1 comments about Geometria Plana Y Del Espacio Y Trigonometria.

1. I Like this book because it does explain the fundamentals of Geometry in Spanish. So if your are fluent in Spanish this book will help you understand math better.

5 comments about Geometry, Relativity and the Fourth Dimension.

1. Over two millenia ago, Euclid wrote his masterpiece Elements and stated in his fifth postulate that only one perpendicular line could pass through any one point adjacent to another line.
   
   One hundred fifty years ago, it was proven that yet another geometry could be described by asserting that more than one parallel line could pass through such a point.
   
   Building on these ideas, Rucker briefly yet thoroughly surveys the relevant mathematics outside the box of Euclidian geometry.
   
   It's a fascinating place too because it involves considerations of hyperspace, four dimensional travels and ultimately Einstein's theory of relativity.
   
   Copiously filled with illustrations to help drive home his points, Rucker has produced a book that meaningful helps one visualize and better understand the fourth dimension.
   
   This book is an excellent read along with Choas, Coincidences and All that Math Jazz, The Fourth Dimension Simply Explained, Einstein's own Relativity and Hyperspace by Michio Kaku which discusses all these ideas as well as contemporary string theory (which purports to pull it all together).



2. it is published years before but it is almost new for today and it explain dimensions and shape of space well and clearly .thanx to amazon for sending me timely.



3. This book has presented the most difficult topics of our world with the easiest words. After reading this book many of my questions that I had in my mind for a long time were answered. It's worth thousands more than its price.
   Congratulation to Mr. Rudolf Rucker for his great book.



4. To understand relativity, it is necessary to understand geometry, specifically how a straight line can be curved. For nearly everyone, any attempt to understand four-dimensional space begins with understanding how a three-dimensional creature would appear to a two-dimensional one. One of the earliest and still the greatest of all introductions to going up a dimension is "Flatland" by Edwin A. Abbott. Quite naturally and sensibly, Rucker starts with Abbott's rendition of the properties of Flatland.
   Rucker then moves on to the idea of curved space, where the shortest distance between two points is a "straight line", which is curved by the properties of the space. The space that we occupy is curved by the presence of matter, as Einstein claimed in his relativity theories. Furthermore, movement causes shrinkage in the direction of the movement and the slowing of time, which causes time to become just another dimension of space. As counterintuitive as this may appear, Einstein's relativity theory has been verified over and over again to a large number of significant figures.
   One of the best things about this book is that Rucker has included problems at the end of each chapter. These problems reinforce the concepts of the chapter; it is unfortunate that no solutions were included.
   In this book, Rucker steps the reader through all of the background material necessary to understand relativity and four-dimensional space. With few exceptions, the accounts are understandable to anyone with an understanding of college algebra.



5. I found the book to be both educational, in that I learned great deal about geomtery and the history of diemsions from this book, as well as being fun to read. Both interesting and intellectually stimulating--I find this combination rare. I recommend ths book to anyone interested in the field.

5 comments about Calculus and Analytic Geometry (9th Edition).

1. I recall seeing this book in my living room when I was seven or eight. I'd pull it off the shelf, open to chapter one, look at the word "limit" and feel my mind go blank.
   
   Well, six years later, I took AP Calculus AB and I made a five on the test. Many factors went in to that. This is one of them.
   
   This book feels very comfortable to read. The margins are very wide and there is a good space between the examples and the text.
   
   Professors Finney and Thomas wrote so clearly. Precise and concise. Whenever I didn't understand what was going on in my AP class, this helped to clarify it better.
   
   There are ample problems for you to practice and apply what you've learned.
   
   It's a shame that this isn't really sold anymore. This is the best. If you can get a copy of this, keep it. So good.



2. This is a good, thorough, well-written text. There are no errors that I have found. We used it for a few years for Freshman college calculus.



3. I used the 5th edition.
   This is a standard textbook. It is a bit more detailed than the competition (because it was written by mathematicians, I guess).
   Calculus books all cover standard topics. Where they differ is in the quality of the exposition and the bag of tricks they teach. You want a book with both, and this might be it.
   It's on the recommended reading list of the Mathematical Association of America as the first choice - as in "***" - for introductory Calculus (i.e., not "advanced") (see: http://www.maa.org/BLL/calculus.htm).



4. The approach to the self study of calculus that I've adopted involves working through every problem in each section. This text is excellent for that technique. The authors present the material in sections that usually build upon on another and can be read in about thirty to forty five minutes while working through the examples. The writing is for the most part clear and easy to follow. Every now and then a section seems disjointed. The progress through differential and integral calculus of functions of a single varible (Chapters 1-7) is very thorough and smooth. The approach is both intuitive and mildly rigorous so that the student is not left thinking that calculus is not without rigor. The emphasis is on applications hence engineering and applied physics students will benefit most from this text. The introduction of transcendental functions is divided into two parts with most material in chapter six. The best aspect of this text is the problem set found at the end of each section. The authors have worked hard to build into the problems the material put forward in the text of the section. The problems are designed to reinforce both calculations as well as to provide stimulation to deeper thought (Theory and Examples section). Problems are divided into sets of problems reflecting the divisions in the section. These problems start as computational exercises and progress into applications and thought problems. Every concept can be looked at from different perspectives and the problems are designed to bring out this understanding of the ideas that are presented in the text. The emphasis is of course on the board applications of the concept. In addition the problems vary in the type of function so that there is a constant review of the techniques of approaches to solving the functions.
   An average classroom problem set for a section would be fifteen to twenty problems that would take about an hour and a half for the average student. To work all the problems in a section (range 40-100) takes about six to eight hours.
   I bought this text four years ago and a new edition has since been published. I am very satisfied with both the content and approach. I can pick up most texts on the subject and find that my working knowledge as learned from Thomas and Finney is more than adequate to follow the study.



5. To easily make myself understood in the most important of technical and scientific circles, I have so far found two scenarios that work. One is first thinking through a concept on an emotional, gut, intuitive level (body as vector, particle, motion, mass, or what not) and then just sharing with the APPROPRIATE, well-behaved AUDIENCE. The other approach is to prepare a massive amount of simple, yet accurate terminology and then make brief, accurate statements to the APPROPRIATE, well-behaved AUDIENCE. For either situation where such scientific explanation is needed, this book has proven itself again and again.
   
   Whether I'm imagining myself traveling along a three dimensional surface or learning exactly how the words "rise" and "run" can be used in relation to the concept of a derivative, this book makes it easy to look up concepts by index or table of contents and then review, refresh, and better understand the terminology and symbols.
   
   Of all the required course materials purchased during my 6 year pursuit of a bachelor's and master's, out of the 5-6 thousand dollars spent, this book has proven itself the most worthy, above all others. In comparison, the rest of the required course materials come off as part of some sort elaborate book-store, department, publisher money-making kick-back scheme. With only a few exceptions.
   
   Bottom Line: For learning about functions of one or more variables including calculus vector analysis, this is THE one-stop shopping experience. The selection and presentation topics are its best features. (If a little more depth in problem solving is truly needed, just pick up one of the great Schuam's like the one on Vector Analysis.)

5 comments about A Comprehensive Introduction to Differential Geometry, Volume 1, 3rd Edition.

1. Michael Spivak begins these five volumes stating his modest aim to write the "Great American Differential Geometry book." He surely has. Instead of listing the numerous subjects Spivak treats clearly and beautifully in these volumes, I'd like to call out the delightful travelogue style in which they are written, using history, anecdotes, and opinion to explain, illuminate, and, when possible, motivate the gleaming modern edifice. Spivak's opinions are sprinkled lightly here and there like easter eggs. How could you not love a math book that uses the subtitle "The Debauch of Indices," or dismisses Eric Temple Bell's history as "supercilious remarks of questionable taste"? Also, don't miss the annotated bibliography in volume 5. The fact that legions of professionals refer to these books in their original *typewritten* format [1st & 2d editions] is a further testament to their quality. The third edition is typeset using TeX and, though beautiful, still manages to retain a little of the quirky typewritten appearance. One quibble: I was disappointed to see that this edition did not use Richard Bassein's bizarre artwork [think 70s psychedelic] for the covers; I admit that this stuff weirded me out originally, but have grown to love it -- where else could I see fuzzy trolls in crowns made from Enneper's minimal surface?

   Let Spivak take you "All the Way With Gauss-Bonnet."




2. This book is the first volume of the 3rd edition in a five volume series on differential geometry. The emphasis on this first volume is the study of differential forms and de Rham Cohomology Theory. Spivak also considers two 'bonus' topics: integral manifolds & foliations and Lie groups.
   
   You'll need some prerequisites to get started. For the differential topology material (including Sard's Theorem and Whitney's 2n+1 Embedding Theorem), I recommend Hirsch's Differential Topology. For results on determinants and symmetric groups, I use Hungerford's Algebra, now in its 12th printing. For the general topology material (Hausdorff spaces, Urysohn metrization, etc.), I recommend Munkres' Topology (2nd Edition).
   
   Spivak begins this volume with a review of topological manifolds in Chapter 1. The author provides the basic definitions and gives lots of examples of surfaces and other manifolds. The discussion of manifolds and surfaces continues in the Chapter 1 Exercises. (The author routinely used the exercise set to continue the thread of discussion.) Quick mention of the surface classification theorem is made, although for the proof of this, you'll need to look in Hirsch or Munkres. The reader gets to have fun gluing topological handles onto and cutting disks out of the 2-sphere.
   
   Chapter 2 reviews some of the basic concepts from differential topology, including the fundamental Whitney Embedding Theorem and Sard Critical Point Theorem. Basic properties of smooth maps are also studied.
   
   Chapter 3 studies the general vector bundle and specializes to the tangent bundle of a smooth manifold. The author is keen on the idea that the reader 'grok' (i.e. understand intuitively) the tangent bundle and the associated induced maps and commutative diagrams. The notion of orientability is also introduced.
   
   Multilinear forms and their tensor product are studied in Chapter 4. This is a key building block in the construction of de Rham cohomology. The author gets side tracked a bit with a discussion of differences in classical/modern notion.
   
   Chapter 5 is a very nice chapter on vector fields. Instead of just appealing to results from differential equations (as is usually done) to build integral curves and the flow of a vector field, Spivak establishes these needed results from differential equations using a very accessible integral equations/fixed point argument. Once the flow of a vector field is show to exist (locally), Lie derivatives and Lie brackets are then studied.
   
   Following the integral curves & vector fields material in the previous chapter, the author detours a bit and studies the problem of integral manifolds of dimensions other than 1 along with applications to foliations in Chapter 6. Spivak establishes a basic version of the Frobenius Integrability Theorem and uses examples to motivate the result before diving into the proof.
   
   The basics of de Rham cohomology are established in Chapter 7 and Chapter 8. Alternating and skew-symmetric forms are discussed, although is may be easiest to establish some of the needed results on the symmetric group of permutations after reviewing Hungerford's Algebra. Differential forms and their wedge product are defined, and Frobenius' Theorem can now be restated in terms of differential forms. Two versions of Stokes Theorem are established and this result is applied to integrating forms on manifolds and studying properties of the degree of a proper map of between manifolds. The formal definition of the de Rham cohomology groups is given and some basic calculations are carried out.
   
   The author does something curious with one of the main results of de Rham cohomology, namely the homotopy-invariance property. He starts this with a discussion section in Chapter 7 (not a called out theorem) in which contractible manifolds are show to have zero cohomology in all dimension by an explicit calculation showing all closed k-forms are exact. The results that the author establishes in Chapter 7 for this `one-off' calculation are precisely what are needed to show the more general result that homotopic maps induce equivalent homomorphisms of de Rham cohomology later in Chapter 8.
   
   Chapter 9 is a very nice chapter covering several foundational topics of Riemannian geometry; include the Riemannian metric, geodesics, the exponential map, geodesic completeness and tubular neighborhoods.
   
   Chapter 10 is a short chapter on Lie groups and is something of a detour from the main thread. The author uses the material as a source of application of the material from the first nine chapters.
   
   Returning to de Rham cohomology in Chapter 11, more foundational results from algebraic topology are studied, including exact sequences, Poincare Duality, the Thom class and the index of a vector field.
   
   The book contains many wonderful geometric diagrams which help motivate the material. In most cases, the author is very careful to highlight theorems, propositions and lemmas. Occasionally key results will be 'buried' in a series of discussion paragraphs, which makes referring to these results later on somewhat difficult. The author never, ever calls out or highlights any of his definitions. This can be somewhat frustrating, especially when trying to track down one of these definitions. Fortunately the index to the book is reasonably good.
   
   



3. If you want a book that is rich with examples then this is it. The proofs are, for the most part, clear and concise, thus a person who is learning the material without the aid of an instructor can follow the logic. However, the author could have spent some more time developing topological ideas (thought he does have an appendix section that does a fair job of it) within the flow of the first chapter. I personally find appendices to be too distracting and tend to slow down the flow of the material in a particular chapter. Other than that, this is a great book if you want to learn differential geometry and the theory of smooth manifolds.



4. Spivak's text gets a lot of good reviews, and it is a fine text. In fact, it's one of the best I've ever seen. Read a few other books on the subject, and you'll agree that this is a massive improvement on them. So why only 3 stars? Because there's a much better text on the subject: John Lee's "An Introduction to Smooth Manifolds". This book outshines Spivak's in so many ways. Sure, Spivak is great at motivating major developments in the theory (for instance, he really helps you understand why we need to define a tangent space and why it is the way it is), but he fails pretty bad when it comes to developing some actual theory.
   
   Reading Spivak's text is like taking a stroll, a fresh break from the usual mathematics textbook style. But you also hit a bunch of brick walls on this stroll. It'll be a great discussion, and then you'll come to a theorem. You'll have no idea what its for (some of the time) and you'll struggle to work through its proof (most of the time). Furthermore, the organization is... well, there is no organization! As a result, Spivak can seem to droll on. Lee isn't as good at giving the overall big picture as well as Spivak, but he does everything else exceptionally. Leave Spivak for bed time reading, but do your real studying out of Lee.



5. A lively, terribly ambitious tome on differential geometry. It was meant as a guided tour through the jungles of geometry, from a historical perspective. It is neither easy to read nor altogether successful in it's aim, but it IS comprehensive, masterful, and absolutely unlike all the others. It's kind of a legend since virtually every mathematician seems to own a copy. Full of pictures and history. Reads like a novel.

5 comments about Schaum's Outline of Trigonometry.

1. This book is great for people with thick glasses with tape on them who are presidents of their computer programming clubs but for the rest of us this is not too great. This book is heavy on the theory but lacks application which MOST people need in their classes. No teacher is gonna ask you what a Apollo Axil Centripital Angle(no such thing) is. Too confusing but very organized. These people did a great job in explaining theory. I would not sugest this anyone to use this independently.



2. This book as mentioned in the title is horrible. It is incomplete in many areas, case and point, curve graphing. In many cases the book does little more than introduce the topic and give somewhat bland math questions. This book will not help you through a normal course because it is somewhat babified.
   Now back to the incompleteness. Half-way through the book trigonometric function graphs are introduced (y=sinx and so on). The book very briefly describes aspects of each periodic function in a somewhat scattered manner. All of the information that is given fits on about one 8 1/2 X 11 piece of paper, somewhat terse isn't it?
   This book is not for beginners and is most likely not even for people that would like to brush up on trigonometry. For a more comprehensive edition of a trigonometry tutorial you must turn elsewhere because this book will leave you asking what? huh? how? Perhaps one of the better trigonometry titles out there, and believe me I say this reluctantly because it is also deplorable, is Trigonometry the Easy Way. In conclusion if you have this book return it or if you can't use it only as a way to reinforce trigonometry ideas.



3. I am observing that my test scores on tests on tests involving trigonometery are increasing, thanks to this thin aid. It is thin, yet it is good. That is almost impossible. This book is one of the best created. Trigonometry is hard, and is mentioned and applied almost everywhere. This book is comphrehensive and is both easy and advanced. Everyone should have this. You can have knowledge of math, science, and computers with it.



4. I needed to brush up on my Trig for a calculus class that I'm currently in. While reading the text I found it hard to follow what the author was getting at.
   
   At certain times magic equations would pop out of thin air and you would have to stew over them for hours at a time trying to figure out what the heck they meant.
   
   This book is definately not for a beginner or someone who's looking for an easy quick overview after being away from the subject for abotu 6 years.



5. I used Schaum's for a Summer School Trigonometry course. It doesn't replace a textbook but it covered all the necessary topics effectively and provided a good alternative to derivations and problems found in the text. I relied on it as a backup and a good check on the work being covered in class.

5 comments about Analysis on Manifolds.

1. This is an extremely readable introduction to the subject of calculus on arbitrary surfaces or manifolds. The author develops the subject from the beginning assuming only basic calculus and linear algebra - and then introduces concepts of integration and tensor analysis as the book progresses. Each segment is accompanied by a series of problems that does well to reinforce concepts. All in all, a good introduction.



2. This book covers a natural extention to my course on analysis in R^n--only content similar to first one sixth of the book got treated at the end of the course. Having read first half (just before manifold) in a continuous fashion (span of nearly a week for 4 hours-ish p.d.), I find this one exceptionally clearly-written, (unlike some point in Spivak's Calculus on Manifold), and in content it serves as a detailed amplification on Spivak's (Sp seems to try to keep the proofs elegant and concise more than possible, making a couple of important theorems render indigestible).

   Other noticeable features are:

   1) Mistake-free.

   2) Proofs are truncated into stages with explicit objectives in each, making them well-structured on paper and easy to recall in future, and in this way techniques in proofs become highlighted into some elementary theorems (to get most job done) so that the scope of applications are much widened.

   3) Motivations scattered throughout the book for integrity.

   4) Examples given illustrate as counterexample of how theorem fails with some condition changed or missing.

   5) The level of presentation is uniform throughout the book: strictly speaking, only a good single-variable analysis course (Rudin will do, and also helpful to refer to the overlapping topics) and some motivation are needed, all essential concepts of linear algebra, topology are introduced afresh and uniquely and in the favorable context: either indispensible in later proofs (can act as a practice of it) or results proven motivate its introduction and properties, though some knowledge beforehand can help you to appreciate more, and focus on mainbody.

   6) Each proof is not necessarily the shortest in methods, you may say, but looks most natural and appropriate at this level. Actually, most time it's quite concise whilst, in main theorems, all details are laid out without undue omission. (In contrast, some authors waffle lavishly between substance, but say bare minimum (sometimes unjustified) when it comes to proofs.) Length is also due to partition of proof into stages, which is way clearer in mind than a gluster of dense but appearingly short arguments. And richness and details of proofs themselves are good for getting hang of techniques.

   All in all, Munkres is clearly a master, while reading it, you just feel it cannot get any better. Clarity, style, and organisation put the book far above its peers, and an undeniably outstanding first course in multivariable analysis and manifold alike.

   Although exam-irrelevant, I will surely continue the journey of reading it, in a belief that it'll serve as a solid step-stone to embark on diff geometry or GR with ease, which is my original purpose. hope you can share my enjoyment.




3. One thing i like about this book is the way Munkres presents the counterexamples : why theorem 5.11 wont work if we ease one statement from the hypothesis. Also, the material is accessible and the exercises hard -- both of which, IMHO, are important benchmark for a good math text.

   However, compared to his classic textbook of topology, Munkres did not perform as well in connecting with the readers. The text is very hard to read, and is not suitable for self study. This is useful only as a class text, or as a reference for those who already knew (or passed) the subject.




4. I ploughed through this book years ago. I just noticed that a couple of reviews were only posted this year.
   I thought I would do the same.

   This was a great read by the way.

   I suspect that everyone who picked up this book at some point was looking for a way to circumvent Spivak's terse exposition. I don't blame them.

   ..and then Browder came out with his analysis text. So with advanced calculus in view, these (more or less) recent publications make the subject even more accessible to undergraduates.

   ..and now Spivak doesn't look so hard, all of a sudden.

   Munkres presentation is certainly original. Motivating examples are bountiful, and the figures are excellent.

   The perfect prequel to Boothby.

   Enjoy.




5. I've just finished all but the last half of the last section, which deals with abstract manifolds, and I've done most of the problems in the book. It is important to note that the book only deals with manifolds that are subsets of euclidean n-space.
   
   Anyway, the book is well-written. It demands some maturity and basic linear algebra, analysis and topology. I found only two misprints which are basically of no consequence. Figures abound and are excellent. I've got only two complaints:
   
   (1) The author never mentions that the set of all C^r scalar maps on an open set in R^n is closed under sums, products and quotients. This is used constantly in the latter parts of the book but is never proven. The proof can be found in Spivak's book. The first time this fact is needed is in the proof of the inverse function theorem (det(Df(x)) is a continuous function of x if f is C^r), and also during the construction of a partition of unity. There are more subtle points than this that are left to the reader, but I feel that it should have been proven or given as an exercise if only for the sake of completeness.
   
   (2) The book isn't hard (though it isn't totally easy), but the very last section on abstract manifolds seems harder to read than all the rest of the book, because the author does less to elucidate things here of all places, where more elucidation is needed. He's trying in several pages to generalize results on euclidean submanifolds obtained throughout the whole book to abstract manifolds. I feel that the exposition ought to have been much more thorough here, or much more informal, or that this section should have just been completely omitted.
   
   Nonetheless I feel I'm now ready to take a course in abstract differentiable manifolds. The problems in the book are good, and there are only at most ten or twelve problems in every section, so the reader isn't overburdened as reading the text well and carefully is a task in its own right.
   
   I've profited considerably by completing this book and I highly recommend it.

1 comments about Calculus: Early Transcendental Functions.

1. As a math instructor at a small college, I am occasionally called upon to teach calculus. Therefore, I examined this book for possible consideration as a textbook in our three-course sequence. At over 1000 pages, it certainly has all the material needed for the three-course sequence we offer at Mount Mercy. The first chapter (number 0) of 72 pages consists of a review of precalculus topics. I consider this to be about right in terms of the amount of review material that should be included. However, if I were teaching the class, I would spend around a week on this material. In my opinion there is a reason for prerequisites and the most important one is so that you can cover the material of the current course, not review what should have already been done.
   Chapter number 1 is an introduction to limits, but the approach is intuitive rather than formal. In my opinion, there is not enough of the traditional epsilon-delta approach to the structure of limits. The remainder of the book is largely more of this "intuitive" notion of calculus. Theorems are stated but rarely proven, most of the time there is a statement of the new technique followed by a series of worked examples. While this approach works well, there are times when there is just no substitute for the complete proof of a theorem when it comes to understanding exactly what the technique really is.
   Therefore, if your approach to calculus is to have the students engage in "plug and chug" exercises, then this book would be an excellent selection for a textbook. However, if you are like me and feel the need to inject some occasional rigor, you will either have to provide it yourself or use another book.

2 comments about Differential Manifolds (Dover Book on Mathematics).

1. This book treats differential topology from scratch to the works by Pontryagin, Thom, Milnor, and Smale. The best thing with this book is that you don't too often have to read between the lines. That is, the exposition is detailed and user-friendly, so I highly recommend this book. If something is to be blamed, the price is horrendous.



2. Don't be deceived by the title of Kosinski's "Differential Manifolds," or by claims that it is self-contained or for beginning graduate students. In fact, the purpose of this book is to lay out the theory of (higher-dimensional, i.e., >= 5) smooth manifolds as it was known in the '60s, namely, the techniques of handle decompositions, framed cobordism including the Thom-Pontrjagin construction, and surgery (sometimes called spherical modification). Offhand, I can't think of another book that covers all these topics as thoroughly and concisely, and does so in a way that is readily comprehensible.
   
   The first 4 chapters are an overview of the basic background of differential topology - differential manifolds, diffeomorphism, imbeddings and immersions, isotopy, normal bundles, tubular neighborhoods, Morse functions, intersection numbers, transversality - as one would find in, e.g., Guillemin and Pollack, Milnor's "Topology from a Differentiable Viewpoint," or Hirsch's "Differential Topology," albeit at a higher level and with much less explanation. As the author himself states, with some understatement, "The presentation is complete, but it is assumed, implicitly, that the subject is not totally unfamiliar to the reader." Although I would dispute somewhat the notion that it is complete, as several very important results on immersions and isotopies of Whitney and Haefliger are cited and used repeatedly, but not proved, since, as the author explains, it would have taken the reader too far a field. The reader should also have a good knowledge of algebraic topology (Dold and Spanier are frequently used as references), as well as the classification of bundles over spheres as found in Steenrod.
   
   Since the purpose of the first 4 chapters (about 75 pp) is to develop the machinery of differential topology to the point where the results on handles, cobordism, and surgery can be proved, several topics are briefly touched upon that are generally not encountered in introductory diff top books, such as the group Gamma of differential structures on the m-sphere mod those that extend over the m-disk or the bidegree of a map from a product of spheres to a sphere, in addition to the aforementioned results of Whitney and Haefliger, but just enough is given so that they may be used in later proofs. Most perplexing is Chapter V, on foliations, which has only a tenuous connection to the preceding material and absolutely none to the following. It seems that the author just included it because he felt that knowledge of the subject was essential for a topologist, not because it was necessary for the purposes of this book; it certainly could be skipped, but is worth reading as a brief introduction to foliations.
   
   The heart of the book is Chapter VI, where the concept of gluing manifolds together is explored. Normally, connected sums are defined by removing imbedded balls in 2 closed manifolds and gluing them along the spherical boundaries, but Kosinski instead constructs, explicitly in local coordinates, an orientation-reversing diffeomorphism of a punctured ball and then uses that to identify punctured balls in each manifold. Similarly, handle attachment is defined, rather than by just attaching a handle to an imbedded sphere in the boundary, but instead by again explicitly constructing an orientation reversing diffeomorphism of a (in the 0-dim case) punctured hemisphere and then identifying it with the normal bundle of a point in the boundary of the manifold. In this way, one automatically constructs smooth manifolds without having to resort to "vigorous hand waving" to smooth corners. The downside to this method (which is likely to be unfamiliar to modern readers) is that much time is spent constructing explicit formulas for handle attachments, e.g., in local coordinates, but after Chapter VI the details of these maps are no longer needed.
   
   The last 4 chapters are the most interesting, as all the tools developed in the first 6 chapters are used to prove results such as the existence of handle decompositions for manifolds; the classification of handlebodies; the h-cobordism theorem, proved much easier than in Milnor's "Lectures on the h-cobordism theorem"; the Poincare conjecture for dimensions > 4; Poincare duality (for smooth manifolds only); the Morse inequalities; the existence of Heegard diagrams; the equivalence of the aforementioned group Gamma with the group of differential structures on the sphere and with h-cobordism classes of homotopy spheres (Theta); the Pontrjagin-Thom isomorphism; results on stably parallelizable and almost parallelizable manifolds; conditions under which surgery can eliminate homology in the middle dimension of a framed manifold that is closed or has a boundary that is a homotopy sphere, thus leading to corollaries about when a manifold is cobordant to a highly-connected manifold (such as a sphere); and the computation of some of the aforementioned Theta groups. As you can see, a lot of important results are derived, whose proofs are complete except for a few technical lemmas that are cited.
   
   Most chapters conclude with a section titled "Historical Remarks" or just "Remarks," that explains the history of the development of the subject, including many references. The author himself, now almost 80, had in hand in some of these developments and was personally well-familiar with the giants of 20-c. mathematics who discovered them, such as Thom, Bott, Milnor, Smale, Whitney, Wall, Browder, Morse, etc. The text is also interlaced with exercises, most of which are relatively straightforward.
   
   The book concludes with a new appendix, written last year by John Morgan (my former thesis adviser), on Perelman's proof of the Poincare conjecture. It's just an overview of the proof and feels really out of place, the only connections being that it concerns the Poincare conjecture in dim 3, whose proof for dimensions higher than 4 is one of the highlights of this book, and also that Perelman's proof involves a kind of surgery. This appendix does little to enhance the value of the book.
   
   The book is not without it faults, however. In addition to the above observations about it being too advanced for an introductory text and the incongruity of Chapter V, there are the usual batch of typos: an arrow pointing the wrong way in a diagram on pg 231; a wrong sign in the second displayed equation on pg 102; the switching between indices 0,1 and 1,2 on pg 93; the reversal of the equations for the equator and meridian, as well as the words themselves, on pg. 212; 1/2 in place of epsilon 3 lines above eqn (2.2.6) on pg 128; missing bars over the h in many places in pp. 110-11, as well as omitting the -1 exponent for g in one place; etc. There are also errors of exposition, such as reversing the order of the i and j terms in the definition of M1 and M2 on pg. 211, which leads to factors of +/- missing from subsequent formulas, that fortunately do not impact the results, but do waste the reader's time; this category would also include Case 2 on pg. 214, whose proof is identical to that for Case 1 after a framed surgery and thus unnecessary, or even the 2 possibilities for m, listed in the first sentence for both Case 1 and Case 2 on that page, that are in fact identical, as well as an extraneous condition on n on pg 171. A more serious omission is Theorem X,5.1 (in the notation of the book), which should have been stated in 2 ways, one of which being analogous to Theorems X,4.5 and X,3.4 for use in proving the corollaries 5.2 and 5.3.
   
   Probably the worst mistake is when the term "framed manifold" is introduced and defined to mean exactly the same thing as "pi-manifold," without ever acknowledging this fact, and then the terms are used interchangeably afterward, with theorems about framed manifolds being proved by reference to results about pi-manifolds, and even with the redundant expression "framed pi-manifold" being used in a few places. Moreover, "framed cobordant" is then defined in Chapter X to mean something different than it meant in Chapter IX.
   
   Another group of complaints that I have is with the system of references. First of all, the chapter numbers do not appear in either the running heads or the theorem numbers, so when a result is cited in a previous chapter, the reader must flip back and forth through the book to find it, remembering the chapter numbers for each chapter, or must go back to the table of contents to locate it. Moreover, many theorems from earlier chapters are used without comment, or a reference is made to a theorem when in fact a corollary is being used (or vice versa!). Sometimes a theorem from another source is cited as the justification for a statement, when in fact the author is directly applying a theorem from his own book that just happened to use that other author's result in its proof - citing his own theorem, by number, would save the reader a lot of effort. And then there's the important imbedding theorem of Haefliger that he frequently cites, even though he never actually states what the theorem says! (I had to read Haefliger's paper to verify that it actually could be used to produce the results that Kosinski wanted.)

5 comments about Visual Complex Analysis.

1. This text is innovative to say the least. It has been very helpful to me in my work as an engineer and graduate study in engineering by providing a great deal of insight into difficult topics. One should note that this text includes topics more theoretical than those included in texts such as Brown and Churchill.



2. Like an a Jack Nickelson impersonator, the impersonator can sometimes outdo, maybe even overdo, Jack.
   
   Tristan Needham is impersonating Richard Feynman here. In fact, he does Feynman better than Feynman! Feyman was known for creating graphical represenations from various branches of calculus in a time when such diagrams had become taboo even in physics! Blame it on the pompous retards, once again they had nearly ruined something for everyone else by trying to imposing arbitrary and ultimately false order upon various physical situations. Their objective this time? It was visualization.
   
   Before reading this book, I would definitely have my head firmly grounded in the fundamentals of applied math. I recommend 4 months or so of serious self-study in What is Mathematics by Courant and Complex Variables by Francis J Flanigan.



3. I bought the book after the 2 chapters on complex analysis in "the road to reality" by Roger Penrose left me scratching my head. This book covers much of the same ground, but at a more gentle pace (600 pages vs. 10 or so!). As I'm making my way through it, I feel it does genuinely justice to its stated goal.



4. This book offers the most intuitive and thorough coverage of the relevant concepts for complex analysis I have ever come across. Tristan Needham needs to write more books!



5. "This informal style is EXCELLENTLY JUDGED AND WORKS EXTREMELY WELL... Many of the arguments presented will be new even to experts, and the book will be of great interest to professionals working in either complex analysis or in some field where complex analysis is used."
   ---David Armitage, Mathematical Reviews
   [from the book of the back cover]

5 comments about Trigonometry.

1. I enjoyed this textbook, especially the way some subjects were so well explained. Not only does this text cover a good bit of material, but it also reveals the way in which the author thinks about this subject. I have noticed, both with this text and the previous two which I have commented on, that certain aspects of the subject become much more transparent or understandable when reexamined with a keener mathematical ability than, at least I possessed, when I was first exposed to these subjects in high school. I had no special interest in math at that time. The limited reexposure one has to trigonomety and geometry as one learns new areas of mathematics in college doesn't seem to do justice to these foundation areas of math.



2. The book is well written with clear descriptions, many examples and plenty of diagrams. The book also contains a large number of exercises and therein lies my gripe. There are absolutely NO SOLUTIONS. For self study this is of little use and I have had to revert to the Schaum's Outline for Trigonometry for practice. Two stars is perhaps a bit harsh but I think it important that potential purchasers notice that no solutions are provided for any of the exercises.



3. Don't waste your time. It's books of this sort that bring disinterest and sleep to the eyes of teenaged minds. To think that the work of Euclid (now freely available on Google Books) has degenerated to this third-hand rendition of the foundation of natural existence is awe-inspiring. Like all other books in this class, the discussion never connects the reader to the idea that the symbolization of the relationships of the orientations of the boundaries of certain forms are universal and utterly fundamental. Instead, we get tossed a few line drawings, graphs, number types mixed with graphic symbols, and condescending ministration on whether we got it right or wrong. The author's idea of connecting trigonometry to other fields of math is to state profound meanings like "trigonometry is a part of precalculus, and is related to other precalculus topics". In one exercise, the author commands: "Using a protractor, measure the angles of the triangle as accurately as you can. Do your measurements add up to 180 degrees? Let us now turn our attention to circles." The missing parts of the discussion being: 1) Who invented the protractor? 2) Why do protractors always work, or do they? 3) Why bother measuring with a cheesy plastic nomograph when its the RELATIONSHIP that's of primary importance? 4) Should I be all OCD about measuring accurately because that's what Trigonometry is all about? 5) My measurements don't entirely add up to 180, why is that? 6) Why 180 degrees, and not 37 lurkmons, and can't I make up my own system of relationships? 7) Then what makes the relationship of orientation of certain extensions or boundaries universal? 8) How is it possible to add numbers to shapes, or did you just miss out on presenting to me a massive chunk of the development of the arithmetization of geometric thought? 9) Why are we doing all this? Is there a progression of thought process involved or should I just keep memorizing an apparently jumbled collection of methods extracted from all modes of mathematical approach at face value? 10) Is this a thinking sort of course? Or do I just follow instructions like a drone? Naaah... Let's just skip to circles. No wonder people despise how math is taught, and also, the teacher.



4. So I bought this book for some Trig class I signed up for at the last minute. I didn't know what to expect. I skimmed through the book and saw triangles everywhere. I figured that a few triangles should not be so difficult to figure out- the measurements and all. It's a skimpy little book, but there sure is a lot of info crammed in there. I studied hard. I had dreams of triangles floating around, suspended in the air. I could not get triangles off my brain. Day and night- triangles and more triangles. I think I just got sick of looking at triangles come final exam, because I went out and got hammered to get them off my mind just hours before the exam. I'm not BSing you either. To make a long story short, I took the exam while under the influence of alcohol. It worked, the booze got the triangles off my brain, but the timing was not good because I had to think about triangles in order to finish the darn exam. I had a perfect GPA until that Trig exam. I did manage to pull off an A- on the exam, which surprised the heck outta me. Now I try to not let triangles get the best of me anymore. I'm angry at them, but at the same time I understand them. Triangles deserve respect. Don't be boozin' before an exam. This was a good book and I recommend it highly. But some advice first; don't be square and let the triangles shape your mind- think outside the box and you'll do fine, circle the correct answers if you can, and come at problems from different angles.



5. Books
   This is a very good Trig book. However, there was no offer of an answer book. I will teach a course using this book. Can you give me a site where I can get an answer book? With the answer book I could give 5 stars.