Customer Reviews

Principles Of Applied Mathematics (Advanced Book Program)

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This book presents various mathematical principles in an organization I have not seen before. It starts with the idea of a transformation, then goes on to relate eigenvalues and eigenvectors to general spectral theory, explain how the need for closed function spaces naturally leads to Lebesgue integration (I know about Lebesgue integration before but I didn't know why it was needed), and show how the definition of certain inverse operators leads to distribution theory. This is a very natural way of organizing these principles. While other books, such as Strang's Intro to Applied Mathematics and Rudin's Real & Complex Analysis, provide you with one mathematical "toy" after another (Fourier series, Lebesgue integration, etc.), Keener's book tells you why you need the toy before giving it to you.

This is a book with very broad coverage, which is also its strength. It happens to be my textbook for a graduate course when I worked toward my phd degree in engineering. I had a hard time through it, although I had a high grade in this course. Even though, I don't think this book is well written.

As my title said, every chapter in this book can be written as a book. I am not exaggerating. It's true. Since the author condenses so much material into a small book, there is sacrifice certainly. For example, the coverage is not thorough for a specific topic, the proof is too short or even not given...etc. To overcome this, usually you need to consult other books to get a more clear understanding. In addition, I had a bad experience that some examples have nothing to do with what he has just said above. That drove me crazy. The author mentions in the preface that he intends to seek a balance on application and theory in this book. I don't think he gets the job done.

In spite of so many drawbacks, there are still bright sides. For example, the broad coverage is good for me to get into or acquainted with some topics. When you are not understanding what he said, try to find a reference book. That usually helps. In addition, the motivation part is good. He tells you why you need this, why that way doesn't work...etc.

Simply speaking, this book is kind of opening a door for you and then you need to work out the rest not depending on it but by yourself.

If you need a book for self study, this is not the one. If not for that course, I wouldn't force myself reading through it. I think Logan's "Applied Mathematics" is much better for self study at the expense of narrower coverage than this one. This one is better for course use accompanying instructor's good supplement.

When an author invites a reader to purchase and read his book, the assumption one makes is that the book provides a new or a different insight into a familiar or difficult subject or alternatively, that it opens new vistas and level of comprehension into the subject matter. First, writing a book on advanced applied mathematics is quite an endeavor and thus, I applaud the author for embarking in such an effort. However, the book overall provides panorama, but very little detail and even less new vistas into critical areas. Thus, there does not seem to be a compelling reason for this book to have been published, because most of the topics have already been adequately covered in other reputable sources. Clearly, the author set out with a lofty and challenging goal; to present clearly and compactly the theory underlying a very broad range of complex mathematical tools. In fairness needed a fuller and more coherent development to accomplish this, that is simply not present in the book. However, there are some bright spots, for which the book deserves commendation and at least two stars. The best written chapters, albeit too short, are Chapters 1-3 and Chapters 6 on Complex variables. But the treatment is neither compelling or original. In chapter 6 the author introduces the reader to orthogonal polynomials and special functions, but leaves the reader with such a limited view of these topics as to make the effort at teasing the reader's curiosity quite frustrating. The book is generous with bibliography at the end of each chapter and a great source for further research. Nevertheless, it would be an improvement if in future editions the author points out to the reader which pages of the suggested readings the reader should be focused on for the chapter in which it is being cited as a reference. The author engages in serial titillation by introducing the reader to exotic topics such as Sobolev spaces but quickly retreats from this topic by stating that many of the operators in Sobolev space lack "self-adjointness" and thus of not much practical use for applied mathematics. (See page 66-67). Similarly, he also introduces the reader to Mellin and Hankel transforms but, does not explain where, why and how they are used. More critically it does not explain when one of these particular transforms are better suited than Laplace and Fourier Transforms. So what is the point of bringing the subject to the attention of the reader in such an incomplete manner? The hurried and desultory treatment of Green functions is too compressed beyond belief for it to be of any significant value to the reader. The same goes for the discusion of Sturm-Liouville systems. The physical examples provided throughout the book are also too condensed and for the most part difficult to follow and relate to the theory the author just discussed that ostensibly equips the reader to understand the example.

The discussion of Laplace and in particular Fourier transforms seems not to logically and seamlessly emerge from the previous discussion of the theory of operators. A few words on the proofs provided throughout the book; some of are so abbreviated or lack explanation that it is a frustrating experience to attempt to follow to their conclusion.

In sum, I commend the author to either expand the length of chapters or consider choosing less topics and develop them more fully and coherently so the book attains depth and eminence.

The book is surprisingly comprehensive not only in scope but also with respect to well selected details, for such a broad sweep within the compact exposition. Every chapter is worth an attentive reading even if one by and large knows the subject. The overall benefit consists in a well balanced picture of the key ideas and techniques within a major part of what is called applied mathematics ( e.g. tensors, differential forms etc are not included), and also of every of the specific subjects treated in individual chapters. Good to have at hand and probably to learn from scratch because of the clarity of exposition.

This book doesn't clearify anything for you. No examples, No further explanation. It only keeps introduce various theories to you. It can compact theories that other books take 2-3 pages to explain it into 5 lines! I think you can imagine. Obviously, this book is not suitable to be your first book (not the second also). Buy it if you are sure that you are smart enough to understand it!