1 comments about Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (Undergraduate Texts in Mathematics).

1. This is the 3rd edition of a popular reference on the subject. There was some production error with earlier version of the book. Even with the latest version, the authors have provided 14-page worth of corrections for the 1st printing of the 3rd edition. See www.cs.amherst.edu/~dac/iva/3ed1.pdf.
   So buyers may be better off waiting for a corrected later version from the publisher.

No comments about General Investigations of Curved Surfaces: Edited with an Introduction and Notes by Peter Pesic.

2 comments about Geometry - Plane, Solid & Analytic Problem Solver (Problem Solvers).

1. A great book loaded with proofs for geometry (SAS, CPCTC, etc) and geometic problems (Pythagorean Therom).



2. I bought this book right after I started a class in Linear Algebra where the first chapter had us working some geometric proofs using the rules for vector addition, subtraction, and scalar multiplication.
   
   Although this book strictly focuses on geometry and does not use vectors in its proofs, it did clearly illustrate proofs that were easy enough to think through in terms of vectors.
   
   The book is well-written and begins with a review of Methods of Proof, Lines & Angles, Perpendicularity, and Triangles before getting into specific geometric forms.
   
   900+ pages and 900+ problems with a summary of theorems and properties at the end of the book. Some reviews at Amazon contain complaints about REA Problems Solvers, but I have always liked them and I really like this one. They are not intended to replace a textbook, but to give lots of examples over-and-above what a textbook would include.
   
   For example, a few years ago I was studying College Algebra and I was having trouble understanding logarithms. So I reviewed the examples in the REA Problem Solver for Algebra and Trigonometry and worked some problems along with the REA book and then I saw how simple they really were. I have purchased REA Problem Solvers for Calculus, Differential Equations, and Linear Algebra as well and have found all of them to be helpful with some being more useful or better-written than others.
   
   One of the complaints about these books is that they are all written using a Courier-style font. This is true, but there are still Greek symbols, integral signs, exponents, etc (e.g. you do not see 10-squared written as 10^2 - they use a superscript for the exponent). Another complaint that I have seen is that they contain errors. I have not noticed any, but that does not mean there are none present. I do not review these books cover-to-cover, but take a look at those topics that I need to review, so I may not have encountered any errors but they could still be present.
   
   Purchased new, the books may seem a bit pricey if you are on a budget (nowadays the books have a list price of $30.95 for some subjects and $25.95 for others, with discounted prices ranging between $17.95 and $30.25). I always buy them from the affiliated 3rd-party booksellers on Amazon. I got my Geometry Problem Solver extremely cheap $3.15 plus $3.99 for S&H - total of $7.14 and the book arrived in a couple of weeks and was in excellent condition (like new!).
   
   Most of the time I have spent in the $12-$15 range (incl S&H) for these guides. Even if I only were to find help on a single topic within the book, that, to me, is still a bargain. Especially so if you consider the price relative to the cost of tuition and textbooks. If you are really price conscious, most college libraries have copies that are rarely checked out. Until my copy was shipped, I used the one in our library and saw that it had last been checked out in 2004!
   
   p.s. I am 54 years old and went back to night school and started studying math 3 years ago with College Algebra, Trig, PreCalc, Calc 1,2,3, Diff Eq, and now Linear Alg. I have had REA guides for all of the classes and have found them useful in every case.

4 comments about Mathematics and its History.

1. This is an overall good text. It offers a very in depth history of many many mathematical ideas. It gets quite technical at times, which can be a good or bad thing, depending on what you are looking for.



2. Stillwell covers a lot of ground in a short undergraduate text intended to unify various mathematical disciplines. Naturally, _Mathematics_and_its_History_ begins with the early Greeks and in particular geometry (which is how mathematics was typically expressed then). The development of algebra and polynomial forms is described followed by perspective geometry. The invention of calculus and the closely related discovery of infinite series provide the backdrop for short biographies of prominent mathematicians (mostly dead white males to multicultural deconstructionists). The development of elliptic integrals (used in solving functions with specified boundary conditions such as a Neumann problem found in fluid mechanics). The treatment then diverges to physical problems including the vibrating string and hydrodynamics, together with a note on the renown Bernoulli family. Then Stillwell returns to the esoteric in complex numbers, topology, group theory and logic with some comments on computation at the end. Some mathematicians may find the overview to lack comprehensiveness, but the book's brevity for each topic and biographical notes present a balanced approach to the more casual reader about this important field of study and how it developed.



3. It is a very good book. It has presented very clearly some difficult-to-understand relationship especially the link between algebra and geometry. It is a very good balance - history, Mathmatics, biography all mixed very well together. Highly recommended.



4. This is a brilliant book that conveys a beautiful, unified picture of mathematics. It is not an encyclopedic history, it is history for the sake of understanding mathematics. There is an idea behind every topic, every section makes a mathematical point, showing how the mathematical theories of today has grown inevitably from the natural problems studied by the masters of the past.
   
   Math history textbooks of today are often enslaved by the modern curriculum, which means that they spend lots of time on the question of rigor in analysis and they feel obliged to deal with boring technicalities of the history of matrix theory and so on. This is of course the wrong way to study history. Instead, one of the great virtues of a history such as Stillwell's is that it studies mathematics the way mathematics wants to be studied, which gives a very healthy perspective on the modern customs. Again and again topics which are treated unnaturally in the usual courses are seen here in their proper setting. This makes this book a very valuable companion over the years.
   
   Another flaw of many standard history textbooks is that they spend too much time on trivial things like elementary arithmetic, because they think it is good for aspiring teachers and, I think, because it is fashionable to deal with non-western civilisations. It gives an unsound picture of mathematics if Gauss receives as much attention as abacuses, and it makes these books useless for understanding any of the really interesting mathematics, say after 1800. Here Stillwell saves us again. The chapter on calculus is done by page 170, which is about a third of the book. A comparable point in the more mainstream book of Katz, for instance, is page 596 of my edition, which is more than two thirds into that book.
   
   Petty details aside, the main point is the following: This is the single best book I have ever seen for truly understanding mathematics as a whole.

2 comments about Essential Topology (Springer Undergraduate Mathematics Series).

1. I have a major in math, many years ago. I have moved into economics, but miss the elegance of math, hence I decided to revisit some old topics, and started with topology. As a student we used lecture notes and no real textbook, so my choice now was this textbook. It is a pure pleasure to read. I wish we had used it as a text book when I studied.
   
   The topics are well motivated. Crossley does a good job in explaining why we should care about these particular lemmas and theorems. The proofs are usually elegant. I find the estetic pleasures a good math book should provide.



2. I have never seen such a beatiful explanation on continuity and its relations to series and sets. Now I understand why, when mathematics is lousily explained,everything seemms to be so hard. I recommend strongly this book for someone for self study on topology. Hope the author can write on other topics of mathematics.

1 comments about Complex Analysis with MATHEMATICA®.

1. What a wonderful and delightful book. The author has taken a well-worn topic and infused it with insight and energy. Best of all, the author communicates simply and clearly. He has brought graduate complex analysis to the masses; I wish that I had a book like this as an undergraduate. It is also a great read: the kind of pick you could pick up and go cover to cover with. However, you'll probably want to be at your keyboard trying out his many examples. The unique thing about this tome is that it is well-written, it is mathematically ambitious and it is an invaluable reference for how to use Mathematica. Also, as we have come to expect from Cambridge, this book has excellent production quality.

3 comments about The Topology of Fibre Bundles. (PMS-14).

1. A nostalgic but still attractive book on (homotoy theory of) fiber bundles. This book is not very accessible as it predates the development of modern machinery of algebraic topology, but is worth reading.



2. This book supplies a lot of intuition and background that more modern texts seem to assume of the reader. Steenrod's writing is meticulous and extremely clear. My opinion is that one can learn just as much out of this seemingly outdated text and probably even more than from the modern texts.

   ... True, more slick machinery has been developed since Steenrod's time, but those big machines are hardly transparent. Steenrod assumes very little of the reader; he even has a quick course in homotopy groups, although he assumes the reader knows the basics of homology/cohomology. Perhaps most importantly, since many of the ideas in the book were new at the time, he doesn't assume that the reader is already comfortable with those ideas. All together this makes a very accessible book indeed.




3. For those individuals who want an in-depth, insightful, and solid understanding of fiber bundles this book must be read. In spite of its date of publication, it still is of considerable value in this regard. Modern treatments of fiber bundles are very formal and the underlying motivation gets swept away in the thirst for rigor. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. Therefore a mastery of fiber bundles is essential for entering any of these fields. But fiber bundles are fascinating objects in and of themselves, and studying them for their own sake needs no apology.

   The author does use some antiquated notation, but that is not really a hindrance to the study of the book. The reader will no doubt have some background in differential geometry and topology before attempting this book, so the appropriate translation to more modern notation should be straightforward. Once started, and with a little thought adjustment to the idiosyncracies of the author's writing style, the reader will find a plethora of neat examples and insights into the subject. In particular, part 3 on the cohomology theory of bundles is exceptionally valuable in that it gives the reader a detailed overview of the origin of what are not called Stiefel-Whitney classes. The theory of characteristic classes has of course advanced and matured extensively since this book first appeared, but all of the modern treatments are lacking in that they do not give the reader an appreciation of the fundamentals of the subject. Indeed, the construction of the obstruction to the construction of a cross-section to a bundle is the starting point for many of the ideas in obstruction theory that one finds in differential topology. And yes, the procedures the author uses can be "cleaned-up" and made more concise, but the price one pays in such an endeavor is the loss of an appreciation of the concepts behind the scene.

   Since the book is a monograph, there are no exercises, and this is probably the only minus to the book. Also, some knowledge of the German language would be useful to a reader who has it, since the author makes references to papers written in German and much of the terminology in the book shows its roots in the German language. One good example of this is the Reidemeister theory of cohomology groups based on a bundle of coefficients, called Uberdeckung by Reidemeister.

   There is no question as to why this book remains in print, and it will no doubt continue to be well into the 21st century. IT is a good example of the idea that something new may not be something better. After finishing it, the reader will be amply prepared to enter into the continually-evolving theory of fiber bundles and their applications, all of which are interesting and important.

1 comments about Master Math: Geometry (Master Math Series).

1. The best presentation of geometry I have ever seen. The topics are presented in a logical manner so that they build, are in context, and make sense. It explains logic and proofs in a way students can really understand. Definitions are provided in the beginning so you can orient yourself and understand the jargon of geometry from the start. It presents concepts three ways: a description, a picture, and a description of the picture. It makes learning so easy! There are plenty of real-world and fun examples and tidbits of information that makes learning fun. It is clear, concise, and the topics have a flow and context that makes is easier to learn the material whether you are taking geometry for the first time, are older and need a review, or are taking higher level math, science or engineering classes and need to quickly look something up and understand it. Learning geometry does not need to be a frustrating experience! Everything you need for basic geometry is in this book! Master Math: Basic Math and Pre-Algebra, Master Math: Algebra, Master Math: Trigonometry, Master Math: Pre-Calculus and Geometry, and Master Math: Calculus are also fantastic!

3 comments about Geometry: A Self-Teaching Guide.

1. This review is based on the first chapter.
   
   The authors/editors did a poor job reviewing this book for errors. I checked the publisher's website and they do not provide online corrections. Being a self-teaching guide it contains self-tests, but when the answers are wrong it is very confusing to the learner.
   
   The discussion on trapezoids is confusing because they only illustrate isosceles trapezoids (but just call them "trapezoids" ) this gives the impression at all trapezoids are isoceles with base angles of equal value.



2. Geometry: A Self-Teaching Guide. This is not a bad text, but as a previous reviewer noted, it does have many errors (and I was also confused by the trapaziod section); on the other hand, most recent high school and college texts suffer from the same lack of craftsmanship. A final complaint is that there is no mention of proofs, which I was hoping to learn. I do like the fact that the book is straightforward and, once you learn how to account for the editing errors, it will give you a good grasp of the practical aspects of geometry very quickly. With this book and a good grasp of algebra you will be more than ready for trig.



3. This is a great little text with numerous illustrations but with a lot of errors. If you have taken geometry in the past or else are singularly hardheaded you will not be thrown by some of these mistakes that appear even in the self test sections. It is a great review, but remember to trust your own mind while studying the material.

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.