No comments about Mathematical Methods for Engineers and Geoscientists.

5 comments about Partial Differential Equations: An Introduction.

1. I have never commented on a book, up until now... and I do so only because I don't think that this book gets enough credit.
   
   People have complained Strauss may not have explained some proofs in as much detail as he could have, people complained that he didnt give enough examples, I think this is more of a problem with the readers than the writers. If you need someone to hold your hand through every step and detail, I think you should reconsider why you are studying what you study.-
   
   I am an undergraduate at NYU, one of the best research institutes for PDE's. I thoroughly enjoyed reading this book, it gives an amazing description of what PDE's are, how to solve them, and how they are used in science. One thing I REALLY enjoyed about this book was it did not do what many other books do: first dive into seperation of variables and focused only on that. Instead Strauss shows how to solve first and second order equations without boundary conditions, giving a very elegant prose doing so!
   
   However, I think much of the problem that people are having with this book is that it's not a "one-size fits all." (Which I don't think any book can be!) If you are a Scienctist or Engineer and just want to learn PDE's to solve problems in science.. find another book, because this book is not the book for you.
   
   That being said, if you are Mathematics student or interested in a more deep study of PDEs this is really a good book for you. You definitely should have taken Calc. 1-3, Linear Alegbra, ODE, and I recommend one semester of Analysis (for function spaces) before tackling this book, that is what I had, and I loved this course.
   
   PDE is a difficult subject/course and Strauss does an amazing job at explaining it, if someone like me can get PDEs so well from this course, than I seriously believe that complaints about this book is due to fault in the readers and not the writer.



2. I have only read bits of this book, but every time I read it I come to the same conclusion. I think the previous reviewers have highlighted the key problem with this book. A previous reviewer wrote: 'If you are a Scientist or Engineer and just want to learn PDE's to solve problems in science find another book, because this book is not the book for you.' There is some truth in this. However, in the preface Strauss wrote: 'This is an undergraduate textbook. It is designed for juniors and seniors who are science, engineering or mathematics majors.' It would appear that the book is simply not very suitable for the wider audience it was intended for - in particular it is too advanced for less experienced undergraduates. Also, despite containing numerous applied examples, these examples are dealt with so briefly that they would make little sense to a scientist or engineer who has not already studied the applied material in depth on other courses. This is somewhat inevitable as the book attempts to include a wide multitude of examples, and so its strength is also its weakness. This book is not a 'one size fits all' but was clearly advertised as one and ends up simply belittling itself in trying to be one thing whilst actually being another. It is a very good reference, however, and is very valuable to scientists and engineers who have already studied much of the material in an applied context, but the text is somewhat disjointed as it reads more like a catalogue of PDEs rather than as a 'how-to-do-it from first principles' manual. The book looks attractive, but every time in the past that I started reading it I soon put it down again! However, I have begun to read the book seriously from cover to cover. Maybe a taught course that is constructed around the book would work, indeed it is a valuable source of examples, but this is not a good book for self-taught study, except perhaps to those with more maths than a typical junior undergraduate. I would however recommend it as a must in the library of anyone who deals with PDEs on a frequent basis, or who wishes to teach the subject. In short - great for the right audience but the right audience is not exactly the one advertised in the preface! (This is a common failing of many textbooks). Since I began to seriously read this book from cover to cover and I am finding it fairly straight-forward, but then I have studied PDEs before. However, those who do not like the book should be comforted, because at first I never liked it either - but give it a chance and read it carefully and remember that many of the examples will not make complete sense until you have studied the science behind them in other courses. In this sense, the text is a valuable reference to senior science and engineering graduates. I think for someone new to the topic the text probably skips the odd crucial sentence of explanation. However, for an advanced mathematics, science or engineering undergraduate the strength of this book is that it puts everything together and so is a valuable reference and valuable to consolidate what you have studied on other courses. However, because it is so all-inclusive, it would need to be several hundred pages longer if it were to describe everything clearly from first principles (even assuming competence with ODEs). So, if you still don't like it, then come back to it in a year or so. I shall update this review if, as I continue with the text, my view changes.



3. As an intro to PDE book, it's simply put, terrible. The author must have forgotten that he was writing a book aimed at undergraduates. No simply examples, not a single one. The book jumps straight into theory right from the first chapter and beyond. The exercises involve very intricate proofs and there are no simple computational exercises, none. If you've taken math analysis or some kind of easier PDE course, then maybe you'll like the book, but if you're like me and you go from ODE's to PDE's than the transition is way too rough.



4. I've spent the past seven years or so working on analytical and numerical solutions to the various PDEs that price financial derivatives. My focus has been very much on getting and extending useable answers. When it comes to PDEs specifically, I'm mostly self-taught, but my background in real variables and functional analysis is solid (one and two graduate semesters, respectively, at Pennsylvania and Harvard).
   
   In some ares of mathematics, a single, classic text whose exposition is top-notch, or that inspires despite its exposition, manages to cover the field well. For example, I'm thinking of books like Halmos (Measure Theory), Rudin (Real and Complex Analysis), or Segal and Kunze in real variables and integration theory; Lax or Reed and Simon I (Functional Analysis) in functional analysis; Lang (Algebra) in algebra; and Kelley (General Topology) or Milnor (Topology from the Differentiable Viewpoint) in topology. (Full citations are in Listmania; see my Amazon profile.)
   
   I've yet to find a single reference for PDEs that addresses all of my questions, but several books taken together manage nicely. I jumped around in these books when I was learning the subject, and I'm convinced that such cherry-picking is the best approach for PDEs, since the field is so broad in theory and applications. The downsides, of course, are expense and potential confusion from conflicting notation and approach.
   
   Ignoring just for the moment the vast area of approximate solutions by discretization and perturbation techniques, here's who seems to be best for what, when the problem involves linear PDEs:
   
   :: Need quick intuition: Farlow, Myint-U and Debnath, Brown and Churchill;
   
   :: Need more theory: Stakgold (Green's Functions), Evans, Folland;
   
   :: Need help on modeling: Strang (CSE), Stakgold (BVPs), Haberman (Applied PDEs), Farlow;
   
   :: Don't understand how concepts relate: John, Levine, Garabedian, Strauss, Carrier and Pearson (PDEs);
   
   :: Can't find tough enough exercises: Carrier and Pearson (PDEs), Kevorkian;
   
   :: Need inspiration or deep intuition: Courant and Hilbert (both volumes), Zeidler (Nonlinear Functional Analysis 2A [Linear Monotone Operators], Applied Functional Analysis [especially AMS 108]).
   
   I've ranked books very subjectively within each category on a composite of things like relevance, completeness, clarity, and ease-of-use. And I should stress that I'm no doubt ignoring many fine favorites purely through unfamiliarity.
   
   WHERE DOES STRAUSS FIT? I repeat, all of these books address each of the needs in some measure, but no one is adequate for all. The terse treatment and broad coverage in Strauss are great for tying concepts together and revealing their logical relationships. This is especially evident in the superb Chaps. 1 and 2-3, as well as in Chaps. 9 and 10, which treat the Cauchy Problem and BVPs in space, respectively.
   
   Chapter 11's discussion of eigenvalue problems, and particularly their asymptotics, is remarkable at the book's level but nowhere near that in Garabedian or especially that in Courant and Hilbert, which is the original synthesis of work beginning with Weyl to which Courant and Hilbert each contributed in important ways. (The notes to Sec. XIII.15 of Reed and Simon IV [Analysis of Operators] have the history of Dirichlet-Neumann bracketing, the main technical advance.) Both of Stakgold's works also discuss this problem but not as well as Strauss.
   
   I've done very little teaching (and I wasn't very good at it!), so my views should be taken with a grain of salt perhaps larger than usual. If I chose Strauss as a text, I'd have to believe either that my lectures would fill in the gaps that Strauss so clearly has or that other books on my syllabus could take up the slack. If you're having trouble learning "from Strauss," the problem may lie not with the book, but with an incomplete course, since Strauss is, in many ways, only a good set of summary notes. Again, it's good as far as it goes, but it doesn't go the whole way; that's why you need the other books.
   
   CHOOSING A SINGLE REFERENCE. If I were packing for a desert island, I'd take Levine and Garabedian, since everything I need can be backed out of their presentations with some effort. In many ways these books can be thought of usefully as a set. Both are written at an intermediate level, meaning that techniques less sophisticated than those involving function spaces are fair game. Levine spends 700 pages on separation-of-variables, Fourier analysis, and transform methods, applied to parabolic and elliptic equations in general and the diffusion (heat) equation in particular. Garabedian picks up just where Levine leaves off to treat the Cauchy Problem for hyperbolic equations and the Dirichlet and Neumann Problems for elliptic equations. His book is also roughly 700 pages in length and like Levine's is a model of clarity.
   
   Although both books have been available for some time, basic approaches to the three classic, second-order, linear equations and their variants--the gist of a first course--have changed so little in the past 50 years that publication date may not be as much a factor in selecting a single reference as it would be in some other areas. Indeed, it's well worth reading Fourier's original memoir on heat conduction, possibly modulated by a modern treatment like Carslaw and Jaeger (Conduction of Heat in Solids) or Crank (Mathematics of Diffusion). Levine (Chap. 13) also contains a technical precis of Fourier's original approach.
   
   If I found that I needed greater depth, meaning function spaces, I'd turn first to Courant and Hilbert or to either of Zeidler's state-of-the-art books. The treatment in C&H is profound and downright majestic. Many have spent a productive professional lifetime in these books, and C&H-2 comes close to being the sort of reference I describe in the second paragraph. Because of its age, it's possible to see in the books' discussion much of the intuition of Hilbert and Sobolev spaces that was later covered with layers of rigorous abstraction. Zeidler's discussion of that abstraction is simply the clearest that I've found anywhere. It's extraordinary that any author works as hard as Zeidler to convey mathematical ideas, and for this reason his books are among my favorites across all topics.
   
   TRANSITIONING TO DISCRETE APPROXIMATIONS. If I also took Gil Strang's new book to ease the transition to discrete approximations and eventually building and evaluating numerical code, I'd forget the rest of the list without worries. Indeed, as Courant mentions in the preface of C&H-2, there was to have been a brief third volume dealing with discrete approximations for existence and construction of solutions. Strang would stand nicely in its stead, if not for existence, then certainly for construction.
   
   Actually doing numerical work is another matter entirely, of course, and I've given some idea of the books I've found useful for these problems in my brief review of Chung's book on computational fluid dynamics.
   
   HOW ABOUT PERTURBATION SOLUTIONS? Finally, if I closed my eyes and pretended that I'd always separate variables, so I only needed to worry about perturbation solutions of hairy ODEs, I'd toss in Bender and Orszag and feel pretty good about analytical approaches. (If you're lost in B&O, and it does have its moments, try Holmes, which is a more accessible survey at less depth. And if you need to begin at the beginning, go to Lin and Segel [Chaps. 6-7. 9, and 11], which treats the ideas you need, before you get buried in algebra.)
   
   If I just couldn't bring myself to make that assumption, I'd take Kevorkian and Cole (Multiple Scale and Singular Perturbation Methods), which deals in part with perturbation solutions of PDEs directly, and Verhulst, which is a bit longer on intuition.
   
   The brave might also consider Van Dyke (Perturbation Methods in Fluid Mechanics), which deals specifically with singular perturbations of the Navier-Stokes equations, their many variants, and the other equations of fluid mechanics. To go this route you'd have to believe that you could adapt Van Dyke's results to whatever problems you ran into, which can be real work. Hinch's short book is useful as a complement.
   
   ONCE MORE...WITH FEELING! If I were learning things from scratch again, I'd sleep with Farlow under the pillow and Garabedian under the bed, regardless of what textbook my instructor had chosen. For the careful Farlow raises at least as many questions as it answers--purists have my guarantee that they'll hate it. You need to supplement Farlow with greater depth in your areas of interest, and in most instances Garabedian cleans things up nicely without doing violence to the concepts.
   
   An added plus is that since the books are reprints, published by Dover and AMS Chelsea, respectively, their cost is quite reasonable, even though Garabedian is beautifully printed and library-bound (would you believe sewn-in signatures and useable inside margins?).
   
   This review is a lot longer than I'd first intended, and its recommendations are in many ways idiosyncratic but certainly worth their cost. Partly, I think that's a function of the field itself. There seem to be as many approaches to learning PDEs as there are backgrounds and interests. The diversity of sources is likewise broad, and their quality is quite high. The beauty and power of the subject have lured many first-class mathematicians, like a striking number of the authors mentioned above, into writing basic texts. In the end someone's treatment will answer your questions, pretty much no matter what they are, if you just have the patience to look around. After enough looking, of course, you'll find you can answer many of your own questions.
   
   One of the things that makes real-world PDEs in whatever field such fun is that getting an answer is all that's important. It doesn't matter what books you use, what willing help you receive from whom, or how you reformulate a problem to make it more tractable, as long as a result that answers the real question (sometimes a rather elastic notion) eventually emerges from your efforts. There's much to be said for learning the field in the same no-holds-barred way, and I hope my remarks can get you started in that direction.



5. you should first read about the Green functions in some other books to appreciate Strauss's way of explaining things. He is clear, and easy to follow. The way he presented the material is not so advanced, Nevertheless the concepts are beyond undergraduate. This book is one of the best books in PDE's I've seen, specially chapters 7-11 and most specifically he did an awesome job for the Green functions.

2 comments about Schaum's Easy Outline: Calculus.

1. I'm a graduating senior and i just bought this book to get myself ready for college this year! It's an amazing book and is easy to understand.



2. This book should help you in reviewing your notes that you took in class. As the title of the book suggests, this is a crash course for general calculus.

   Figures and graphs in the book should help you understand many different problems and how their solutions are derived. You should expect to solve many of the problems in the book yourself, on paper, also. That is the only way to understand mathematics; by doing it yourself. This book should help you get started in the right direction because of the ample examples present in it.

   Do not expect to get rid your other calculus books and use this as the sole text for reviewing Calculus. This book is meant to be a companion to your other calculus textbooks, not their replacement. As good as it may be, you still need detailed explaination of rules and theories about Calculus, which are not explained in this book.

4 comments about Introduction to the Calculus of Variations.

1. This book is a more thorough treatment of the calculus of variations than some of the other books on the subject that you could buy. It's bit more expensive... I especially reccomend this book if you like Sagan's other book "Boundary and Eigenvalue Problems in Mathematical Physics". If you like his style of writing, and I do, he again shows nice touch of explaining the material in writing and then with examples. A good book with a nicely written and more thorough treatment of the subject (unless you want a full price text that is)..



2. I used this book as a compliment to a Theoretical Mechanics course. The section on Hamilton's equations is especially well written. Although probably more mathematically rigorous than anything I needed, the style is so silky smooth that anyone interested in mathematical physics will surely enjoy it. And it will be a surprise for many to find that the "proof" of equivalence between the Lagrangian and Hamiltonian formulations presented in most texts is half incomplete.
   The book is complemented by good examples, clear notation and quite a number of graphics. Of course, proofs and arguments are absolutely rigorous, but well explained. This is a mathematics text, after all. I strongly recommend it, as well as any other of Sagan's books.



3. This book is neither too naive nor too brief, still I was able to read more than 80 pages in one day, simply because it is written very clearly. An excellent introduction in Calculus of variations, very concise and very clear. Some context is devoted to control theory, which can be considered as a generalization of calculus of variations.
   The part on field embedding and Hilbert invariant integral is perfectly wriiten, actually the best that I have ever encountered.



4. I really like ("plain") books that are written in the definition, lemma, proposition, theorem, and examples style. Sagan does enunciate propositions, theorems, etc. but he does that after giving proofs for the things he will state latter. I simple don't like this style. Another point that I really dislike in Sagan book is the notation. He doen't change the notation, for instance, when he talks about a function y and the y coordinate. This is a little bit boring. Moreover, the exercises aren't stated clearly (I always have to look at the text, many pages before the exercise, to know what the exercise demands to be done). Besides this the book is very complete. A good but that could be better.

No comments about Nonlinear Analysis in Chemical Engineering (McGraw-Hill chemical engineering series).

5 comments about Linear Differential Operators.

1. Some mathematics and physics writers stand head and shoulders above the rest. Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos. A joy to read, if you are both mathematically and verbally inclined.



2. A very intuitive (geometrical) exposition of matrix calculus, adjoint problems, bilinear identity and Green's function (and more). If you really want to understand these concepts, read this masterpiece!



3. This book has material I've found in no other book. Lanczos is a pleasure to read -- his writing is clear, elegant, and entertainingly opinionated. I've liked every book of his that I've read.



4. Somebody writen:
   "Some mathematics and physics writers stand head and shoulders above the rest. Goldstein...Liboff...Morrison...Morse and Feshbach...and Lanczos. A joy to read, if you are both mathematically and verbally inclined."
   
   I think some mathematics and physics writers stand head and shoulders above even Goldstein...Liboff...Morrison...Morse and Feshbach. It is the case of Lanczos and Dirac.



5. As the other reviewers have said, this is a master piece for various reasons. Lanczos is famous for his work on linear operators (and efficient algorithms to find a subset of eigenvalues). Moreover, he has an "atomistic" (his words) view of differential equations, very close to the founding father's one (Euler, Lagrange,...).
   
   A modern book on linear operators begins with the abstract concept of function space as a vector space, of scalar product as integrals,... The approach is powerful but somehow we loose our good intuition about differential operators.
   
   Lanczos begins with the simplest of differential equations and use a discretization scheme (very natural to anybody who has used a computer to solve differential equations) to show how a differential equation transforms into a system a linear algebraic equation. It is then obvious that this system is undetermined and has to be supplemented by enough boundary condition to be solvable. From here, during the third chapters, Lanczos develops the concept of linear systems and general (n x m) matrices, the case of over and under determination, the compatibility conditions, ...
   It is only after these discussions that he returns (chapter 4) to the function space and develops the operator approach and the role of boundary conditions in over and under-determination of solutions and the place of the adjoint operators. The remaining of the book develops these concepts : chp5 is devoted to Green's function and hermitian problems, chap7 to Sturm-Liouville,... The last chapter is devoted to numerical techniques, amazing if one think that the book was written at the very beginning of computers, which is a gem by itself.

1 comments about Variational Analysis (Grundlehren der mathematischen Wissenschaften).

1. A winner of the prestigious Lanchester prize in 1997 (the very year it was published), this book is an instant classic. Written by two eminent researchers in the field, it is a handy reference on convex analysis, duality, optimality conditions, set-valued mappings, epigraphical convergence and variational problems.

   Resulting analysis is applicable to a wide variety of problems in mathematical optimization, operations research, management science, economics and engineering. New and unpublished structural results have been derived in this book and presented in the light of new developments in this area. To give an example, "Variational Analysis" advances some of the results in the princeton classic "Convex Analysis" (an almost necessary read for any researcher in mathematical optimization) also written by the first author of this book. To my knowledge, there is no other book at this time which compiles all the results in this area under one hood.

   The book follows a theorem-proof format for most part. Thus, it serves more as a reference than a textbook. A complete reading would be daunting given the length and density of material in the text.

   This book is worth buying if you are a researcher in mathematical optimization. To quote the citation of the Lanchester award:

   "In all, this book should serve as a landmark in the technical progress of optimization, an essential technical tool of operations research and the management sciences."

1 comments about Schaum's Outline of Calculus for Business, Economics, and The Social Sciences.

1. Since i'm studying for my first actuary exam, i was looking for a book that can help me pass the test. I didnt have to read a whole book to understand the problems. Schaum's Outline of Calculus for Business, Economics, and The Social Sciences helped me solve problems fast and easy! i recommend this book for people who need to learn in less time!

1 comments about A First Look at Perturbation Theory.

1. I am a first year graduate student in physics and have just recently gotten interested in classical perturbation methods. I have read the text and have found it to be a very nice, straightforward read. Series expansions to the roots of polynomials are covered and various techniques for solving differential equations are also introduced. It is a great book and I highly recommend it! The title accurately describes the text however: the book provides only a first look at these topics. It is a great book for those like me who would like to get an idea about what perturbation theory is good for, but it is not appropriate for those looking for a thorough text.

5 comments about Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics (3rd Edition).

1. This was the book that I learned Complex Analysis from. Definitely made the subject accessible to pretty much any reader. Plenty of exercises: some more theoretical, some more applied. It skillfully straddles the gap between being a theoretical math book and a math book for people with more applied aims (such as engineers). Most topics are covered thoroughly, though certain more complicated subjects such as winding number are left out for simplicity.

   This book definitely prepared me for tackling the dense, theoretical, and exceptional "Complex Analysis" by Ahlfors. I'd recommend it as an introductory book for anyone trying to get into the subject who is intimidated by Ahlfors, as well as for anyone who is only interested in the essential commonly-applied tools.




2. First let me say that this book was an introduction to the subject for me. After reading the first six chapters, and working through most of the problems, I have to say this book is great. I highly recommend this to anyone who is learning on there own. In particular, the chapter on residues is excellent. The chapter on series is also good, although I would have liked more worked examples for proofs involving uniform convergence. Also, a little more emphasis on the Arguement would have been nice. Nevertheless, 5/5 for this one, it is extremely well written and the authors really provide motivation for the theorems to come. This is definitely one of the best math books I have read. Great buy, worth every penny.



3. This book was not exactly introductory level but if you have some familiarity with concepts, it will serve as a good reference book. Very concise but contains many good examples.
   
   I used this book in conjunction with "A First Course in Complex Analysis"
   by Dennis Zill for a graduate level course, which is more of an introductory text than this book.
   
   I recommend using both for your first course.
   
   Another reference: Search for "Complex Analysis Modules by Mathews") on google. This served as a great online reference and has a corresponding book: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006 by John H. Mathews and Russell W. Howell. Although I did not read this book, the author has put up wonderful online notes from this book, which I did use.



4. Complex Analysis is always there in every applied math document of engineering context. The reason I bought the particular book was that I stumbled on some old forgotten Conformal Mapping techniques in Digital Filter Design and needed some good reference to go through...I ended up reading the whole book from first to last page as it managed to capture my interest and distract me from my original purpose for a couple of happy months. So if you are planning to stick to the foundations beyond your studies and course exams, then THIS BOOK IS FOR LIFE...the subject is very extensive and tricky but the book manages to present completely all the necessary elements in the right pace and volume that keeps the application-oriented reader's attention focused while keeping at the same time -in my opinion- the right level of mathematical strictness. All the most essential theorems and formulas are nicely placed intro frames so underlining is not that necessary. Last but not least there is a wealth of examples and illustrations that make it a very friendly tool for anyone about to take course exams or some old engineering graduate seeking a quick reference like myself.



5. There are many books on complex variables, but this surely rates well as an introduction. It is great for self study. It bridges the gap nicely from calculus. The problems at the end of the sections are of a rich and varied type and do enhance your learning experience. This book deserves a second look.