Calculus for the Painstakingly OCD: Elements of Real Analysis
Written: May 04 '03 (Updated May 04 '03)
|
Product Rating:
|
|
|
Pros: Some of the best math text writing I've ever worked with.
Cons: Some of the worst math text writing I've ever worked with.
The Bottom Line: Elementsis a good Real Analysis textbook for a upper-level college course in introductory Analysis, though it can be frustrating to work with.
|
|
|
| avepythagoras's Full Review: Herbert S. Gaskill and P.P. Narayanaswami - Elemen... |
Elements of Real Analysis by Gaskill and Narayanaswami is an introduction to the mathematical world of Analysis, the greater science that includes Calculus, Topology and other 'infinite' mathematical topics.
In the words of a professor/friend, "Real Analysis is calculus done right." I, on the other hand, always thought Real Analysis was "Calculus for the the painstakingly OCD." But corny math jokes aside, Real Analysis is a difficult subject to learn, and it is the culmination of four years of college level mathematical training. For many math students, it is usually their first major 'proof-based' course. And yes, this subject is all about proofs. It’s not like Calculus, where you are given some sharp tools to manipulate and graph functions, where proofs for many theorems are 'beyond the scope of the text.' Real Analysis is the axiomatic development of the very foundation that makes Calculus so effective mathematically, providing a rigorous, logically complete exposition of the Calculus. It also introduces students to the rigors of pure mathematics, and the expectations of graduate level research. Proofs are very much the scope of this text.
Because of the difficulty of the subject matter, it is important to find an adequate, self-contained, and thorough textbook that keeps both the student and professor in mind. I have a hard time working through math textbooks that assume too much, as if they are writing only to the 'uberstudent', someone with the most subtle mathematical ingenuity. Math students are human, were not all Galois or Riemann or Wiles. Some of us don't see the 'immediately obvious' subtleties of particular proofs. As hard as this subject is to learn, I would assume the difficulty of teaching this subject would be equally as infuriating. It requires an unusually talented and astute mind to capture the central themes and issues of the logical movements within the proofs, and making such things accessible to a ragged band of lazy, jaded and bored college seniors who are more concerned with finding jobs or grad schools than proving a particular function to be uniformly continuous.
Elements of Real Analysis was the text we used for my college Real Analysis course. It was a very painful course at times, mostly because Elements is a bipolar, mixed bag of textual clarity, at moments very helpful and very well-written. Elements at its best provided insightful dialogue and concise, understandable proofs. In fact, at times Elements was one of the most user-friendly math text-books I have ever worked with. But mostly, it was a muddy, near opaque test of my personal patience and willingness to spend hours trying to understand particular proofs, not because the proofs were difficult, but because the language was so obscure and hard to follow. Also, many of the most important definitions are left with no examples, thus many learners are left to drown, with no means of salvation, save their own slightly buoyant math textbooks.
This textbook was written to accommodate a one year or two-semester course in Real Analysis. The first four chapters represent the core methodologies of Real Analysis, and probably will comprise the first semester. The final six chapters provide further theoretical applications, working through the specifics of Calculus, and developing more complex analytical notions around the core learned from the earlier chapters. Whereas the first four chapters are mandatory and should be developed linearly, the final six chapters provide more flexibility and can be tackled in any particular order. This gives the educator more freedom, once the students have mastered the basics.
Chapter 0-Basic Concepts
Chapter 1-Limits of Sequences
Chapter 2-Limits of Functions
Chapter 3-A Little Topology
Chapter 4-Differentiation
Chapter 5-Integration
These were the chapters I covered in class, and hence, my review of this text is based specifically on them, though I have personally worked through chapters 8-Infinite Series and 9-Transcendental Functions myself. While I don't normally review books that I haven't finished, in this case, because most students and professors will only use the first five chapters, I feel that my review of this text will be helpful enough.
Chapter 6-Infinite Series of Constants
Chapter 7-Sequences of Functions
Chapter 8-Infinite Series of Functions
Chapter 9-Transcendental Functions
This text also has an extended appendix providing ample discussions of Set Theory and Boolean algebra.
The first two chapters are flawlessly written and the axiomatic development of calculus is driven by open discussions of the various theorems and definitions. No stones are left unturned, and the authors assume little about the student audience. While difficult, I found these chapters very easy to work with, the explanations of the logical processes behind the theorems are beyond adequate and offer insights deeper than most students would find under their own means. This helps nurture the student, and makes for easy self-study. As the text progresses the writing quickly turns opaque. The theorems become harder to work through, the writing becomes more obscure, and the authors begin to make unfounded assumptions about the student/readers' previous knowledge. Proofs in the second and third chapters are shortened, and much is left out because 'it’s immediately obvious.' I've never understood why many math writers assume this; I've found that little, if any, math is 'immediately obvious.' It may be obvious to the writers and that mythical uberstudent, but not to the common math student. I've found 'it’s immediately obvious' really means 'sorry, I'm too lazy for clarity.' For the most part, this text fluctuates from clear, concise and user-friend to frustrating, audacious and opaque. For some reason or another, maybe they go over their bought of laziness, chapters 5 and 6 see a return to clarity and adequate writing.
The worst writing in this text, from what I've worked with, is chapter 3-A Little Topology. Many times, while working through this chapter, I gave up frustrated and annoyed and went to the library to find another text instead. This is sad, because Topology is a very intriguing and contemporary topic, a topic that has the potential to inspire and motivate future prospective researchers. Topology is also a very difficult and heavy subject, not to be taken lightly. Elements makes the mistake of taking it too lightly. And it is a very painful experience. Too many key definitions are left without further explanation or examples; it’s hard to understand proofs without adequate explanation of the definitions. Because of this, chapter 3 becomes very confusing, especially when the authors expect you to fully comprehend their muddy an imprecise definitions.
Even with its many flaws, I would recommend this textbook to a professor or enterprising self-studying mathematician. Theoretical mathematics isn't supposed to be easy, and while this textbook can be painfully difficult it is not always so. I enjoyed working through many of the chapters in this textbook, when writing is good, it is exceptional, some of the best mathematical writing I have ever seen in a textbook. But be warned, this text lacks homogeneity, and can be very obscure and difficult to work with.
Recommended:
Yes
|
|
|
|
Epinions.com ID:
avepythagoras
|
|
Location: Gainesville, FL
Reviews written: 38
Trusted by: 14 members
About Me:
Should be back soon, maybe...
|
|
|
- Add to My Yahoo!: 
- Add to Google Homepage: 
© 1999-2009 Shopping.com, Inc.