No comments about Calculus: Early Transcendental Functions.

No comments about Geometry for Christian Schools (2nd Edition).

4 comments about Rotations, Quaternions, and Double Groups.

1. This book has the best explanation of Clifford algebra
   that I've ever seen.
   The coverage of dihedral, space groups, quaternions and even projective
   diagrams is just very ,very good.
   He introduces "The Rodrigues programme" which is
   a very good angular approach to quaternions
   that actually predates Hamilton's quaternions
   but has been overlooked.
   He doesn't spare on word definitions and explanations.
   The index is fully operational
   and definitions are for the most part complete and understandable.
   In other words he actually doesn't just pretend to teach
   group theory, he actually does!
   If you have always wondered about SU(2) and quaternions, this is the book
   that you "need" to read.
   Roger L. Bagula



2. I came to this book with a good understanding of matrices, tensors, complex numbers, quaternions and some quantum mechanics. But I was unsure about spinors, and I hoped this book would help. It didn't.
   
   Much time is wasted is confusing and unnecessary quibbles. Each rotation can be represented by either of two quaternions. But which one? Far too much is made of this dilemma. Each rotation has two poles. Far too much is made of this too.
   
   The author's plan seems to be to create as much confusion as possible, and then show how quaternions can clean it all up, like a superhero at the end of a movie. Much better to *start* with quaternions and never let the confusion arise in the first place.
   
   Dismal.



3. This book displays great erudition. The reference section lists 140 books or articles about quaternions and rotations, going back to the 19th and even 18th centuries. The author shows off his command of this literature, taking a very thorough approach, bringing up subtle points that experts on the subject might not have fully grasped. Unfortunately, for a non-expert like me, the result is often bafflement.
   
   Some passages just seem obscure, for example on p 28 "rotations are an accident of three dimensional space. In spaces of any other dimensions the fundamental operations are reflections". There is no further discussion of this point, which is far from obvious to me.
   
   My background is in computing. I was looking for a general introduction to quaternions and their applications. This book did not meet my objectives. It is inexpensive and well produced but the contents too inaccessible.



4. This book will be difficult for "paper mathematicians" because it describes concepts of orientations in multidimensional configuration space that requires visual / spatial ability. It does so with with a depth and granularity that can be appreciated by those who work with these "tools". This small book is not a quick read, nor a general overview. It takes time to ruminate on these words to gain understanding of increasingly important behaviors in classical and quantum physics, as well as modeling the complexity in systems. But, these rewards require an investment of time and thought.

4 comments about Lectures on Classical Differential Geometry: Second Edition.

1. I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling. For ten bucks, you get a great summary of a wide range of topics in "classical differential geometry" -- the stuff geometers were interested in one hundred years ago. Today it's gauge and string theory -- but the topics discussed in this book are timeless, and many have seen remarkable renaissances in recent years. It is a wonderful little book ... I am using it to teach a basic differential geometry course next year.



2. Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.

   However, Struik can't be used to understand what is happening today. For these purposes,books by O'Neill and do Carmo would be more appropriate. The discussion of manifolds and coordinate charts, the discussion of connection forms, differential forms, covariant derivatives, exterior derivatives, pullbacks and pushforwards can be found in these texts. This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g. Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').

   The price of the book makes it an attractive purchase.




3. This is a survey of classical i.e. early 20th century differential geometry and not a more "modern" abstract treatment.



4. While it is quite true Dirk Struik's work is on classical differential geometry, the older methods and treatment do not necesarily imply obsolescence or mediocrity as some readers or reviewers suggest in their evaluations. Classical Analysis is still an important branch of Mathematical Analysis. So classical approaches and topics should not be dismissed as a waste of time, useless, outdated or even invalid. Remember Andrew Wiles' recent attack on Fermat's Last Theorem and his ultimate proof of its validity, an event that made headline news. That is a quintessential classical problem in mathematics (i.e., in number theory), only recently resolved. So remember: CLASSICAL Differential Geometry is part of the title.
   
   First of all, this book is very readable, being that it requires no more than 2 years of calculus (with analytic geometry and vector analysis) and linear algebra as prerequisites. Exposure to elementary ordinary and partial differential equations and calculus of variations are highly desirable, but not absolutely necessary. There are numerous carefully drawn diagrams of geometric figures incorporated throughout the book for illustration and, of course, better understanding. Topological methods are not used in the book, and the concept of manifolds not mentioned, much less treated. So this is an older work that bridges the very foundational and applied aspects of differential geometry with vector analysis, a field and body of knowledge widely used nowadays in the sciences and engineering and exploited in applications such as geodesy. For those insisting on modern approaches and want to omit studying foundations and historical development, please read up on other books such as O'Neill and Spivak. (Also, there are tons of other newer works, i.e., on "modern differential geometry", I am unfamiliar with. They are probably availble for browsing in college bookstores.)
   
   The author begins by leading the reader from analytic geometry in 3-dimensions into theory of surfaces, done the old fashion or classical way, i.e., utilizing vector calculus and not much more. Along the way, he takes the reader through subjects such as Euler's theorem, Dupin's indicatrix and various methods for surfaces. Then he continues with developing important fundamental equations underlying surfaces, e.g., Gauss-Weingarten equations, looks at Gauss and Codazzi equations, and proceeds to geodesics and variational methods. He includes a somewhat detailed treatment of the Gauss-Bonnet theorem as he progresses. He ends up with introducing concepts in conformal mapping, which plays an important role in differential geometry, minimal surfaces and various applications, one of which is geodesic mapping useful in geodesy, surveys and map-making. He does all of it with clarity and focus, including problems or "exercises" as he calls it, in under 240 pages - brevity that is rare in many mathematical books and works these days.
   
   For those with a mind for or bent on applications, e.g., applied physics (geophysics), applied mathematics, astronomy, geodesy and aerospace engineering, this book would be an excellent introduction to differential geometry and the classical theories of surfaces - being that one need not worry about abstract analysis and topological aspects of mathematics. Perhaps the title should be "Topics in Classical Differential Geometry" or "Introduction to the Theory of Surfaces in Classical Differential Geometry". But one must keep in mind that Dirk Struik is an old MIT hand and contemporary of Norbert Wiener, also at MIT, and Richard Courant (and many great German-educated mathematicians) who lived and worked in the early to mid-20th century, a long time ago and before computers became commonplace, an era in which total abstraction in mathematics and physics was not quite widely emphasized, but clear concrete thinking was important. A good friend of mine and co-worker who studied at the University of California, Berkeley, told me he had great respect for the classical geometers such as Struik and Eisenhart, understanding that they built ideas from a scatch and wrote in such a way that readers can discern the physical origins of geometry, in particular of differential geometry, a subject that supposedly started with Gauss during the early or mid-19th century when he performed survey work for his government in Germany. (The term "torsion" introduced and sed by Struik in the first few chapters of the book comes from classical mechanics, and is commonly employed in mechanical structures/structural engineering nowadays.)
   
   I for one am an aerospace engineer. There were one or more occasions where I consulted the book for formulas and expressions of curved surfaces and spheroids in my work of flight navigation (flying over the ellipsoidal Earth, as one example). I am sure that are other areas, e.g., space engineering, where classical methods of differential geometry embodied in Struik's book can come in handy.
   
   The only problem I have with the book is that the "exercises" do not come with solutions, but I do not think that is a major drawback unless one uses it as a textbook for a course that requires assignments and drill exercises.
   
   Judge for yourself by borrowing this book to read, i.e., if you are interested, can tell whether you like or dislike it on the first pass, and for what reasons one way or another. Find out for yourself.

1 comments about Geometry of Conics (Mathematical World) (Mathematical World).

1. There is a definite dearth of modern books dealing with geometrical conics, that is to say using the methods of classical euclidean and projective geometry to derive their properties. In this respect Akopyan's book should be warmly welcomed.
   
   A few other points pertaining to what used to be called Modern Geometry, such as cevians, symmedians, Lemoine and Brocard points, Simson lines, and some of their properties are also presented to new generations of readers.
   
   Much of this stuff used to be taught in this way in the 19th and early 20th century (cfr. C. V. Durell's delightful books), but later fell out of fashion. Fortunately a revival of interest in this classical way of teaching geometry can be perceived these days.
   
   I've only read part of the book so far, but I must admit it is a lovely book.
   
   However I find the book a bit beyond "... the reach of high school students", as the pace is rather brisk. Particularly projective geometry definitely deserves a longer and more detailed introduction.
   
   There is a mistake in the definition of parabola in the last paragraph of page 2 (line 3 from bottom), where "equal" should be substituted for "constant".

1 comments about Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica Support.

1. Phil Gregory has managed to condense the most important aspects of Bayesian probability and data analysis into a book that is actually rather practical. This book will give you a solid Bayesian understanding of probability, starting with first principles (Cox's desiderata), continuing to Bayes theorem, an introduction to common probability distributions, and concluding with rather advanced numerical techniques such as tempered Markov Chain Monte Carlo. The book is geared towards a reader who will use Mathematica to work through examples, but can be successfully used by others who prefer cheaper and more practical computational frameworks. It's not a flawless book by any means---first of all, although the book purports to cover frequentist alternatives to Bayesian methods, the frequentist coverage is very shallow and inadequate to give the reader enough background to either use or really understand frequentist usage. The chapter on maximum entropy techniques is woefully incomplete, and doesn't include general Jeffreys priors (derived from the Fisher information) or really explain the various issues associated with defining the entropy for continuous distributions. The section on deriving priors with uncertain constraints actually doesn't give an answer to how to handle uncertain constraints! But on the plus side this book answered a number of questions that have long puzzled me, such as why frequentists marginalize over nuisance parameters by minimizing the likelihood function with respect to them instead of integrating over them (it's a dodge that only really works for Gaussian-like distributions), and how to handle the enormous numbers of parameters that Bayesian calculations can generate through the use of Markov Chain Monte Carlos. I wish the book had been longer or more detailed, but if you want to learn Bayesian analysis and don't care much about understanding frequentist statistics, this is an excellent place to start.

No comments about Precalculus: A Graphing Approach.

4 comments about Introduction to Geometry, 2nd Edition.

1. This is Coxeter best book. Introduction to Geometry covers a wide range of topics and is the first book that I will look at for any geometry topic. It is now a little dated but only in the topics that it does not cover. Like all of Coxeter works each topic is clear and to the point. If you only buy one book on geometry this is it.



2. A better title for this book would be "Advanced Topics in Geometry". The chapters are pretty much self-contained.

   This book presumes a thorough, rigorous knowledge of high school geometry such as you might get in a college geometry course designed for future teachers along with considerable mathematical maturity.




3. This is a wonderful book. It isn't for mathematical beginners, but it isn't opaque either. It requires a student to think, experiment and to learn by puzzling things out in one's mind rather than simple memorization and regurgitation. Nor does it follow the all too common modern method of over simplifying things to allow people to pretend they have learned math while only dabbling in a few basic topics.

   This book is amply illustrated with many exercises (answers are provided at the back for all the exercises). The book also has some humor and wit with the quotes it distributes throughout the book to help liven things up.

   There is also a list of helpful references and an index. When reading the book, don't be afraid of going to a dictionary or the web or some other math books for clarification of some terms or more basic concepts. It is essential to have everything clear in your mind before moving on or you will stumble. As in all math, it is like a building with the next stage being built on the present one which is built on the previous one. You can't skip steps very successfully very often.

   This is a great volume to have in your library, but even better to work through.




4. This is a book that can lead you into the beauty of geometry. Mathematics is not constructed with knowledges and techniques, but the wisdom and ideas. This book does reflect it.

5 comments about Taxicab Geometry: An Adventure in Non-Euclidean Geometry.

1. Some years ago, I was employed by a company that built mapping software. One of the projects I worked on was the design of features that allowed for the computation of the shortest path from one position to another following only roads. This form of travel is similar to the taxicab geometry in that movement is restricted to lines. The only difference is that roads can be placed at any location or angle whereas the lines in taxicab geometry are equidistant and perpendicular. Think of it as the geometry of graph paper. As I constructed the program, I was struck by how so much of classical Euclidean geometry does not apply. Yet, the geometry is generally easier to understand because it is almost always how we move from place to place.
   The phrase non-Euclidean geometry generally conjures up thoughts of curved space and Riemannian geometry. However, in this delightfully simple book, a natural non-Euclidean geometry is developed in great detail. Very little mathematics is needed to understand the geometry, if you can mark and understand the points on a grid of graph paper, nearly all of the topics will make sense. A large number of problems are included at the end of each chapter and solutions to many appear in an appendix.
   The problems cover topics such as finding the point(s) of minimum distance between two or more points and what the taxicab analogues of circles and ellipses are. Determining the point of minimum distance between three or more points is a hard problem in standard geometry, but fairly simple in the taxicab geometry.
   Geometry is the godfather of abstract mathematics, being first used to codify the parceling of land and the construction of cities. In this book, you will learn how to minimize functions based on the restrictions of traveling through cities, a task with many applications in the world.



2. I thought that this book would be about geometry of exotic (but real) places in outer space (such as a black hole, for example). Instead, it turned out to be a lethally boring book, full of math problems, that was LESS interesting than my geometry book in high school!



3. Before purchasing this book, realize what it is. This is a book about non-euclidean geometry. Specifically, a specialized form of non-euclidian geometry affectionately referred to as taxi-cab geometry. This is not a table top book, but is a book for mathemeticians and those interested in mathematics. Others need not apply (regardless of how interesting the topic is). This is an excellent introduction to non-euclidean geometry because it strips away common misconceptions about the nature of non-euclidean geometries. This text is excellent for grade school children and those who would like to branch into more advanced non-euclidean geometries like hyperbolic.



4. Very simplistic treatment, with some results left for the reader to work through exercises. The chapters are almost non-existent, with all the book being mainly exercises.



5. I use the ideas in this book in my mathematics teaching in high school. Students learn to think of the world as Euclidean through most of their instruction; Taxicab Geoemetry gives teachers a very straghtforward way to introduce non-Eucliean Geometry. Admittedly, this book is not thorough, and it is very open ended (which I consider to be positive). Nevertheless, for its intended audience it is outstanding.

1 comments about The Map of My Life.

1. Professor Goro Shimura has written an entertaining and illuminating autobiography, which serves also as a story of the development of mathematics in our time. This is a major book that tells the very interesting tale of Shimura's life, starting with his ancestors, seventeenth-century samurai retainers of a feudal lord in the area of Tokyo, where his family has lived ever since. It continues through childhood in Japan, travels throughout the world, a stay in Paris, and his taking his professorial position at Princeton University. Shimura is one of the greatest mathematicians of our time and a key developer of the ideas that led to the proof of Fermat's Last Theorem. The Shimura-Taniyama Conjecture was the overarching result proved by Andrew Wiles in the 1990s, which led to Fermat's Last Theorem as a corollary. Throughout this entertaining book, Shimura reveals many interesting details about other famous mathematicians of his time: Andre Weil, Carl Ludwig Siegel, and Claude Chevalley. There are original letters reprinted here, and other material that helps paint a complete picture of mathematics and its development in the twentieth century. Highly recommended!