5 comments about Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature (A Touchstone Book).

1. This book was OK---but it had more fractals in nature and not so much in the way of computer-generated fractal art which is what I was looking for. Not bad if you can find it used.



2. This is a fantastic source of images on the subject of fractals, but not a great source of learning. Most books on math and science are difficult for the general reader; few authors (like Isaac Asimov) can make complex things easily understood. But the author of this book is, in my opinion, doing the public a disservice by oversimplifying the subject. The explanations underestimate the public's ability to think, and even include a number of things which are either dead wrong or made-up! The subject of fractals is still new, and there are recently more books available to explain fractals to the general public. Again, this is a great source of images, if that's what you're looking for, but look for another source if you want to undersatnd and appreciate this incredible and important topic.



3. This book says absolutely nothing. It has a few good pictures (the best one is one the cover by the way), but the text is utterly worthless and uninformative. My favorite quote from the book is "Nonlinear means not linear." Really? Don't waste your money. Now I understand why I found it at the used bookstore.



4. This book brings a comprehensive and visually intriguing approach to the study of fractal geometry and the chaos theory. Through thought provoking imagery and discernible explanations & comparisons, John Briggs has sparked my curiosity where I now look more closely at the world around me. I believe this book is intended to captivate those with the ability to visualize and appreciate the aesthetics and interconnectedness of the arts, sciences and the natural phenomena that surrounds us. An insightful & visually stimulating read!



5. Technically this book is good. The images, however, are staggeringly beautiful. I think Dr. Briggs teaches esthetics, which is the study of beauty. Of any book on fractals I have read, this is by far the most beautiful. I really liked this book, so much I donated it to the library of my alma mater.

5 comments about Geometry Demystified.

1. because of the topic coverage. I picked this up because, though a math teacher, I really don't like Geometry. Never have. This book covers some of what a high school student would need in just a few chapters. So, Why aren't I recommending it to students needing help? Because the author completely skips proofs, which is what most students are having trouble with. It also has no chapter dealing with circles and theorems related to them. What it does cover it covers in a highly interesting and original way (why does a stool have three legs instead of four?). It is also filled with topics not covered in a high school geometry course, but which are very interesting on their own. Given the authors other books' titles, this is perhaps "geometry for electricians and hobbyists". If you are someone with bad memories of geometry, but you would like to try revisiting it, then this is highly recommended. It would also be a good outside source for students doing well in Geometry but wanting to read about some higher level topics (including 3 - 4 - and higher dimensional geometry.) The book has loads of multiple choice test questions, so you can see how well you are understanding what you are reading, but it has no detailed solutions in the back--just the correct answers. (techinical point: readers should know that the author teaches polar coordinates "backwards" from the way we teach it in Trigonometry. The form is (r, theta), NOT (theta, r).)



2. This book presents geometry in a straightforward way. Emphasis is on the facts, without getting sidetracked in proofs, (although the purist might object to the fact that proofs are not given). The drawings are relevant and straightforward. The book is well organized and proceeds logically from beginning to end. There are conversational problems with answers in the text, and lots of multiple-choice test questions with answers in the appendix. The test questions are especially good, because they resemble the standardized tests schoolchildren are forced to take these days. This book, along with with a standard school textbook, should make high-school students highly proficient in this subject, and get them ready for more advanced courses such as calculus and trigonometry. Note: I also have the chemistry, physics, and trigonometry books in the Demystified Series and have found them to be of comparable quality.



3. For me, I liked this book since it had tests at the end of every chapter to see where you were in that particular topic. I normally just went through each test and tested myself with the multiple choice questions, then gave my answers to a friend to mark (which was easy on their part so that was another plus, since it's simply multiple choice.) If I had less than perfect (yes, I strive to work my best!) then I go back (without looking at what the answers were to the questions I got wrong) and then checked over the chapter again until I think I knew the new answer.
   
   Any how, I think this concept is good; and marking multiple choice questions are very simple to do as well so it doesn't take up too much time from a friend or family member.



4. I took geometry when I was in high school, and my son will be taking it next year so I was looking for something with which to brush up. This book is not it.
   
   First, it does not have any proofs. This was a huge part of geometry 20 years ago, and I can't believe that they no longer teach proofs in today's geometry.
   
   Second, it goes way beyond high school geometry. The last few chapters cover geometry in 3 dimensions as opposed to just planar geometry, and then 4-Dimensional geometry using time as the 4th dimension, and then touch on how n-dimensional geometry would work. I found this really fascinating, and thus 3-stars, but not exactly the 'geometry demystified' for which I was looking.



5. Well, typically most geometry self-teachning books focus on the topics typically covered in a high school geometry class, which I might add are quite simplistic in nature. I skimmed through this book at my local bookstore out of curiousity (I've had good luck with the demystified series, especially physics, by the same author), and what really caught my attention was the coverage of the elementary high school topics in a lean, mean six chapters with the rest covering hyperspace, warped space, polar coordinates, and basic vector mathematics.
   
   The quizzes are pretty good for reinforcing the concepts...they actually make you think (gee, what a concept)! It'd be nice if there were more of the problems, however (but not as many as Glencoe's books do). The tests really give one a sense of being in a classroom, especially since there's a final exam as well.
   
   This book is a gem, and for $13, you can't really go wrong.

3 comments about College Geometry: A Problem Solving Approach with Applications (2nd Edition).

1. This is one of the few introductory level texts I have seen that gives some of the real flavor of mathematics, without being too challenging for beginning students. The initial section on problem solving is modelled on the famous book by Polya, "How to solve it," and has many simple but thought-stimulating problems. The following sections develop plane and solid geometry with many illustrated problems and interesting historical notes. The final chapters carefully introduce geometric proofs. There are also review sections on simple algebraic manipulations and basic logic, as well as a short section on the implications of alternate parallel postulates. Overall, the text has a well thought out development of basic skills and concepts, and enough interesting tidbits from more "advanced" topics to challenge the imagination of any student.



2. This book is pitched at an extremely low level
   quite beyond anything in the 'math for poets'
   category - often dropping below even that of high
   school. Indeed, the book compares unfavorably
   with the canonical hs text by Jacobs. To give
   just one example, it takes the authors 273 pages
   to get to the idea of cross multiplication [a staple
   in the repertoire of any decent middle school
   student]. In particular, math majors as well
   as anyone interested in the subject should
   steer clear of this and consider instead books by
   Pedoe, Court, Coxeter, etc. If you are looking
   for a problem oriented approach to geometry, try
   the relevant offering in the Schaum's series
   [acknowledged masters of this approach].
   In the meantime, let's not sacrifice any more trees
   for products as weak as this.



3. One of the problems a number of math students face is learning how to think about the problems they face. They simply never develop the necessary tool set that will allow them to understand what the problem is asking and what they should do to attack it. Once they have an answer, they are not sure if they have found the correct answer. This is a fine BASIC text for college and high school students who want to get a handle on dealing with geometry. If you have a deep mathematics background and are looking for an advanced college text on geometry, this is probably not for you.
   
   However, if you want to learn the basics on how to think about geometry and a lot of help on how to solve a variety of geometric problems, this is a terrific text and will be a big help. I enjoy the way the text engages the student from the very beginning and asks him or her to THINK. It isn't a bunch of material to memorize. What the authors do is build the student's understanding through problem solving. If the student will take the time to work the problems and not give up on the problems he or she finds difficult, the understanding will come and will be more ingrained in his or her thought processes than would happen through memorization.
   
   There are lots of geometric drawings, as one would hope, and there are a number of applications of geometry to real life and that should help the student, as well. Again, this is meant as a basic geometry text and can be suitable for a good high school student as well as non-majors in college that want to get an introduction to the basics of geometry.

3 comments about MP: Basic Mathematical Skills with Geometry.

1. The book and the publisher's MathZone show a nice attempt at integrating the power of the Internet with a traditional maths text. The material is for high school readers. You can of course treat the book just as a conventional text, and refrain from accessing MathZone. In this respect, the book is well polished, being in its 6th edition, and very logically internally consistent. As befits Euclidean geometry.
   
   Now if you do want to use MathZone, what to do at the website? Perhaps the most fruitful approach, if you are disciplined enough, is to take those tests offered there. In addition to doing the exercises in the book, of course. The tests are a valuable metric of how well you comprehend the material. The authors and publisher have put a lot of time into MathZone. Go for it!



2. I have the third edition which I used in college back in the mid- 1990's. Of all the math books that I have used, This text is my favorite. The authors of this text really make math fun and easy to learn. I highly recommend this book to all who just haven't understood math and hate it. This book will make math fun.



3. I am still waiting for my product. I emailed the seller asking when I should expect it to arrive. The delivery dates were between July 22 to approximately August 13. The seller emailed on August 13 to say she was out of town and asked if I still wanted the textbook. I replied to the email stating that "Yes, I still need the book". I emailed her two more times since then, no response from her and still no book. I will wait until August 25 to see if the book arrives. If not, I will refute the charges.

1 comments about Very Special Relativity: An Illustrated Guide.

1. Years ago I took a course in symbolic logic. Our professor gave us some proofs created by Bertrand Russell. I loved working out the proofs on my own, and then checking my work against Russell. His clarity of thought was startling after my floundering.
   
   At the end of the course, my professor gave me a copy of Russell's ABC of Relativity, Revised Edition, which I read and re-read for years. (I'm currently using the 4th Revised Edition edited by Felix Pirani, 1985.)
   
   As a general reader, I don't have a deep understanding, but Russell provided familiarity with the fundamental concepts of Einstein's theory. This beautifully graphic book has enhanced that understanding.
   
   Sander Bais uses elementary geometry to illustrate his explanation of fundamental concepts like time dilation. The text and diagrams also illustrate the difference between Newtonian physics, in which time was universal, and Einsteinian physics, in which the speed of light is universal.
   
   The text appears on one page; a spacetime grid appears on the opposite page, with red, yellow and blue arrows illustrating the text. [I wish a CD disc was included to animate the diagrams to aid my studies.]
   
   Russell's ABC helped deepen my understanding of each pair of Bais's pages. For example:
   
   Russell: "If people could leave the earth and travel about for a time and then return, the time between their departure and return would be less by their clocks than by those on the earth: the earth, in its journey round the sun, chooses the route which makes the time of any bit of its course by its clocks longer than the time as judged by clocks which move by a different route. This is what is meant by saying that bodies left to themselves move in geodesies in space-time."
   
   Bais: "It is comforting to see that w{minute} = w if v = 0, and maybe less comforting to see that w{minute} approaches 0 as v gets close to c."
   
   [Imagine here a spacetime grid with red, yellow and blue arrows.]
   
   If those three paragraphs make any (but not too much) sense to you and if you would like to learn more about relativity, I urge you to pick up a copy of Bais's book. I believe it will enhance your understanding of this important subject.
   
   As Russell concludes in his book: "What we know about the physical world, I repeat, is much more abstract than was formerly supposed. ... The final conclusion is that we know very little, and yet it is astonishing that we know so much, and still more astonishing that so little knowledge can give us so much power."
   
   Robert C. Ross 2008

3 comments about Sacred Mathematics: Japanese Temple Geometry.

1. For anyone who truly loves mathematics, this book is a must have.
   Simply put, the book tells the story of sangaku, geometry problems which were painted in color on wooden tablets and displayed at Buddhist temples and Shinto shrines throughout Japan. Most of the sangaku were composed by people from all walks of life-priests, farmers, children women, samurai, etc.-between 1600 and 1900. Approximately 900 of the old tablets have survived and even today one is occasionally found at an abandoned temple/shrine. Tony Rothman has assisted Mr. Fukagawa Hidetoshi, a retired Japanese high school teacher, who is one of the world's foremost experts in sangaku, in producing a beautiful book. Various chapters discuss Japan and temple geometry, the Chinese foundation of mathematics, Japanese mathematics and mathematicians of the Edo period. In addition, the book contains over 200 sangaku problems ranging from very elementary to extremely difficult. The book also contains extensive excerpts from the diary of Yamaguchi Kanzan, a Japanese mathematician, who treked through Japan during the 1800s collecting sangaku problems. Finally, there are chapters on East and West, Japanese attempts to handle differentiation and integration, and inversion. The book contains numerous diagrams which accompany the problems and there are 16 color plates. In summary, this book captures a beautiful form of vanished mathematics which was artistic/religious in nature. Mr. Fukagawa Hidetoshi and Mr. Rothman are to be congratulated for producing a superb book which tells the story of this vanished mathematical/religious art form. Buy your copy today. This book contains enough history, mathematics, art, and religion to keep one's intellect perplexed for years.



2. I am always interested in what Tony Rothman has to say. He is the real deal, teaches physics at Princeton, Harvard, etc., who comes up with revolutionary insights you just can't find anywhere else. SACRED MATHEMATICS is a revelation and a tremendous challenge, another brilliant one in this writer's repertoire.
   
   I began my Rothman studies after reading INSTANT PHYSICS, which pretty much brought me up to speed in what had always intrigued yet baffled me. Then I was amazed with his majestic DOUBT AND CERTAINTY followed by the jaw-dropping, myth-busting EVERYTHING'S RELATIVE. I couldn't get enough so I started backtracking and discovered the Pulitzer Prize nominated A PHYSICIST ON MADISON AVENUE and SCIENCE A LA MODE, where he maybe first established his continual theme of treating science with the skeptical irreverence it often deserves. In between, I discovered articles in SCIENTIFIC AMERICAN, DISCOVER, ISAAC ASIMOV'S SCIENCE FICTION MAGAZINE and THE NEW REPUBLIC, not to mention some weighty scientific papers and reports. Finally, I found his science fiction novel, THE WORLD IS ROUND, with which the movie industry might finally have the tools to do justice.
   
   Tony Rothman is a great and gifted writer and SACRED MATHEMATICS is a beautifully illustrated book of art, religion, history and geometry. I see from his web site that a novel about The Great Seige of Malta is next. I anxiously anticipate that and hope that both APOCHRYPHA and the plays there mentioned will soon be published.
   
   I strongly recommend SACRED MATHEMATICS and, in fact, everything written by Tony Rothman to anyone, who in a world too often full of nonsense and lies, cherishes instead reality and truth. Rothman's voice is beautiful and unique.



3. The last (for the moment) title of Fukagawa&Rothman is really excellent. Not only the printing is superb, but the mathematical content is also outstanding. Strongly recomended to every lover of geometry...

2 comments about Multivariable Calculus with Analytic Geometry (5th Edition).

1. i am using this book in my cal 3 class at this very moment. there are many mistakes in the text and the answers in the back of the book are sometimes wrong or they are factored in a weird way. the text is very easy to understand and with all it's shortcomings i dig this book



2. The book did have some irritating mistakes in the back (no more than the average math book), but I give it a thumbs up for being pretty clear and readable throughout. Some problems I had using the book though are:

   -it doesn't provide adequate examples, too few and not on the same level as the problems
   -towards the end, it loses some coherence by splitting up related ideas into many formulas and notations.

   Overall word, the book teaches pretty well, but a good teacher is really needed to wade through some of the book.

5 comments about Computational Geometry: Algorithms and Applications.

1. Pro:
   (1) Each chapter begins with a practical example. For example, the chapter computing intersections of lines starts with a discussion of a map-making application that goes into enough detail to see how the algorithms they present would be useful. This is a considerable step up from the common practice in algorithms literature of motivation by way of vaguely mentioning some related field (i.e. "These string matching algorithms are useful in computational biology"). This book does a much better job of motivating the material it presents, but if you're primarily interested in the abstract problem, these sections can be skipped.

   (2) Each chapter is relatively self-contained. Feel free to skip ahead to subjects that interest you.

   (3) Surprisingly readable. Unlike most technical material, one can read an entire chapter in a single sitting without missing much. Generally, each chapter will develop a single algorithm for a single kind of problem.

   (4) It's very up to date. This second edition is less than two years old, it includes some new results in the field.

   Con:
   (1) Algorithms are only given in pseudocode. The emphasis is on describing algorithms and data structures clearly and completely. If you're looking for a "cookbook" with code to copy and paste into an application, perhaps O'Rourke's "Computational Geometry in C" would be a better choice.

   (2) There are many important advanced results that are not discussed in the main text. An obvious example is the first chapter, which describes a well-known convex hull algorithm that takes O(n log n) time but algorithms that are faster for most inputs are mentioned only in the "Notes and Comments" at the end of the chapter. Someone interested in lots of gory details would be well-served to combine this book with Boissonnat and Yvinec's more detailed and mathematical "Algorithmic Geometry".




2. This is one of the really few computational geometry books available. It fills a niche and does it decently. However it could be better:
   
   1. The chapter layout is not very good. There are many "revisiting this" and "we saw in chapter so-and-so".
   
   2. The mathematical proofs are often written in a single paragraph full of "English" interspersed with mathematical notation, instead of the tried and true way of numbered equations and one-per explanations. This makes for disconcerting reading.
   
   3. The book in general could have done with more math and code, and less "English", not to mention more and better diagrams -- they tend to be sparsely detailed (ie. a picture is worth only a hundred words). The arrangement of diagrams also needs to be better: some are in the margins, some are in the middle, again not easy and intuitive to follow.
   
   Hopefully a future edition will address this issues.



3. The authors amass an impressive array of algorithms related to finding geometrical properties. Where these algorithms are performed on a computer. The book itself does not advocate any particular programming language. The algorithms are given in pseudocode, and you are expected to manually convert these to code in your choice of language. Given the calibre of the discussion in the text, which suggests that the readers are quite experienced, then this manual step should be easy to most.
   
   There are numerous contexts in which the text might prove useful. Ranging from graphics to GIS to robotics. Thus, there is an entire chapter on the planning of robotic motion. The robot can in general translate and rotate.
   
   Each chapter comes with an exercise set. Which helps make the book suitable as a graduate or even undergraduate text.



4. This book is extremely well written, easy to understand, and actually is the standard text for Computational Geometry classes, as far as I know. The only thing I didn't like about it was that there seemed to be a few errors in some of the pseudocode. But, it's to be expected when publishing a textbook, and I think it'll probably be cleared up in future editions.
   
   Overall, great book. I'd recommend it to anyone taking graphics or a computational geometry class.



5. The authors did a great job of introducing the reader to all the important aspects of the field of computational geometry while keeping it simple and understandable.

4 comments about The Pythagorean Theorem: A 4,000-Year History.

1. The Pythagorean Theorem could rightfully be called the 'Crown Jewel of Mathematics'. For from its truths and intellectual spawn come all the wonders of our modern word--high rises, automobiles, cell phones, interplanetary probes, you name it! Unfortunately, the last serious book on this subject was written over 80 years ago by an Ohio school teacher, Elisha Loomis. Enter Dr. Eli Maor! He has written an absolutely marvelous book about 'The Crown Jewel' that will captivate anyone with a good high school mathematics background. Read it and behold a wonder!



2. Eli Maor is a fine mathematician who has produced some wonderful books on math topics for a general--well, let me say, educated--readership. His book, Trigonometric Delights, is my favorite. It is very interesting and engaging. I want to say "for an educated reader" again, though it seems rather redundant. Why would anyone who didn't know anything about trig and have an interest in the subject even bother to pick up the book? Still, as someone who spent more than ten years in high school math classrooms, I also found his work useful to interest and inspire my students (and myself).
   
   Since the class I taught most often was geometry, I was very happy to see this book on the Pythagorean theorem. I have to admit, as an avid reader on the subject, I was familiar with much of what's here; particularly, the historical development and the more "Euclidean" applications of the theorem. On the other hand, he developed some proofs and problems I hadn't seen before which I found quite interesting.
   
   Overall, however, I didn't find this book quite as engaging as some of his other work. I got the feeling he started off wanted to write a book that would have more universal appeal than some of his other titles. I mean, after all, nearly everyone knows what the Pythagorean theorem is, or has at least heard of it. But there wasn't nearly enough of the "simple" stuff and the last half of the book really goes quite far afield into mathematics without which someone without a pretty decent background in the subject will have a difficult time; particularly since the development is rather sparse in what feels like an aborted effort to keep things simple. Even some of the earlier demonstrations and proofs are a bit difficult if you don't have the background in Greek mathematics which, unfortunately, is often lacking these days.
   
   Still, as someone who loves geometry and has a pretty good background in it, I found much here to like. Any reader who feels confident in their mathematical ability will probably find much here to like too.



3. I loved e: the story of a number, both the story and the mathematics in it. But for some reason this book does not catch the same spirit. It doesn't have the exciting thread of a story that makes you want to turn to the next page, and the many different proofs make it feel like it's a patchwork of items forcing itself to support the topic rather than a natural inspiring thread that helps you see the growth in the mathematics. I found it disappointing.



4. XXXXX
   
   "To this day, the theorem of [Greek mathematician] Pythagoras [which states that the square of a right-angled triangle's longest side or hypotenuse is equal to the sum of the squares of the other two sides, written in the language of mathematics as (c^2 = a^2 + b^2) or, more commonly, (a^2 + b^2 = c^2)] remains the most important single theorem in the whole of mathematics. That seems like a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that it is translated to numbers...In fact, the numbers that compose right-angled triangles [called Pythagorean Triples such as (3,4,5), (28, 45, 53) and (65, 72, 97)] have been proposed as messages which we might send out to planets in other star systems a test for the existence of rational life there."
   
   The above quotation is found in this fascinating book authored by history of mathematics professor and author Eli Maor. (Note that the above quotation was not said by Maor.) It catches the importance of this deceptively simple theorem, a theorem children's writer Lewis Carroll (who was also a mathematician) called "dazzlingly beautiful."
   
   What did I learn from this book? Answer: there's a lot more to the Pythagorean theorem than (a^2 + b^2 = c^2)!! Maor may be the first author who has examined all the mathematics, history of mathematics, and physics books and collected just the material directly and indirectly related to the Pythagorean theorem.
   
   The result is that Maor has brought the long history of the Pythagorean theorem back to life. Sometime around 570 BCE Pythagoras proved (notice I said "proved" and not "discovered") a theorem about right triangles that made his name immortal. He also pondered the workings of the universe and tried to relate its workings to the laws of musical harmony. In the subsequent centuries, this theorem was used and developed by others such that it has become central to almost every branch of science, pure or applied. After twenty-five centuries, this theorem was expanded and thrust into four-dimensional space-time by Albert Einstein to formulate his own picture of the universe.
   
   Yes, there is simple mathematics in this book. To understand it, all you will need is some high school algebra and geometry and a bit of elementary calculus.
   
   Do you have to follow the mathematics found in this book? NO. Personally, I found that you could skim, even skip the mathematical parts and still not lose the essential flow of the main narrative. (Actually, the more difficult mathematics is relegated to the book's appendices.)
   
   Throughout the book are diagrams and even some pictures to enhance its main narrative. As well, there are eight pages of colour photographs found near the book's center.
   
   A feature of this book is that it contains "sidebars." These are brief sections (there are ten) found at the end of some chapters that usually focus on some aspect of the Pythagorean theorem. My two favourites have the following titles: "The Pythagorean Theorem in Art, Poetry, and Prose" and "Four Pythagorean Brainteasers." You don't have to read each sidebar.
   
   Another feature of this book is its chronology. It more or less summarizes the main events in this book in chronological order. This chronology begins in the year 1800 BCE and ends in the year 1996.
   
   Finally, a note on the book's cover picture (displayed above by Amazon). It shows the detail or "zooming in" of a beautiful larger 1649 picture called "Allegory of Geometry" by artist Laurent de la Hyre (displayed on this book's inside back flap). The book's cover picture zooms in on several geometric figures, the one on the top left showing Euclid's proof of the Pythagorean theorem.
   
   In conclusion, this book is essential for anyone that wants to be familiar with the four thousand year history of the Pythagorean theorem. I leave you with some actual lines from Gilbert and Sullivan's "Pirates of Penzance:"
   
   "I'm very well acquainted, too, with matters mathematical,
   I understand equations, both simple and quadratic,
   About Binomial Theorem I'm teeming with a lot o'news,
   With many cheerful facts about the square of the hypotenuse."
   
   (first published 2007; list of colour plates; preface; prologue; 16 chapters; epilogue; main narrative 215 pages; 8 appendixes; chronology; bibliography; illustrations credits; index)
   
   <>
   
   XXXXX

5 comments about Optimization by Vector Space Methods (Series in Decision and Control).

1. Optimization by Vector Space Methods, by David Luenberger, is one of the finest math texts I have ever read, and I've read hundreds. Many years ago this book sparked my interest in optimization and convinced me that the abstract mathematics I had been immersed in actually would be applicable to real problems. Since then, Luenberger's book has inspired several of my graduate students. I merely lent them my copy, and Luenberger did the rest; he drew them in by carefully laying the foundation for an elegant theory, with just the right mix of formalism and intuition, and opened their eyes to the beauty and practicality of abstract mathematics. Anyone with an interest in higher-level mathematics (beyond multi-variable calculus, say) would benefit from exposure to this finely-crafted book. I daresay, the rampant math anxiety that is so prevalent in the West would be substantially reduced if more authors would take such meticulous care in presenting their material.

   The format of Luenberger's book is also extremely appealing in a way that I cannot quite put my finger on. The typography and illustrations are inherently crisp and inviting; they draw you in. There is nothing at all superfluous or gratuitous in this book. It is utterly to-the-point, methodical, and above all, clear. The techniques are developed starting from an elementary treatment of vector spaces, then proceeding on to Banach spaces and Hilbert spaces. Along the way, Luenberger introduces convexity, cones, basic topology, random variables, minimum-variance estimators, and least squares, among many other things. There is a recurring theme of duality, which can be used in a way analogous to the inner product of a Hilbert space. In particular, the familiar projection theorems of Hilbert spaces can be echoed in simpler normed linear spaces using duality, which Luenberger motivates and covers beautifully.

   The book also covers some of the standard fare of functional analysis, such as the Han-Banach theorem, strong and weak convergence, and the Banach inverse theorem. However, Luenberger never wanders too far off into abstract nonsense; around every corner lay tantalizing application of these ideas to optimization. Luenberger first explores optimization of functionals then covers constrained optimization, which builds upon concepts such as positive cones and Lagrange multipliers. The optimization methods themselves have endless applications in fields such as computer vision, computer graphics, economics, and physics. Indeed, the list is effectively endless as optimization techniques pervade math and science.

   I'm certain that the appeal of this book is helped immeasurably by the inherent beauty of the subject matter. Hilbert-space methods are lovely in themselves--they possess a structure that engages one's geometric intuition while at the same time admitting convenient algebraic properties. Once you are in the habit of phrasing problems in abstract settings such as Hilbert spaces, it forever changes how you look at things; you cannot help but look past the clutter to the essence of a problem (or, at least try very hard to do so). While this material is not nearly as abstract as, say, category theory, it nevertheless hits a high point in mathematics--a point more people ought to experience.

   If you've had some exposure to optimization methods, or need to apply them in the context of computer vision, graphics, or finance, to mention just a few areas, then I urge you to take a look at Luenberger's fine book. It too hits a high point in clarity of mathematical writing. Combine beautiful theory with endless applications and lucid writing, and you have a winner of a book.




2. This book is a timeless classic, filled with extraordinarily powerful mathematics and applicable to just about every serious subject area. Luenberger did a masterful job of writing a book that will "unravel the spaghetti" seen in most other books. The visual perspectives he provides to seemingly abstract ideas are the key.



3. The exposition is pretty clear and the book has a good number of worked non-trivial examples. At $40 this would be a great book, but $100 for a PAPERBACK book written 30 years ago is a bit ridiculous. The first 1/4 of the book is also a (very) basic introduction to functional analysis which, if you have had any contact with this subject before, you will probably skip making the book quite short.



4. Professor Luenberger unites many areas of optimization using a few principles from functional analysis. The explanations are clear and the proofs are compact and elegant. This book is your tool for understanding the deep connection between linear programming, convex optimization, game theory, optimal control and series approximation (e.g. Fourier series).
   
   Luenberger's book has over 1300 citations as of March 2006. In my opinion, the material in this book is essential for any graduate student or professional who intends to contribute to the literature in optimization or optimal control.



5. Although Luenberger's book is probably the only treatment out there that combines optimization with applied functional analysis, I have to say that this book does not do a very thorough job of explaining the abstract connection between these two very different fields. The book is well written in a grammatical sense, however, one of the major shortcomings I have found is that 99% of the proofs Luenberger gives throughout the text are either incomplete, left to the reader, or just plain confusing. (for example, the proof given for the Hahn-Banach Theorem is pretty much useless). Furthermore, the examples the author gives are mostly one-liners and do not offer any kind of clarification to the important theorems given in the text. Also, Luenberger does use concepts from Lebesgue and Riemann-Stieltjes integration and mildly says at the beginning of the text that it is not necessary for the reader to have a background in these fields to be able to understand this textbook. This is a very dangerous assumption, since most of functional analysis is based on Lebesgue measure theory. As a result, Luenberger's treatment of Lebesgue integration is merely "washed over" and, in some parts, is flat out wrong.
   
   Overall, don't buy Luenberger's textbook expecting to be able to learn optimization and/or functional analysis just from this book. I would recommend consulting Rudin's Real and Complex Analysis and Wheeden's Measure and Integral before pursuing this book. Luenberger's book is very much a serious applied mathematics book written by a non-mathematician which merely brushes over very deep results in optimization theory.