No comments about Dynamical Systems VIII: Singularity Theory II. Applications (Encyclopaedia of Mathematical Sciences).

No comments about Numerical Approximation of Partial Differential Equations (Springer Series in Computational Mathematics).

2 comments about Single Variable Calculus, Volume 2.

1. Pretty good with respect to sample problems, theoretical explanations of topics, and good application problems as well.



2. The problem with this book is their explanations: they often explain nothing, and their example problems, which are often ridiculously simple while the accompanying problems are ridiculously complex and difficult, leaving you with little idea of how to solve the problem or even what they want you too accomplish. Add in the assumption factor: the book references equations or mathematical concepts not in the book and simply 'assumes' you have the knowledge, and the shortcut factor (examples sometimes condense several steps into one line, leaving no way to separate what did what) and you have a bad way to learn math.
   
   Here's the biggest clue. I often resort to Google to learn the math when attempting the assignments. If I use other sources to learn the math, why should I (or you) ever buy this book?

No comments about Abstract Harmonic Analysis: Volume 2: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups (Grundlehren der mathematischen Wissenschaften).

2 comments about Ordinary Differential Equations in Theory and Practice (Classics in Applied Mathematics).

1. This is a very good book in that it treats both the numerical and analytical aspects of ODE's in a complementary manner. Also, the examples used give insight to how ODE's are used and solved in mathematical models



2. This book is a classic and was my second exposure to differential equations at the university. It's written very nicely, which fits the two authors, who I both had as professors, and does what it is supposed to: give insights into the theory of ODEs and what classical methods there exist to solve them. It disusses the different types of ODEs accurately and clearly, and then continues on how to solve them. The book does not include any algorithms you can easily code or snippets of code, which can be argued to be both good (so no reference is made to a particular programming language) and bad (not really practical). I believe that this book can be of value to many scientists and engineers, who want to understand what happens and what is important when solving ODEs.
   
   One negative thing: the book does not include any geometric methods for Hamiltonian systems, although it briefly reviews these dynamical systems near the end. But as an introduction it's the best you can buy.

1 comments about Real Submanifolds in Complex Space and Their Mappings.

1. The book presents an excellent discussions on some of the most active research fields in the subject of several complex variables. It includes a fairly complete list of references and an accurate account of the historical backgrounds of the materials.

No comments about Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, Volume 7 (Monograph Series on Nonlinear Science and Complexity) (Monograph Series on Nonlinear Science and Complexity).

No comments about Perturbation Theory for Linear Operators (Classics in Mathematics).

5 comments about Basic Calculus: From Archimedes to Newton to its Role in Science.

1. If you are deeply curious about the amazing ability of mathematics to define, describe and predict the physical world and its behavior (incl. the solar system) you will be thrilled with this book. It concentrates on the essence of the matter, basic calculus, and includes real-world applications set within the context of some of history's most important scientific questions.

   The author clearly demonstrates that he not only possesses a great curiosity, fluency, and appreciation for the subject but also thrives on imparting these things to others. He has provided a great deal of supplementary information on his web site including a detailed description of the contents, scope and focus of the book.

   The Solution Manuals ARE available from the author simply by e-mailing him at: hahn.1@nd.edu

   Most calculus books make some compromise in presenting the material. In the case of a thoroughly rigorous text, that compromise most often means sacrificing historical context, intuitive understanding, and real-world application (even though the book may be "exercise-rich" with contrived examples). Basic Calculus successfully navigates a difficult (and different) course, focusing on these commonly sacrificed areas and effectively presenting the pearls of calculus knowledge without delving too deeply into eye-glazing minutiae. In lucid and interesting style, it accomplishes exactly what it sets out to do - it imparts the essence of the matter, in context.




2. This splendid book aims to develop calculus from within its rich historical context and to demonstrate its power across a range of disciplines. The author succeeds admirably. Two hundred pages devoted to key ideas in the history of mathematics and science lead smoothly into calculus as we know it today. The remaining three hundred plus pages cover the usual topics, but with attention given to an extraordinary spread of interesting problems in science and business. The explanations of concepts and notation are as lucid as any I have encountered in a basic calculus book.

   Because one of the distinguishing features of Basic Calculus from Archimedes to Newton to its Role in Science is its historical dimension, something should be said about the criticism of one reviewer that the book oversimplifies the history by using modern notation. Yes, Hahn does tidy things up. (Very nicely, I might add.) But what else can anyone really do? As Hahn notes, Leibniz's cryptic first work on calculus - Nova methodus pro maximis et minimis, itemque tangentibus...calculi genus - bewildered even his friends, the brothers Bernoulli. These famous mathematicians found Leibniz's article "an enigma rather than an explication." Hahn could try to unriddle the Nova methodus for us, explaining in detail all the fuzzy concepts and strange notation that baffled the Bernoullis. But that hardly seems the thing to do in a basic calculus book. Better to do just what Hahn does - seize on the essential ideas and use everything now at a mathematician's command to bring them into a clear light. Hahn has an excellent sense of just how far to go. The result is a truly extraordinary book that will amply reward readers looking for something special.




3. How can we say that introductory courses in calculus at universities are meaningful if the students are never involved in math as professionals? What motivation can we offer them for studying it? This textbook, by Alexander J. Hahn, provides an outstanding answer to the question backed up by the author's precious teaching experience at the University of Notre Dame.

   After reading the text, the reader will start to see calculus as a gift by our ancestors that helps us to analyze practical daily problems: calculus as a culture to be passed on to the next generation. Firstly, as the author says, "this text could as well have the title The Story of Calculus." As we read it, we find ourselves reliving history with the great persons like Archimedes, Descartes, Leibniz and Newton. We feel the activity and wisdom of the characters close-up, and we even experience their joys and sorrows as if they were our own. In a way, this book is a historical novel. It shows what calculus looks like as a critical tool that has helped to clear up the mysteries of the universe. Secondly, "the purpose of this text is to demonstrate its broad and formidable informative power." As the author explains, calculus enables us to designing telescopes, to read nuclear clocks, to design suspension bridges, and to understand the interior ballistics of rifles, the rocket equation, gravity, and the expanding universe. Economic subjects, such as banking, CPI, market mechanisms, cost analysis are also covered with full explanations. Books with such range and depth are rare indeed. It is easy to understand why the author received an award for teaching excellence.

   Of course, "the emphasis is always on the careful development of the mathematics and information that it provides", and most of the topics of first-year calculus courses (including differential equations) are covered (but partial differentials and double/triple integrals are not). The exercise section of each chapter contains advanced explanations of historical, scientific, and mathematical topics, and is organically integrated with the text. The total number of the problems in all the 15 chapters is close to 700. With its many figures and illustrations, as well as full derivations of the equations, this text is also suitable as a supplementary or a self-study manual.

   I strongly recommend Basic Calculus to those who have doubts about "the usual math training" which sometimes makes us feel like machines (not humans), as a rare and engaging view of mathematics from a different angle. I have found the contents of Hahn's textbook ideal for my students in general physics and calculus courses at Hosei University, Tokyo, and I am now completing a translation of this book into Japanese in collaboration with my colleague Professor Ichimura.




4. What this book is not is a traditional calculus text. It covers a lot of traditional topics, but not in a familiar way. It is not terribly rigorous, nor does it need to be. It is designed to fill the first two semesters of calculus. There are a LOT of books that do this in the traditional way, that is they scare the life out of the student :-). It is my belief that this book will take a lot of the mystery out of calculus, since it develops the subject in the context of applications. I also think that most students will find the approach engaging. There are plenty of practice problems at the ends of the chapters, and some are quite challenging.

   The focus of this book is not to present calculus as a theory, a thing which most students are simply not prepared for at this level. Rather it is to present calculus as the pragmatic development of methods to solve certain classes of problems. In this regard it does a fantastic job. Along the way the students's algebraic, geometric, and trigonometric skills are all tested and firmed up.

   The notion of the limit, such a mystery to most freshmen (and, truth-be-told, to many upper-level undergrads) is given a strong intuitive thrust right from the beginning.

   If you want more problems, get the Schaum's outline book and read them side-by-side.




5. Basic Calculus: From Archimedes to Newton to Its Role in Science
   is a beautifully done text. It is very clearly written and
   logically organized, tracing the development of calculus with
   many interesting examples from the physical world and man's
   quest to understand the physical world. The text is concise
   and so readily understood as to be elegant. Finally, all of the
   solutions to the exercises are given at Professor Hahn's internet
   site. Its address is www.nd.edu/~hahn/ One way to remember the
   website is that the letters "nd" are for Notre Dame, where Dr. Hahn teaches.

5 comments about Complex Variables.

1. Flanigan's book starts at the beginning, and it covers some central aspects of complex function theory, elementary geometry, harmonic and analytic functions.
   
   The central topics are (in this order) calculus and geometry of the plane, harmonic functions, complex numbers, integrals, power series and analytic functions, and the standard Cauchy-and residue theorems, ending with a brief chapter on conformal mappings.
   
   The book was published first in 1972, but reprinted since by Dover. It is suitable as a text or as a supplement in a standard course in complex function theory, late undergraduate level, or beginning graduate. While it contains the standard elements in such a course, we note that a systematic treatment of power series comes relatively late, in Chapter 5, beginning on page 194. Some readers might want to begin with that. Flanigan concludes with the Riemann mapping theorem.
   
   Of other Dover titles on the same subject we recommend the books by Volkovyskii et al, Schwerdtfeger, and Silverman. Review by Palle Jorgensen, August 5, 2006.



2. The two basic facts about analytic functions are these: they satisfy the Cauchy-Riemann equations and they are conformal. These are two lanes of a two way street between complex function theory and potential theory. The Cauchy-Riemann equations imply that the real and imaginary part of the function are conjugate harmonic functions. Harmonic functions are functions that satisfy the Laplace equation, and they thus describe steady-state heat flows and such. So facts about heat flows translate into facts about analytic functions. For instance, if no heat is generated inside a circle then the temperature at some interior point will be some sort of smeared out average of the temperatures along the circumference, so the maximum temperature in the disc must be somewhere on the boundary. This carries over to analytic functions: the maximum of the modulus of an analytic function on a disc must be attained on the boundary, and, if the function is never zero we can invert it and find that the minimum of the modulus must be attained on the boundary. And from here we obtain a quick and easy proof of the fundamental theorem of algebra: if a polynomial is never zero the minimum of its modulus on a disc must be attained on the boundary, but as the disc is taken larger and larger, the modulus on the boundary of course goes to infinity. QED. Thus we have a sort of physical proof of a very formal mathematical theorem. And there's plenty more where that came from. Integrating along a closed loop sort of corresponds to integrating the heat flux across the boundary, and poles correspond to heat sources, so if there are no poles inside the loop the influx and the outflux will be equal and the integral will be zero, and in general the net flux will be determined by the strength of the sources (i.e. residues of the poles). All this because the Cauchy-Riemann equations turned analytic functions into physics. But we promised a two way street, although admittedly there is less traffic in the opposite direction (flows around obstacles could have evened the score but are omitted). The key here is that harmonic functions are conformally invariant, and analytic functions are conformal, so an analytic function applied to a harmonic function produces a new harmonic function. An indication of the usefulness of this fact is this: the Dirichlet problem for the disc is easily solved by the Poisson formula but remains hard for a general domain, but because any domain can be mapped to a circle by an analytic function we can, in principle, solve the general problem by simply mapping the circle solution to our new domain. In conclusion, we very much applaud the idea of a harmonic function approach to analytic functions, but we also feel that this book is a bit stiff and does not sufficiently exploit the power of the intuitive and geometric ideas involved; we strongly recommend Needham's wonderful book for these aspects.



3. Just like finding two solutions for the same problem gives additional insight, Flannigan was able to give me an additional insight to the whole subject of complex analysis with his approach that is drasticly different from any other book on the subject I know.
   
   Used this book during Mathematics Ph.D. studies to prepare for a preliminary exam in complex analysis. The unorthodox approach helped me get another angle of the subject. In particular I would note the introduction of harmonic functions before analytic functions and using "real analysis" techniques to prove "complex analysis" theorems like the maximum principal and the Liouville theorem for harmonic functions. Before the number "i" is even introduced, you already know these theorems for analytic functions once you define them as a pair of harmonic ones.
   
   The student friendly tone of the author was a blessed interchange from the standard graduate books like Ahlfors, and for a fraction of the cost, it makes a wonderful buy for a self study book for the complex Ph.D. exam.
   
   I would not assign it as the course book for undergraduate students taking a first course in complex analysis (which is what it is intedned for) though. It would be frustrating for a student to ponder through Green's theorem and real analysis material, which is by no means introductory, for 100 pages or so, when what he or she needs and/or wants to be doing is to deal with the algebra and geometry of complex numbers.
   
   Overall, an awsome book if you already tasted the subject and want to get a better feel for it. If it's your first time, stick with the traditional books.



4. Check this sentence from the preface:
   
   "The Cauchy Integral Theorem is thereby an easy consequence of Green's Theorem and the Cauchy-Riemann equations. Goursat's remarkable deepening of the Integral Theorem is discussed, but is not proved."
   
   Such an upfront motivation of physicality in Complex Variables or Analysis is more than a rare find, it can only be justly defined as heavensent. A gift from the gods! This miraculous text absolutely deserves its many 5 star reviews. (other readers should still figure out the previous text that has an even clearer presenation of the physical foundations than Flanigan [to think i gave it 3 stars!])
   
   Perhaps the most Physically Intuitive text on Complex Variables Ever (here's the first full paragraph): "We examine the the geography of the xy-plane. Some of this will be familiar from basic calculus (for example, distance between points), some may be new to you (for example, the important notion of 'domain'). We must also consider curves in the plane."



5. The other reviewers have already done a terrific job describing the content. I'll just add that while a profesional/accademic mathematician might find this book a bit informal, as an amateur I really appreciate its appeal to intuition and the author's tendency to review the meaning of terminology for a few of it's subsequent uses immediately following it's introduction. Not a difficult read, but a working knowledge of calculus for one variable is prerequisite. Seems to be out of print as of this date -- I've had good luck with used book purchases through Amazon marketplace from dealers with positive ratings better than 95%.