Customer Reviews

Differential Equations With Applications and Historical Notes

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I've taught upper division students from this book (and the first edition) 5-6 times for over a decade. I remain impressed by the broad range of topics from which the teacher and reader can select. As with his excellent calculus textbook, the author tries to show students how mathematics is a human activity, a subject that developed in response to actual needs and which is still lively and developing. No part of mathematics illustrates this development better than the topic of differential equations, which was invented to solve pressing problems in astronomy. One example: In Newton's time, accurate location of position on the open seas was an unsolved problem, crucial to commerce. New techniques from differential equations led to the ready calculation of tables which, together with the invention of Harrison's sea-going chronometer, effectively solved the navigation problem. Differential equations lie at the core of the physical sciences and engineering and are proving increasingly valuable in biology and medicine. Simmons' book will not appeal to readers who want merely recipes with examples of their use. Such readers might prefer the excellent books from the Schaum's Outline Series. Those readers who want to see vital mathematics well presented, those readers who think that mathematics stops at trigonomerty or the calculus, those readers who want to use differential equations intelligently, and those readers who just like a cracking good mathematics story should get a copy of Simmons' book and read.

Nathaniel Grossman Professor of Mathematics, UCLA

Each mathematic's book has its particularities. I had said before the Tanembaum book was the one better than I had on differential equations, considering its easiness reading , its methodology and its organization, but evidently these three elements can be present in many other books. This is the case of the present book, which I consider excellent. This is a great book, a very well achieved work . It introduces each element interesting the reader for the following topics, with exercises very well planned according to the exposed theory and with answers at the end of the book. Each chapter contains a historical note that will be extremely instructive and stimulants for the reader that doesn't feel pleasure for the exact sciences . Of special mention they are the chapter 6 (Some special functions of the mathematical physics), the chapter 8 (non lineal equations), and the chapter 9 (calculus of variations). The calculation of variations is an introduction to this branch, but I assure you that when reading it, you kept desires of treating other texts specialized in the topic like that of Sagan. You won't find in this book any complication, and you will be satisfied considering that it is a book for undergraduates, an excellent introduction to the differential equations.

Among the book of Tanembaum, Derry Grossman, D'prima, and the introductions of differential equations of C. R. Wylie and Kreiszig in their books of advanced mathematics for engineering, I keep this. If you plan to learn differential equations, if you want to have a really good text as introduction to the topic, then this is the book to buy, you won't lose your money.

This is an absolutely wonderful book to learn differential equations from. The explanations are for the most part very clear and there is just enough detail, not enough to qualify as gruesome! The layout and organisation is also excellent and makes the book easier to read, althoguht I suppose this could be subjective. There are also numerous problems - lots of practice! - and you can check your answers against those in the back of the book. I need differential equations for my discipline and although the class is going rather slowly, I think I shall be able to get alot from the book. For the student learning DE primarily from the book, this is just about all you can ask for. I would give this book a rating for ease of use by such a student similar to or better than the Bostock and Chandler A-level series.

I have read this book in my first year of engineering along with a few others related to calculus. But compared to this book, the other books seem a lot less effective.
The subject treatment has sufficient detail for the really interested people and is sufficiently concise for those seeking nifty tricks and methods to solve ODEs.
Some higher end stuff such as non-linear DEs has also been treated well. Though I didn't know a word about these particular equations, the book taught me well enough to handle a few types of those.
Great book!

One of the best written books on the subject I've seen. As long as you're prepared to follow his anaylsis line by line (no skimming at the back there) you can give yourself some serious mathematical muscle.

He derives the first type of elliptical integral from the motion of the pendulum at page 21 (on my elderly foreign-printed software version I've had for years). That's the level it's pitched at.

This is a good book. Simmons has an eye for what would motivate each step and shows enough explicitly to keep the text flowing nicely. There are lots of fairly easy problems which extend the material in the book. He points the students to other sources before getting into heavy complexity. The historical notes are particularly enjoyable. Simmons has an aversion to using complex analysis, or even just complex numbers. This means that he has to resort to trignonometric identities for many manipulations which would have been done more compactly using Eulers formula.

I would have given this book 4 or 5 stars except for one glaring fact: Large sections are copied without attribution from other authors. When I went to my bookshelf to seek clarification on some points I noticed that large sections of the chapters on Fourier series, and partial differential equations were copied without attribution from the excellent book "Mathematics of Physics and Modern Engineering," by I. S. Sokolnikoff and R. M. Redheffer (McGraw Hill, Kogakusha, 1958). I provide some details below. Although Simmons has inserted some historical and other comments, and shown some minor manipulations more explicity in a few cases, errors, sloppiness, and omissions have crept into the copy which make it less clear and accurate than the original. Sokolnikoff's excellence still comes through Simmons' light reprocessing, and it is hard to give Simmons credit for that. I would be very unhappy if I were to see my work being used without attribution the way Simmons does. One wonders what other books Simmons has copied from to produce this text.

Details

Fourier Series. Simmons pp 246 -262. Sokolnikoff pp 175 -211

Simmons rewords the text, and reorders it slightly, but the flow of the argument, points made, and equations are exactly the same. Simmons leaves out a good section on complex Fourier series because of his aversion to complex numbers. Here are examples of Simmons "light reprocessing":

Simmons: "We begin our treatment with some classical calculations that were performed by Euler. Our point of view is the the function f(x) in (1) is defined on the closed interval - Pi LE x LE Pi, and we must find the coefficients in the series expansion."

Sokolnikoff: "We take the point of view that f(x) in (18-1) is known on (-Pi, Pi) and that the coefficients an and bn are to be found."

An error has crept into Simmons copy. Simmons refers to a closed interval, whereas Sokolnikoff's original refers to an open interval by using rounded brackets. Also, Sokolnikoff's English is better. There are many examples like this.

Simmons fills in a few steps by explicitly giving trigonometric identities, whereas Sokolnikoff merely asks the reader to recall them. This makes Simmons' treatment a bit easier to read.

Simmons' treatment of "Pointwise Convergence of Fourier Series" on pp 293-297 is a complete copy of Sokolnikoff's treatment on pp 204-207.

Partial Differential Equations. Simmons: pp302 - 322; Sokolnikoff pp 431-471

Simmon' treatment of the vibrating string copied part of the derivation from Sokolnikoff, particularly the discussion around Simmons' Fig 48 (Sokolnikoff's Fig 3 in Ch. 6). Simmons copy of this diagram showing the forces on a string leaves out some critical elements making it unclear. It does not show the tension on the element of the string from BOTH left and right, it just shows the rightward tension - acting on the LEFT end of the element. Also it doesn't show the differential dy in the diagram. Quite a muddle. Sloppy. If Simmons had copied Sokolikoff's diagram accurately, it would have been much clearer.

The Heat flow example in Simmons' Sec 48 is a copy of Sokolnikoff's Ch. 6 Section 9 (p455).

Treatment of the Dirichlet problem in Simmons' Sec. 41 is a copy of Sokolnikoff's Ch. 6 Sec. 12 (p 467), except that Simmons shows explicitly some easy manipulations that Sokolnikoff had left as a problem for the student.

This is one of the best books out there for introductory ODEs. This book should be read by everyone who's interested in the physical sciences. It has many applications to physics, and that's one of the main reasons I've decided to take the course. By the way, I took the course at UCLA with professor Grossman (he reviewed the book below). He's a brilliant and a challenging mathematician, which comes at no surprise since he's a graduate of Caltech. Anyhow, I really recommend this book for all mathemticians and physicists. It's a great investment, since it can be a great reference in the future!

This book was very difficult to understand. If it hadn't been for the professor's clear interpretation of the material presented in this book, I wouldn't have been able to follow it at all. Mr. Simmon's limited explanation of the mathematics and skipping critical steps in the examples make for very frustrating reading.