2 comments about Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs).

1. This is probably the most clearly written self-contained book on the basics of differential geometry. The author does a great job explaining the ideas behind purely mathematical 'dry' constructions. On the other hand, everything is defined correctly and precisely. A very readable and useful book with the perfect combination of formal math. and intuition. I would recommend it to students in theoretical physics area, together with the Nakahara's fantastic book.



2. This text is phenomenally easy to read and well organized. The author starts you on a journey by first explaining the importance and power of classifying manifolds namely by certain invariants preserved by certain mappings ( diffeomorphisms ).
   
   For example, like Euler, we could count the number of holes in the surface and using this combinatorial method we are led to homology theory.
   
   Or like Gauss, we could use a differentiation and integration to come up with the idea of curvature as an intrinsic feature of the surface.
   
   Modern approaches use differential forms to represent homology and cohomoly groups.
   
   The author also deals with fibre bundles demonstrating their importance in analyzing manifolds specifically how the number of fibre bundles possible with given Lie groups as structure groups over the manifold can be answered by characteristic classes such as the Chern and Pontrjagin classes. The use of differential forms is indispensible.
   
   Perhaps the most satisfying aspect of this book is that it clarifies the notions of connection, connection form, curvature, curvature form for manifolds and fibre bundles.
   
   There are plenty of exercises to boot.

1 comments about Calculus.

1. The book is a thouroguh cover of calculus, but doesn not have many good example problems to llok at. It is a good book, but if you're a person who needs to seee example problems, this book is not for you.

5 comments about The Golden Ratio: The Story of Phi, the World's Most Astonishing Number.

1. Several years ago I prepared a review for amazon on this book. Since that time there have been many others to contribute. There are those like me who found it fascinating and gave it five stars, others that gave it a 4 or a 3 because they quibbled with the author over some mathematical issues and finally agroup that really hated it and found it boring and gave it only 1 or 2 stars. Some of those in the third group claim to be mathematicians but thought the book had too detailed. I don't see how a true mathematician could not love this book. Here is what I wrote that I still believe.
   
   The book is 253 pages and 10 appendices about a number called the golden ratio. I give it 5 stars. It is a book for mathematicians and non-mathematicians alike. The first question I asked was how can an entire book be devoted to one number. Well Beckman wrote a book about the number pi and certainly that was interesting. There is a lot to say about the geometry of pi and many mathematical and statistical properties it has. Some including the Buffon needle problem are related by Livio in this book. He contrasts pi to the golden ratio (phi) which also has geometric and mystical properties. The quantity pi is a transcendental number meaning it is not the solution of any algebraic equation. On the other hand phi is algebraic as it is the solution to a quadratic equation.
   Other strange properties of phi are:
   1. If you subtract 1 from it you get its reciprocal
   2. Add 1 to it and you get its square
   
   To see the marvelous algebraic and geometric properties of phi you need only scan through the 10 appendices. Scan through the book and the pictures show you the many artistic properties related to phi.
   
   Although algebraic phi is an irrational number. By applying the quadratic formula to its solution (see Appendix 5 in the book) you will see that its solution involves the square root of 5. Pythagoras and his followers in ancient Greece were said to have discovered irrational numbers (a natural consequence when you study right triangles) and hid this knowledge from the populous.
   
   Phi is defined by Euclid as the "extreme and mean ratio". As Livio quotes Euclid " A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". This leads to an equality of proportions that yields phi=1.6180339887 rounded to ten decimal places.
   
   Livio also discusses the relationship between the ratio and our concept of beauty (i.e. the quality of the perfect face). It is also interesting that in his new book on the impossibility of solving the 5th degree polynomial by radicals Livio relates the Galois theory of groups to concepts of symmetry. There he also attributes our perception of besuty to symmetry.
   
   If you have the time read the book thoroughly. Write a review that adds to what has been said if you like. Or skim through the pages and appreciate the artist properties of phi along with its algebraic and geometric properties. Read about fractals and myths. Enjoy this wonderful book!
   



2. I happened to notice that he says Babylonians found the general solution for the quadratic. General solution of the quadratic was given by Bhaskara. The author has not read Fibbonaci's book. Fibonacci himself said in the preface that he learnt new math from India. Fibonacci numbers were found by Hemachandra. there were many other errors...I would not recommend to my students



3. One of the best books I've read. It is an in depth study of the Golden Ratio...the history, purpose, relationship to other concepts. I am intrigued by math, art, and science and found this book very, amusing. You will need a basic understanding of high school math to fully appreciate some of it. Oh, by the way, the author shoots down most other author's claims that the golden ratio has been used in classic architecture and art. Superb job Mario Livio!



4. I bought this book with a thirst to know about this number phi. I did learn about the number phi. However a large part of the book was devoted to instances where various people thought the number phi was present but the author spent considerable time developing the opinion or fact that phi was not influencing this or that particular instance. I got REALLY tired of that.
   
   For me, the first chapter and a half or so and the last two chapters were the meat of the matter for my interest. The book was worth it for the last chapter.
   
   I think that the author would have been better to write a book titled "Why Is Mathematics So Effective?" That seemed to be the central question that really drove the author.
   
   I don't regret reading it. I just feel it wasn't really the book I signed up for.



5. subtitled: The Story of PHI, the World's Most Astonishing Number
   
   that's alot of hype for the irrational number 1.6180339887... . the author barrages the reader with hyperbole. 'mysterious', 'astonishing', 'amazing', 'wonderful', 'beautiful', 'fascinating', 'curious', 'crucial', 'unimagined', 'divine', etc. etc. not just astonishing, but "the World's Most Astonishing Number".
   
   horse feathers. the people who are 'fascinated' by this are the same who freak out when they see 11:11 on a digital clock; the same who have 'lucky' numbers; the same who fear Friday the 13th.
   
   in fact there are more 'crucial' AND more 'astonishing' numbers. how about 0 or 1 or 2 or 10 or infinity? i guarantee you that if we changed our everyday number base from 10 to , say, 13 that the wheels would fall off of this old bus. now THAT is 'crucial'. and 'divine'? please! what could be more 'divine' than 1? maybe 2 :-) . 'astonishing'? 1 is 'astonishing'. it factors into EVERYTHING! it's everywhere and in everything. 0 doesn't factor into anything. these are more 'astonishing' than phi.
   
   Dali knew how to capitalize off of frenzied hype, so he threw together the "Sacrament of The Last Supper" and when phi's superstitious cultists found out that it featured the 'divine' proportion they took care of turning that ugly, mediocre effort into a 'divine' icon.
   
   the author is supposedly a PhD? whatever.

5 comments about A Course of Modern Analysis (Cambridge Mathematical Library).

1. Neville Watson's mother was Mary Justina Griffith, the daughter of the rector of Ardley in Oxfordshire. Neville's father was George Wentworth Watson who was a schoolmaster, but is more famous for his work as a genealogist. He played a large role in the publication of The Complete Peerage, a 13-volume database of the British peerage, generally accepted as the greatest British achievement in the field of genealogy. The first edition was published in London between 1887 and 1898. George and Mary Watson had two children, a boy and a girl, the eldest being Neville.

   Neville was educated at St Paul's School in London where he was very fortunate to have the outstanding teacher of mathematics Francis Macaulay. He mixed with equally outstanding pupils, for Littlewood, less than a year older than Watson, was also a pupil at the school. Having won a scholarship to Trinity College, Cambridge, Watson matriculated there in 1904. At this time there were three young fellows of Trinity all of whom had a major influence on Watson's mathematics. They were Whittaker, Barnes, and Hardy. Perhaps the one from this trio who had the greatest influence on him was Whittaker, despite the fact that he left Cambridge in 1906, two years after Watson began his studies there.

   Watson graduated as Senior Wrangler in 1907 (meaning that he was ranked in first position among those who were awarded First Class degrees), completing the Mathematical Tripos in the following year in the second division of the First Class. He won a prestigious Smith's Prize in 1909, becoming a Fellow of Trinity College in 1910. This was particularly pleasing to him for he had a great love of his College, and throughout his life he collected prints of the College and of previous Fellows.

   After election to his Trinity fellowship, Watson spent four further years in Cambridge before leaving to take up an assistant lectureship in University College, London. From 1918 to 1951 he was Mason Professor of Pure Mathematics at Birmingham. He married Elfrida Gwenfil Lane, the daughter of a farmer from Holbeach in Lincolnshire, in 1925. They had one son.

   Watson worked on a wide variety of topics, all within the area of complex variable theory, such as difference equations, differential equations, number theory and special functions. He is best known as a joint author with Whittaker of A Course of Modern Analysis published in 1915. The first edition of the book has only Whittaker as an author. In 1922 Watson published The theory of Bessel functions which was another masterpiece. Titchmarsh wrote of Watson's books (see for example [2]):-

   Here one felt was mathematics really happening before one's eyes. ... the older mathematical books were full of mystery and wonder. With Professor Watson we reached the period when the mystery is dispelled though the wonder remains.

   One piece of work undertaken by Watson deserves special mention. It involves the problem of wireless waves, which were quickly found to travel long distances despite the fact that theoretically they should not have been able to follow the curvature of the Earth. A mathematical model had been constructed where the Earth was represented by a partially conducting sphere surrounded by an infinite dielectric. Such a model had been used by Macdonald, Rayleigh, Poincaré, Sommerfeld and others. Although Watson was not interested in how best to model the situation, he was, however, very interested in using his expertise to determine mathematical solutions to the given model which others might then check against observations. He obtained solutions to the problem in 1918 which showed conclusively that the model was not a satisfactory one.

   In 1902 Heaviside had predicted that there was an conducting layer in the atmosphere which allowed radio waves to follow the Earth's curvature. This layer in the atmosphere, now called the Heaviside layer, was only a conjecture in 1918 but it was suggested to Watson that, having shown the previous model to be wrong, he now look at the model resulting from the postulated Heaviside layer. Watson showed that if the layer was about 100 km above the Earth's surface and it had a certain conductivity, then indeed the solutions obtained closely matched observations. That Heaviside, and Watson, were correct was confirmed in 1923 when the existence of the layer was proved experimentally when radio pulses were transmitted vertically upward and the returning pulses from the reflecting layer were received.

   Watson undertook a major project by examining in detail Ramanujan's notebooks, extending his results and supplying proofs. In fact he wrote twenty-five papers relating to results in Ramanujan's notebooks, and he spent many hours making a hand written copy in wonderful script of all the notebooks. He enjoyed numerical calculations and spent many happy hours doing numerical work on his calculating machine.

   He was elected to the Royal Society of London in 1919. In 1946 he received the Sylvester Medal of the Royal Society:-

   ... in recognition of his distinguished contributions to pure mathematics in the field of mathematical analysis and in particular for his work on asymptotic expansion and on general transforms.

   Watson was also very active in his support for the London Mathematical Society. He served as secretary from 1919 to 1933, president from 1933 to 1935 and acted as an editor of the Proceedings of the London Mathematical Society until 1946. The Society awarded him their De Morgan Medal in 1947. The Royal Society of Edinburgh elected him to an honorary fellowship.

   We find a little of Watson's personality described in [2]:-

   He was the university's expert on the timetable; students with unusual combinations of subjects usually had to be referred to him for advice, and for many years after his retirement the dates of the academic year were governed by the "Watsonian cycle". ... He took great trouble with the style of his letters and his conversation and enjoyed finding a pungent phrase to express his points of view or his criticism ... he made no secret of his aversion to cars, telephones, and fountain pens. He loved trains - whose timetables were as familiar to him as those of the university lectures - and unusual stamps.

   Article by: J J O'Connor and E F Robertson




2. If I could, I would give this book ten stars. When I first sat down to read it, I couldn't believe what I was seeing. This is the only book I have ever seen on complex analysis (or any scientific field for that matter) in which the authors cover so much material (everything from residues to integral equations to elliptic functions and MUCH more) and yet manage to make the whole text fit into a framework which is relatively easy to follow, even for someone completely new to complex analysis. Moreover, the majority of the many hundreds of excercizes in this book range from moderately to nail-bitingly hard, and encourage a true understanding of the material being covered. I would reccommend this book for ANYONE who has mastered basic calculus and analysis and wishes to begin learning complex analysis and the theory of special functions. The book's coverage of the following topics is especially noteworthy: The gamma function (the book uses the INFINITE PRODUCT as the basic definition), the hypergeometric function (and the confluent hypergeometric function), bessel functions (a field in which G.N. Watson was a leading expert), and the Weirstrassian and Jacobean elliptic functions and theta functions (I LOVED the intuitive development of the theory of the elliptic functions, which is made to parrallel that of the trigonometric functions, which are of course familiar to the reader). I would ESPECIALLY recommend this book for those pursuing SELF-STUDY (although it is NOT for the mathematically weak-of-heart, but no book on the topic is), as it is quite self-contained and readable for a book on complex analysisis. Once you buy it, you won't even think to complain about the high pricetag, because you will be way too absorbed in the math to think about anything else.



3. Excellent for reference and technique. Don't expect calculus on manifolds or forms. But work the "old school" problems and the payoff will be tremendous" For those who want to a more modern (geometric) supplement Spivak is good and there is a book called Visual Complex Analysis that is by Needham (spelling?) that is quite good.



4. Classic text. Good encyclopedic source of mathematics - pure & applied. Definitely not a picture book and takes a bit of digging to find what one needs. Typical abbreviated index found in British texts of that era.



5. The product I recieved is in terrible conditions, very used and dirty, the book must have been a new edition (in blue color) but this one is grey. I send you some photos in another message in order to show you the bad conditions of this book.

2 comments about The Foundations of Mathematics.

1. This book was written for those inquisitive/advanced high school students (6th formers in England) to give them an introduction to the number systems, formalization in mathematics and the concept of proof. Stewart is well-known for his clear writing and this book is an excellent example.



2. Stewart and Tall have written some excellent mathematical texts, and this book is one of them. It deals with the basics of mathematics and treats them in a clear but thorough way. Although it is probably not essential, it is a good accompaniment to anybody starting a mathematics degree.
   
   They take the approach that one should already be familiar with an idea (via examples) before seeing its definition. This works well, although sometimes the effect is that the pace is slow and the exposition is drawn out.

5 comments about Analysis by Its History (Undergraduate Texts in Mathematics).

1. I wish there had been books like this when I was at (high)school! It is one of those rare books that bridge the yawning gap between the popular personalised history books that are so inspiring to the young mind, (eg. E.T.Bell's "Men of Mathematics", Kasner & Newman's "Mathematics & the Imagination" or Kak & Ulam's "Logic and(?) Mathematics") and the terse, somewhat desiccated university text books. This can leave the undergraduate not fully appreciating the motivation for exhaustive rigor and also losing any perspective of where the abstract theorems and lemmas are ultimately distilled from. This book links the historical characters, controversies and challenges with the modern techniques that gradually emerged to deal with the pathological behaviour of sets, series and functions. It would be a mistake to confuse this book, as some of your reviewers have done, with the many first-year undergraduate texts that are available. It could be regarded as a sophisticated high school book that gives a real flavour of how the classical problems are treated in modern rigorous style, or alternatively as a colourful motivational aid to early undergraduate analysis courses. I hope that the publishers encourage similar ventures in other branches of the subject, for instance algebra, differential & integral equations, probability and perhaps even quantum theory.



2. Chapters 1 and 2 treat classical differential and integral calculus. This is a disorganised mess of historical and mathematical tidbits. It's not a great place to learn calculus, but it's good side reading since there are many interesting topics, some of which are often neglected in today's books: continued fractions (!), complex functions already on page 56, an interesting section on differential geometry, Euler-Maclaurin summation, etc. The authors also have the very commendable habit of including charming facsimiles of figures from original works.
   
   Chapter 3 "Foundations of Classical Analysis" and chapter 4 "Calculus in Several Variables" are almost completely ahistorical. The "by its history" part of the exposition is restricted to some scattered superficial remarks, including silly nonsense such as that if Leibniz had know of the intricate progression of theorems needed for a modern proof of the "fundamental theorem" then "he might not have had the courage to state and use this theorem" (p. 239). And in another parodic misuse of the historical perspective, the authors introduce Descartes's folium merely for the purpose of practising the determination of stationary points (p. 322)---of course, Descartes introduced the folium for a much more interesting purpose, but to learn that story we must look for an "Analysis by Its History" book worthy of its name.



3. This books gives a unique approach to Calculus using its historical development. The most notable feature of the book is that the order of topics is reversed from what has become standard in current textbooks. It begins with the analysis of areas and volumes. This is followed by derivatives, continuity, and the notion of function. This is the order in which analysis developed, but not the order one would follow if building understanding of the subject from a foundation upward. Historically, the foundations were laid last.
   
   The book is not intended as a history of analysis. It is rather intended as a textbook or reference in which the topics are presented in historical order. The historical background is intended to give insight into a modern view of the subject. It accomplishes this admirably.
   
   The book is filled with examples, quotes, vignettes, historical background, computer graphics, and copies of original documents. Special topics are interspersed throughout. The book gives us a fresh and envigorating view of Calculus. It is an invaluable resource.



4. I return to this book again and again just out of sheer pleasure. The depth of scholarship of the authors shines through on every page and the choice of historical material is fascinating.Topics like compactness and uniform convergence can here be seen to have arisen out of genuine necessity-they are not (as would seem from other books)mere names in a standard syllabus. If you have any mathematica interest at all, take this book on holiday and sleep with it under the pillow to extract more from it by osmotic pressure overnight.



5. This very interesting book contains very good historical perspectives on analysis. If you want to know how things like trigonometric functions, logarithms, infinite series, differential and integral calculus and differential equations come about (but written from a modern viewpoint), then this is the book for you. It is not a book for casual reading like E T Bell's Men of Mathematics, but the reader will learn a lot of college and undergraduate mathematics along the way.

2 comments about A First Course in the Numerical Analysis of Differential Equations (Cambridge Texts in Applied Mathematics).

1. This is an excellent reference and textbook for someone hoping to go beyond the introduction to numerical DE found in any of the standard numerical analysis textbooks. It is not a research monograph, but is also not easy reading. It has already become a fairly standard reference in the literature because of its complete coverage and further references to more specialized sources. I have used it as the textbook for a graduate course on numerical differential equations. I highly recommend it for that purpose and as a reference for someone doing independent reading.



2. A very informal style of writing with lots of explanation. He doesn't skip large steps like in the old-fashioned terse style of math texts, which makes it very readable, though some readers may not like it. Not very rigorous, but he's upfront about it.
   
   The original version from 1996 has quite a few errors, and the author maintains information on errata on his website. The most recent reprinting has corrected most of these errors. So, even though there is only a single edition, some versions have errors and some don't. So, BEWARE BUYING USED EDITIONS because they will most likely be from an earlier printing and thus have more errors. I assume the new version on amazon is the corrected version.

3 comments about Real and Functional Analysis (Graduate Texts in Mathematics).

1. I've read several analysis books and this is one of the better ones that I have read. It covers a variety of interesting and useful topics and the exposition is clear. It's presentation is a bit more abstract than some others starting with some functional-analytic concepts before doing integration in that framework. However, if you want to study stochastic analysis, getting in this frame of mind will definitely help your understanding of stochastic integration. For a truly thorough understaning of the subject, I recommend purchasing this book as well as the somewhat easier "Lebesgue Integration on Euclidean Space" by Frank Jones - the two together cost about the same as Royden, Rudin, or the terrible book by Aliprantis.



2. It drove me up the wall, in my first course on measure and integration, that integration was first done for positive functions, then for real functions by writing them as a difference of positive functions, then complex functions in terms of real and imaginary parts. Why couldn't you just integrate real-valued functions
   intrinsically, without the silly decomposition into positive and negative parts?
   
   After that course, I found Lang's book. What a blessing to see that you can just integrate in infinite-dimensional spaces right from the start. I can't understand why virtually all books on integration theory still succumb to the "positive functions first" approach.



3. Up to my knowledge, this is the only book that constructs the Lebesgue integral for functions to a general Banach-space instead of the real numbers (thus saving us from the unnecessary and esthetically dissapointing construction through positive and negative functions).
   I don't know how Lang does it, but eerytime you'll pick up one of his books, you'll marvel at the beauty of mathematics !

4 comments about Topics in Matrix Analysis.

1. Horn and Johnson's MATRIX ANALYSIS AND TOPICS IN MATRIX ANALYSIS are true classics (like Knuth's Art of Computer Programming). You will find classic theorems and lemmas in matrix theory and linear algebra here along with their proofs (some of these are not found elsewhere).

   TOPICS IN MATRIX ANALYSIS contains a lot of stuff including LMI's, Kronecker and Hadamard products of matrices and their properties etc. I found this book indispensible when I was studying Semidefinite Programming.

   Both these books are now available in paperback (cost around 30+) dollars each. I have recently purchased both copies and can only strongly recommend them to anyone else.




2. This book is a sequel to, and a worthy successor of, "Matrix Analysis". The latter was directed mostly at methods applicable to solving generic matrix problems. Whereas the present book takes a more focused view. Its topics should be understood as more specialised. Like the case where a matrix might be upper triangular and in positive or non-negative definite form. Or where certain assumptions might be made about a matrix's eigenvalues.
   
   There are some nice theorems proved about the spectral properties of various types of matrices. More to the point, the book has many useful ways to actually find the eigenvalues of such matrices. Where these methods might be more efficient than the generic methods of the earlier book.



3. This book is an excellent reference for researchers in the fields of Matrix Analysis, Numerical Analysis, Theoretical Linear Algebra, etc. I am doing currently some research involving Matrix functions and generalizations of M-matrices and I use this book all the time. Some important features of this book include the facts that 1) it is well-written 2) It is clear 3) It very useful to researchers and graduate students. It is one of those books which you can't stop reading once you start.



4. If you are communication engineer and exploring the world of channel estimation, MIMO, etc. and you need to understand the mathematical approach, you need this book. The book covers the SVD issue, which is most important for many applications.

1 comments about Problems in Mathematical Analysis III (Student Mathematical Library,).

1. THE WHOLE SERIES IS SIMILAR TO THE TRHEE VOLUME DEMIDOVICH BOOK, WHICH I CALL
   
   THE KING OF ALL MATHEMATICAL PROBLEMS!!!!!!!!!
   
   Problems in Mathematical Analysis (Hardcover)
   by g. yankovsky (Translator), B. Demidovich (Author
   Publisher: mir publisher; 4th Printing edition (1976)
   ASIN: B000GTC2GA
   
   YET DEMIDOVICH IS MORE COMPREHENSIVE AND COVER MORE MATERIALS. GREAT FOR PUTNAM ATHLETES!