to see such important concepts introduced so quickly.
   
   In short, this is a great book. Even with only light coverage of supersymmetry (this is covered in detail in volume II) many interesting and up-to-date topics are presented. Clearly the author put a lot of time into thinking about how to make a difficult subject as approachable as possible. Throughout the book he anticipates questions the reader may have, or maybe should have, and addresses them.



5. Dr. Polchinski may know a lot on string theory but he doesn't know that much on how to write a book. I have been struggling with this book trying to learn string theory and it has been a total failure. You may think it's me but is not. I have studied chapters 1 to 4. I will announce some of its bad features: 1-The notation is awful specially on chapter 2 when he defines the infinitesimal variation of a physical quantity in a very complicated way, all formulas are presented in terms of awful excesively complicated expresions that make you feel sick (and I'm not joking), also on chapter two he defines a way for applying Wicks theorem (eq.(2.2.7)) using exponential operators but I finally gave up and did it my way for calculating expression (2.2.13). 2-Many of the results are not derived and trying to understand what happen from line to line is, besides being a mystery, in my opinion hard to say the less.
   3- On chapter 3 I liked the way he calculates de Faddeev Popov determinant in terms of ghosts and you begin to hope that the book is finally going to start getting better but is not, on page 102 and 'till the end of the chapter (page 118) he starts just throwing a lot of equations that you just can't understand where they came from, specially page 105 where he uses the geodesic distance to higher orders but never explains nor show what this expressions are nor what approximations he is doing, nor where they came from. Then again on page 107 he gives a relation between operators regularized by dimensional regularization and by 'polchinski' regularization, at least the second one is defined but the other is not (on curved space)and he just shows some awful equations that no one knows where they come from. This book has been written for someone who already knows a lot on string theory but it is not for someone who is trying to learn string theory for the first time. All in all try instead the classic book by Green Schwarz and Witten or the one by Theisen and this one use it only as a reference.

2 comments about Chebyshev and Fourier Spectral Methods: Second Revised Edition.

1. Prof. Boyd's book is god-sent,

   I discovered it during a self-embarked journey in the blooming world of spectral methods. Unfortunately,
   these very potent and promising techniques have only recently
   escaped the confines of the field of Applied Mathematics where
   they were first developed. Thus, most existing literature is too
   mathematically oriented and rather opaque to the engineer and
   applied scientist, as it does not offer them the basic
   operational knowledge that they would require. This book is
   one of the first to overcome this chasm. It provides
   a survey of all the necessary fundamentals for the application
   of spectral methods to various disciplines of computational
   engineering but also delves deep into various advanced topics.
   At the same time it provides one with sufficient ammunition
   to explore, otherwise intimidating, more theoretically-oriented
   texts. The text, reflecting the author's extensive knowledge
   on the subject, has an unusually flowing writing style to it
   and throughout it are interspersed some quite entertaining
   snippets of the author's humor.

   I recommend this book to all students of spectral methods, regardless
   of level of expertise.




2. Spectral methods, as presented by Boyd, are techniques for numerically solving differential equations. His book is a collection of A LOT of practical information presented mostly through a mathematical frame work. Practical means different things to different people; in Boyd's case, he discusses the details of what happens in putting the mathematics to use (the pitfalls), and when each technique should be used. Supporting numerical methods, such as matrix techniques, are discussed where needed. Example computer code is scarce. Worked examples are inconsistently used, and sometimes abstract.
   
   As a novice to the field, I found the level of presentation a notch too high to be able to put it to use. It was more abstract than applied. I'm not saying it is not informative, only that this is not a good first book on the topic. I might get this as a second or third book.
   
   I give it 4 starts due to two complaints. There are not a lot of illustrations, and moreover those that are included are often too simple or need more annotation. A little more thought should go into them, and there should be a more of them for some of the more abstract topics. Additional thought should go into the organization too. Information at different levels of expertise are scattered throughout so you either (a) need to know the answers already, (b) skip ahead several chapters, or (c) go on an aside in another text.

5 comments about Standards of Brewing: Formulas for Consistency and Excellence.

1. This book addresses a great many of the problems that one encounters day to day in a beer quality laboratory. The writing is in a style that is easy to understand and I thought that the best part was that the book was written with a great sense of humour. I never thought that I would enjoy reading a chapter on "statistics and process control" and yet the author guides you through it in such a way that he leaves you with a clear understanding of the concepts and you enjoy the journey. The author appears to be superbly qualified to write the book from his experiences in the brewing industry.



2. This book is exactly what I was looking for to help me understand many of the terms used in the brewing quality lab. The exercises at the back of the chapters were a great way to review whether I really understood the concepts or whether I had to go back and read the section again. The author's quirky
   sense of humour and how he described his own past experiences in the quality laboratory made me laugh out loud at times and while I was laughing I was learning. A great book that explains why the quality standards are there and how to make good beer by adhering to them.



3. A wonderful book. I learned as much as I laughed. I would recommend this book to anyone that wants to know more about brewing. A++++!



4. This book was thorough and informative about big brewery quality controls. But, be warned if you are a 5 gallon a time homebrewer the book will have little guidance/application for you. Andrew Steele - New Zealand



5. If you are a begging brewer wanting to learn to mash this book is good but there are a lot more books out there that either cover more begging issues.
   This book is simple and short!

5 comments about The Quantum Theory of Fields, Vol. 3: Supersymmetry.

1. Great book, contains a lot of material, will be useful to many as a reference on supersymmetry for years to come. Highly Recommended!



2. If the two first volumes of "The Quantum Theory of Fields" were considered masterpieces in a modern and original presentation of the basics of quantum field theory and its penetration in the recent development of particle physics, with the machinery of spontaneously broken gauge theories, the new volume embraces the wide subject of supersymmetry in Weinberg's typical style, which always means a self-contained treatment of the subject, from its foundations and motivations, to its most recent application as a possible scenario for new physics beyond the Standard Model.

   A complete review is published in CERN Courier, May 2000




3. Finding good introductions to supersymmetry can be difficult. Most introductions concentrate on N=1 supersymmetry in four dimensions, and there the superfield forumlation can be useful. However, when you go to N=2 supersymmetry (e.g. when considering theories in five or more dimensions), component fields can be better. Many times it's a matter of taste. For those cases, you have to go to review articles. Anyway, Weinberg concentrates on N=1 4D supersymmetry and supergravity using the superfield formalism. However, he ventures into the N=2 strong-weak coupling results of Seiberg and Witten, which are now a fundamental part of (supersymmetric) field theory. The text is, as the previous volumes are, a fantastic resource for learning the subject, and as a reference (for things like gravity- and gauge-mediated supersymmetry breaking, as well as the minimal supersymmetric standard model, which are open areas of reserach). As for all modern areas of research, the body of knowledge is stacked higher every year; but the topics covered here stand as solid fundamentals of supersymmetry. For more advanced topics, one is forced to go to the recent literature.



4. The whole current production run of this book has a defect. A glue is bleeding through on the inside of the hard cover fold, front and back. This does not seem to affect the structural quality of the book and is not visible from the outside. If you need this book and get it with this defect, don't bother trying to exchange it.



5. I'm a beginning graduate student in theoretical physics, who learned SUSY from the ground up from Weinberg's text. Weinberg is (in my opinion) by far the best text I could find on SUSY. It is totally self-contained* - every equation can be checked by the reader; the idea's are solidly explained, and the choice of topics is extremely relevant. However, I should say that this is probably not a book for those looking for a quick introduction, or a sketch of the subject (which are valuable in their own right).
   
   In the very beginning (I knew very little SUSY) my impulse was to avoid this book, as the "notation" seemed kind of heavy, there were too many long equations (superfield identities), and it was clear that reading the book was going to be a serious endeavor**. Instead I was looking for a quick fix. However, having found the other sources inadequate***, I gave Weinberg another try. I learned how to read his book (from the point of view of a beginner; a veteran can easily use it as a reference)~ read it actively, checking the equations at the level of looking for typos. I poured in many hard hours, and have a binder full of derivations to show for it****. But as a result one is very well equipped to tackle the literature.
   
   I especially appreciate how Weinberg builds SUSY from the ground up. He makes it come together so logically, and coherently, it is nice to watch, and I feel one is rewarded in deeper understanding.
   
   His treatment is often original and improves in many ways upon the original literature. For example, his treatment of SUSY representation theory and constructions of superfields. Also his treatment of holomorphy arguments is the best I've found anywhere (literature included). His treatment of Seiberg-Witten is his own pedagogically minded retelling of that story ~ it takes a slightly different angle than the original work, and fills in many of the details. Reading the original Seiberg-Witten afterwards was much facilitated.
   
   A word on prerequisites: A basic knowledge of QFT is needed ~ if you have Weinberg's Vol I, II, this is overkill. However, you should be comfortable with the representation theory of the Lorentz group ~ especially spinors. Weinberg provides useful appendices on spinors in Vol III, and has the rep. theory in Vol I (an understanding of angular momentum at the level of say Sakurai ch 3 helps here). To understand the interesting non-perturbative results (chapter 29) you must be comfortable with 1-loop beta functions in YM, and the chiral anomaly (covered in Vol II as well as many other texts).
   
   A caution on typos: There are many minor typos which you probably won't notice unless you rederive the particular offending equation. I know of about fifty (over a range of about 300 pages). The nature of the subject is such that there could have been many more though (lot's of long equations with many indexes). Luckily, the errors often do not propagate ~ subsequent equations are usually typo free. There doesn't seem to be an errata website, which is unfortunate.
   
   Finally, there are a few exercises after each chapter. Some of them seem intellectually gratifying, and some are rather messy algebra.
   
   
   
   * With the exception of some of the MSSM stuff, but this is clearly stated, and totally reasonable.
   ** But alas, for a beginning student, this is the nature of the subject.
   ***There is one fantastic supplement to Weinberg (after you've gone through the first couple of intro chapters), these are Argyres' notes. They nicely cover Seiberg duality which Weinberg doesn't talk about (but he does a great job with Seiberg-Witten).
   **** The meat of the book can in principle be covered in < 2.5 months by a super-dedicated student (skipping the SUGRA chapter) and of course depending on one's incoming background and interest!

2 comments about Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists.

1. I am so happy with this book that I could not wait to finish one chapter and then post a review. This is my initial review but maybe I will extend it.
   
   I am a theoretical physics student and so far I have read one section on lie algebra and the approach is very clear and not at all In mathematical language and notations ( that you are required to master first to understand the underlying mathematics generally).
   
   Robert Gilmore done a very good job on this introductory book which fits with the title. He explains the ideas in very clear and concise way for non mathematical students. First he explained lie groups briefly and then came to lie algebra and explain why this is done. All most all authors forget to mention why they introduced lie algebra. I have many other books on group theory and lie groups e.g. Sternberg, Fuchs & Schweigert , Wu-Ki Tung, Georgi etc and the main point to be noted is that many authors do a good job explaining " how" but they forget to mention "why". This is where Robert Gilmore comes in. It is a pity that he did not write a book on Group theory as a whole including other topics in group theory as well.
   
   My advice is if you need an introduction to lie groups and lie algebra and tired of authors who only try to impress other authors instead of the student then invest on this book.You won't be disappointed and maybe this one goes into your collection.
   
   PS: for student of particle physics, try also Lie algebras from Howard Georgi.



2. This book is intended as an introduction to the topic for students in physical and chemical disciplines. It should not be thought that this book is an abbreviated version of the previous one. The structure of this text is radically different from the 1974 book, which was more a compendium of group theoretical techniques, and presented very actual topics used in physics. This book, preserving the essential motivations, has been written to develop, step by step, the techniques and methods used when groups are applied to describe physical phenomena, with details and explanations that are usually omitted in most textbooks.
   
   The book consists of sixteen chapters containing a large number of problems to be worked out by the reader. The results are presented in a very direct way, avoiding too technical developments and extracting the main facts. This philosophy is very convenient at a first level, because it focuses on the most important points and does not confuse the reader with involved proofs.
   
   In the first chapter, the author presents the historical motivation that led S. Lie to develop the theory of continuous groups, the Galois theory. This serves mostly to motivate the study of Lie groups, but presents no specific interest to the physicist. Chapters 2 and 3 are devoted to the main properties of matrix Lie groups, which are the main object of study in this book and correspond to the types usually encountered in applications. In this sense, the different classical groups are presented as those subjected to different constraints, motivating the geometrical interpretation of these groups.
   In chapter 4 the discussion of Lie algebras begins. First of all, it is illustrated why these structures serve to simplify a lot the analysis, since Lie algebras correspond to the linear approximation to the group at a given point. The exponentiation map is shortly introduced, without going yet into more involved questions like the local determination of the group from the Lie algebra. Important facts like the adjoint representation, the Killing and invariant metrics are introduced. This leads to a first insight into the structure of Lie algebras. This analysis continues in chapter 5, where the Lie algebras of the classical groups are derived from the corresponding constraints. The role played by the Killing form is studied in these examples, constituting a first approximation to the well known characterization of semisimple algebras. Chapter six is devoted to the usual techniques to deal with Lie algebras in physical applications, namely, the realizations by creation and annihilation operators and the realizations by vector fields. Although a very short section, the problems illustrate important topics like the angular momentum by means of Schwinger representations used in Quantum Mechanics. The seventh chapter reconsiders the problem of exponentiation in a more technical way. The limitations of the procedure and the isomorphism problem are developed having in mind the important su(2) case. The main result, the covering theorem, is presented graphically, illustrating quite well the general pattern of the theory. The Campbell-Hausdorff formula is introduced motivated by the non-trivial reparameterization problem. The informal way chosen to present this deep result is quite adequate, since it focuses on the meaning of the theorem instead of presenting a technical proof that it not trivial. Once the basic material has been presented, chapter 8 begins with the systematic study of the structure of Lie algebras. The main types of algebras, abelian, nilpotent, solvable, simple and semisimple are defined using the properties of the adjoint representation. Although not explicitely stated, this corresponds actually to the Levi decomposition. One important point should be clarified here: in section 2.3, the "canonical" form of solvable algebras is presented, according to the well known flag space technique of the Lie theorem. However, upper (respectively lower) triangular matrices are the model for solvable Lie algebras only for the complex base field (the Lie theorem being false in general for real solvable Lie algebras). At no point this crucial point is mentioned, which could lead to confusion to the non-expert. Chapters nine and ten concentrate on the classification problem of complex semsimple Lie algebras. This part is a shortened version of the material contained in the previous book of the author, presenting only the indispensable facts. The graphics of root systems help a lot to understand the general situation and the motivation of the classification of Dynkin diagrams. I miss however some comments on the Cartan matrix, which is the natural link between the (fundamental) roots and the corresponding diagram. The next chapter focuses on the real forms of simple complex Lie algebras. The main idea of its obtention is studied, as well as the main steps of the Cartan method to determine the non-equivalent real forms. The material of this section is crucial for applications, since many important models are based on non-compact Lie algebras. Being a quite delicate question, I agree with the author in the decision of leaving out the notions of inner and outer involutive endomorphisms used in their classification.
   These first eleven chapters cover the main facts about Lie theory that any student in either physics or chemistry should master for a full comprehension of more technical. Chapter 12 reviews Riemannian symmetric spaces, a very important type of manifolds. Here the geometrical role of the exponentiation map is exploited, helping to understand the implications of the choice of real form and its consequences in the geometry and topology of the corresponding manifold. The material is again presented and commented using important examples, instead of developing cumbersome theoretical argumentations, which can be found in the cited literature. The results are complemented by carefully chosen problems of physical nature, pointing out the relevance of symmetric spaces in applications. Chapter 13 introduces a more sophisticated technique, the contractions of Lie groups. This procedure, of essential importance in physics, is developed following the classical method of Inönü and Wigner. How to use contractions in limiting processes of other objects is illustrated in the different sections. However, I believe that focusing only on Inönü-Wigner contractions gives a quite restrictive view of this technique (even if this constitutes a very important class of contractions, as shown by their applications to symmetry breaking).
   Chapter 14 constitutes an introduction to the study of symmetry in physical systems. This is an important part, since many textbooks usually assume the reader is aware of the different notions of symmetries used. To this extent, the author chooses a classical and vital example, the hydrogen atom. The different types of symmetry (geometrical, dynamical, spectrum generating algebra) not only point out the different physical properties to be described by means of symmetry, but also the importance of how to embed a Lie group into another. The detailed description made by the author will surely clarify some aspects that are generally quickly reviewed, and therefore constitute a difficulty for the unexperienced reader.
   The Maxwell equations are derived in chapter 15 using the properties of two fundamental groups in Physics: the Lorentz group SO(1,3) and the Poincaré group. Although it may appear that this chapter is disconnected from the rest, it actually has been placed in the right place. On one hand, the Maxwell equations are connected to the most important physical groups,.and further, these are closely related to the conformal group previously introduced, being a natural link to justify the importance of symmetries of differential equations.
   The last chapter connects with the first in the sense that Lie groups are used to determine whether a differential equation can be solved by quadratures or not. Since this is a large and complicated theory, only the basic elements that show how Lie groups are used to simplify the integration of differential equations are presented.
   
   This book constitutes a very comprehensive introduction to Lie theory in physics, dealing with the most important features that students will encounter. The problems help not only to understand the material presented, but also exhibit real physical situations where Lie groups are used This book further solves some difficulties encountered by beginners in other books, usually written at a more specialized level.

3 comments about A First Course in Computational Physics and Object-Oriented Programming with C++.

1. This book can serve several audiences. It teaches both computational physics and the use of C++ in writing object oriented code. Clearly, if you are already know one of these topics, but not the other, then the book is a natural fit. You can concentrate on what is essentially half the book.
   
   The more challenging task is if you are unfamiliar with both. Well, it is reasonable to assume that you know some physics, say at the first year undergraduate level. And perhaps you have done some programming, in a procedural language like Fortran or Basic.
   
   The amount of abstractions, or rather the level of difficulty in this, is less than in a typical physics text that is explaining Maxwell's Equations or Einstein's Special Relativity. The physics in the book revolves around trying to compute certain numbers in an efficient manner.
   
   While from a programming standpoint, computational physics examples are given as an important use case, to help the student grasp the OO concepts.



2. This book was developed during many years of teaching scientific programming to engineers and scientists in both electrical engineering and physics courses. About 1/3 of the text is accessible to beginning programmers even at a high-school level, while the last part of the book can serve as a second-term undergraduate scientific programming course or as a reference text. While the title indicates that a major focus of the text is computational physics, the book contains problems and examples from numerous scientific and engineering disciplines and can be employed across a wide variety of course offerings.
   
   Because of the practical difficulties faced by beginning students, a first course in scientific programming generally requires very significant personal intervention by the instructor or laboratory assistant. This book effectively removes this issue by providing a common base of free Windows software on CD-ROM that is meticulously documented in the text (the software is also available for Linux). The reader is introduced to programming through numerous assignments containing real-world technical problems. The assignments at first contain nearly the entire program to be developed; as the book develops, however, fewer code sections are provided. This method allows the user to absorb proper program structure while avoiding frustrating and confusing stylistic traps. A solution manual is made available to instructors through Cambridge University Press (see their website for errata) while the CD-ROM also contains copies of all programs presented in the text.
   
   This book presents a compact but completely unified picture of modern programming practice as it applies to scientific programming. The fundamental, underlying principles of the C++ language and scientific programming are stressed in order to simplify retention of complex C++ syntax and of the mathematical and physical content. More involved topics in numerical analysis, scientific programming methods and C++ are presented in an intuitive and easily-understood manner. Examples of the subjects covered are: software engineering principles (UML), numerical analysis, scientific graphics programming, the Standard Template Library (STL), Monte-Carlo methods including the Metropolis and multicanonical techniques, partial differential equation solvers, calling Fortran from C++, C++ program optimization.



3. This is a very good book, the codes are clear and
   written from a computational point of view. It is easy
   to set up the software. I agree with the authors self
   remraks except that he should wirte up some harder examples
   in the end. But still, the best.

5 comments about PI in the Sky: Counting, Thinking, and Being.

1. Barrow, an astronomer at the University of Sussex when this book was published, provides an entertaining and informative account of the foundations and philosophy of mathematics. Do mathematicians invent or discover mathematics? What 'reality' do mathematical entities like pi have? What accounts for what physicist Eugene Wigner has called, in a now-famous paper, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (299)? After an interesting account of the history of counting and numbers, Barrow discusses in succeeding chapters the philosophies of formalism, inventionism, intuitionism, and platonism, a sophisticated version of which he seems to favor. Perhaps most mathematical workers follow what Alfred Korzybski called "the 'christian science' school of mathematics, which proceeds by faith and disregards entirely any problems of the epistemological foundations of its supposed `scientific' activities" (Science and Sanity 748). I commend Barrow because he considers these epistemological questions important and writes about them so engagingly. Barrow's discussions of theories and personalities provide useful background for understanding mathematical foundations. As for Barrow's conclusions, from a non-aristotelian view, the appeal of platonism seems understandable as an example of identification, the confusion of orders of abstracting. Barrow doesn't seem to consider that mathematicians may both invent and discover mathematics. He seems so taken with the effectiveness of mathematics in the natural sciences that the notion of mathematical entities existing solely as high-order abstractions in human nervous systems seems insufficient to him. As Korzybski pointed out, we live in a world of multi-dimensional, ordered structures or relations. It does not seem unreasonable, then, that we can map this world with an exact language of relations, i.e., mathematics. But as Korzybski also pointed out many times, "the map is not the territory."



2. The author's leading claim in this book is that "the only mathematics we know is human mind and brain based mathematics." This claim can be understood in either of two ways, which the author does not distinguish from one another. On one hand, by "mathematics" he might mean the practice or family of practices that go by that name (the sort of thing that a math teacher gets paid to teach). In this sense, it is just trivial that the only mathematics we know depends on humans, just as the only civil engineering or basketball we know depends on humans. But let's be charitable, and try to construe the author's claim in a way that does not reduce it to a mere triviality. Let's suppose that by "mathematics" he means not the practice of investigating mathematical fact, but the body of fact thus investigated. But if that is what he means, then his claim is clearly false. It is clearly false, for example, that the fact that 1 is less than a million depends on humans, their minds, or their brains.

   Thus, the whole book is premised on a fallacy that can be spotted by a second year philosophy major.




3. doesn't the mathematical concept of greater than
   come from a human mind? sure, some birds can count and distinguish between object sizes but can they creatively abstract and apply the concept to solve other physical problems?
   nope. and if you think the whole book is based
   on a false premise, it still has some interesting views, facts and features.
   does it warrant a 1 star?
   i mean you can learn from everything.
   even mistakes.
   i mean i learned from you just now.



4. That may be a silly question. After all, most of us use counting and numerical calculations many times a day. However, the reading matter here digs below the surface, and asks such awkward questions. What is the nature of maths? Would there be any maths if there were no mathematicians?
   
   Starting with theories of counting, and the origins of methods of enumeration, John Barrow plunges headlong into the philosophy of mathematics. Perhaps the book ought to carry a health warning, for it should not be read accidentally. Readers need to have a grounding in some of the great mathematical movements, and discoveries. (Perhaps it is a bit judgmental to even use the word "discoveries"; are mathematical ideas invented or discovered? That topic is part of the subject matter).
   
   I liked the debate, but found the volume hard going. It is not the kind of book to read solidly from cover to cover. A great deal of re-reading is necessary, and picking it up on the train requires a conscious effort to remember what the current debate is about. Some of the arguments are very intricate for those of us who are not mathematicians.
   
   The work of some of the pillars of mathematics are described in varying detail, together with the triple crises that hit maths in the early years of the 20th Century. The optimism of Hilbert on the one hand, or Russell and Whitehead on the other was washed away by the work of Kurt Godel. The Austrian Godel, by the way, has been described as one of the most innovative minds of that century.
   
   There are some interesting insights into some of the characters from the history of maths. Leopold Kronecker did not believe in negative numbers. However, he had been a BANKER. How did he convince his customers that the problems caused by negative numbers (i.e. too little in their accounts) needed to be solved? There were also some disturbing questions raised by the work of Cantor on set theory. This gives rise to a wonderful paradox called "Hilbert's Hotel".
   
   As with many works on philosophy, it is not the answers that are important, it is the questions. Does the entity pi exist, even if there are no mathematicians. Is there really a universal 'pi in the sky', external to any human thought? You decide.
   
   Peter Morgan, Bath, UK (morganp@supanet.com)



5. Might have been a classic if I had understood more of it. This is an extremely deep subject, searching for the source of mathematics. Is it a closed system of symbols that can solve any problem since it is a self-constructed system, as the formalists claim? Apparently not exactly, as Godel has proven that any system of math is contains unsolvable problems.
   
   Is it merely the presence of numbers and definite operations in the mind, learned from human activity? But this essentially limits math, as Barrow points out, to a branch of psychology; it is "finite, shorn of many truths that we had liked, divested of so many devices that were as much a part of human intuition as counting, and divorced from the study of the physical world."
   
   Then there's the Platonist view, that mathematics is an ideal, discovered and not invented. "Mathematics exists apart from mathematicians," says Barrow. This view, teetering on the brink of mysticism, is closest to where the philosophy of mathematics is today, according to Barrow, especially among consumers of mathematics such as physicists working at the extreme edge of science.
   
   Compare this to my review of Frank Tipler's The Physics of Immortality: Modern Cosmology, God and the Resurrection of the Dead

2 comments about Mathematics for Physics with Calculus.

1. Probobly the best summery of mathematical physics an undergraduate could buy. Biman Das creates excelant physics review books.



2. Highly recommended supplemental book for my Calculus-based physics class. It provided an excellent quick review on some topics from a while back, and a great summary and introduction to topics I needed to know for the course but had never been taught. Its not a substitute for taking a calculus course, but it is a great review if calculus was a while back, or if you never covered certain topics. Either way, this book will come in handy throughout your physics courses.

3 comments about Paul Dirac: The Man and his Work.

1. After missing the first collection of essays on this brilliant recluse published soon after his death, I picked up the present version as soon as I was able. It did not disappoint.

   The book is a collection of four lectures given in the subject's honor in 1995 on the tenth anniversary of his death. The final lecture and the latter part of the third are highly mathematical and technical and clearly intended for a professional audience.

   But for me, the first lecture by Abraham Pais is worth the purchase price alone. Pais was not only a contemporary physicist, but also a close friend and as close to a confidant as was possible with such a reticent man.

   Through Pais' eyes, we see a mathematician turned physicist who was very different from the man to whom Dirac is most frequently compared, Albert Einstein. Einstein was a physicist first, mathematician second. Dirac was exactly the opposite. Einstein became a social and political critic, Dirac never strayed far from his study. The two were similar in that both viewed mathematical beauty as primary and both hated the modern remake of quantum mechanics (after the initial theory) for very similar reasons. This last point was interesting as Dirac was the first one to combine all his contemporaries' work on this improved quantum physics into a formal mathematical structure. His resulting equation, called naturally the Dirac equation, is classic Dirac, short and sweet. It combined Einsteinian relativity with the new quantum theory and Dirac considered the result to govern most of physics and all of chemistry. Stephen Hawking, the renowned theoretical physicist, says in his introductory memorial address to the book, "If Dirac had patented the equation ... he would have become one of the richest men in the world. Every television set or computer would have paid him royalties." For this work, Dirac shared the 1933 Nobel Prize with German physicist Erwin Schroedinger. One unexpected consequence of this work was a mathematical conclusion that defined a "negative energy" matter (aka antimatter) solution. Simply put, he had discovered a universe noone had imagined. To this day, we see the effects of this discovery from medical necessities (PET scan imaging-Positron Emission Tomography) to science fiction (Star Trek).

   The quotations and anecdotes Pais chooses are well placed and often very funny. They are also supported by the images of Dirac portrayed in the sketch on the cover and in the few photographs scattered through the first two lectures. They reveal his character well. He saw mathematical and physical realities so clearly that he simply could not understand why others did not see them as well. The photo of him "listening" to future Nobel Laureate Richard Feynman in Maurice Jacob's section is one of the most amusing of the collection.

   In the second lecture, Jacob shows the path of discovery and effect on latter day experimental physics of antimatter. He goes too long in spots but is generally fine.




2. We were ourselves participating in the inauguration of the Paul Dirac memorial in Westminster Abbey. Especially the speeches of Stephan Hawking and Abraham Pais were very touching as they did not only touch Dirac's work but also his personality and life. He was a very complex person and a great physicist. This book reflects that more than others about him.



3. A man Stephen Hawking calls 'probably the greatest British theoretical physicist since Newton,' has got to be a pretty bright man. Paul Dirac wrote the definitive equasion that joined the Theory of Relativity and Quantum Mechanics. Like Einstein before him, his equasion is very simple to express, very complex in its overall impact. It explains things like how television sets or computers work.
   
   This book is not exactly a biography, but more a tribute to him. It is a series of four talks given about Dirac eleven years after his death, upon the dedication of a plack to him in Westminster Abby.
   
   Abraham Pais describes Dirac's character and his approach to his work.
   
   Maurice Jacob explains not only how and why Dirac was led to introduce the concept of antimatter, but also its central role in modern particle physics and cosmology.
   
   David Olive gives an account of Dirac's work on magnetic monopoles and shows how it has had a profound influence in the development of fundamental physics down to the present day.
   
   Sir Michael Atiyah explains the widespread significance of the Dirac equation in mathematics, its roots in algebra and its implications for geometry and topology.

2 comments about Essentials of Mathematica: With Applications to Mathematics and Physics.

1. Great tutorial for those using Mathematica 6.0. It has a variety of different applications to related fields but overall serves as a guide for how to perform various functions in a software environment that may be very new to some.



2. This book uses a late version of Mathematica 5, NOT 6.0. I specifically wanted a book for 6.0 and got this because of the prior review. Again, not 6.0. The book does have good information on using Mathematica.