Customer Reviews

All the Mathematics You Missed: But Need to Know for Graduate School

5 star:		(2)
4 star:		(9)
3 star:		(1)
2 star:		(1)
1 star:		(3)

 

 

I used this book for an opposite purpose to the one the author intended. For me it served to review all the math I *had* learned long ago in school (both undergraduate and graduate), but was starting to forget. The author's informal style and rapid-fire delivery were just right for these topics. The subjects I had truly missed, mainly the more abstract parts of algebra and geometry, I found difficult to follow, though I did come away with some feeling for them. This is not a perfect book. The informal style extends to numerous typos in equations, and modern computer-oriented approaches get short shrift. Nevertheless, I found it a unique resource and a pleasure to read.

This book has a very particular purpose: to recap some basic concepts from undergraduate mathematics so that you get the "big picture". In other words, for every math course you took as an undergrad, this book provides a good outline of the major ideas and how they fit together. But, it is only an outline; nothing more. If you actually missed out on some topic, or your knowledge of a subject is shaky, then this book won't help much. It will only help by providing a bibliography of some other references for that subject.

This book is meant to organize your undergraduate math knowledge, not to supplement it.

With that said, I'll mention a few words about the content of the book. It is quite well written and definitely extracts the essential ideas for your quick consumption. There are a few topics that I personally feel are missing, such as Gram-Schmidt and Jordan Canonical Forms for Linear Algebra, and UFDs and PIDs from Algebra. In general, it seemed like the book leaned a little more towards analysis than algebra, but the vast majority of important topics were indeed encapsulated in their synopsis.

Good for a very specific audience, but otherwise not wonderfully useful.

When I was in graduate school, it seemed that my professors were constantly making reference to a theorem or definition that I had never heard of, or that I had forgotten. The professors would usually acknowledge the possibility that their students were unfamiliar with the cited material, but they would say something like, "Oh, you can pick that up anywhere." Determining the "anywhere" was often a frustrating and time-consuming experience. I often thought that "someone" should write a book condensing all that material that I could "pick up anywhere" into one book. And I just discovered that someone has indeed done exactly that.

One can quibble about the choice of topics in this book. Three of the sixteen chapters in the book are devoted to vector calculus leading up to Stokes' Theorem. Five others concentrate on various forms of analysis and differential equations. Personally I think that perhaps some basic results in Number Theory might have been helpful; others may object to the omission of Algebraic Topology, although I don't think there is much material in early graduate school that depends on a knowledge of results or definitions in Algebraic Topology.

I agree with the previous reviewer who suggests that the book would be improved by the inclusion of answers to the exercises, but that omission doesn't upset me as much as it did her/him.

My biggest criticism of the book is that there is a disappointingly large number of typos. Even though this is a first edition, it should have been more carefully proofread. If a second edition is ever issued, I hope that problem will be corrected.

I found this book to be very helpful as a guide to self-study in mathematics. I did not rely on the chapters for understanding, but rather used them as a topic list for a several year course of study. I used the bibliography to find the best books for study and then later used the chapters as an essential review. When I finished, I felt I had a completely satisfactory undergraduate education in mathematics at a fraction of the usual cost. I now have an excellent library as well.

A previous reviewer pointed out that this book is meant to organize ones knowledge about math- not supplement for lack of knowledge. I agree. It gives recaps of the main ideas, and helps one to see the big picture about various subfields of math. Of course, NO ONE BOOK could POSSIBLY teach all of those subfields with a significant level of detail. So one should not attempt to use it for that purpose.

I'm a math undergrad, starting my senior year soon. I've been using this book to preview areas of math before taking a class in that area. It's been tremendously helpful to me to have an idea about the big picture and the context before grinding into specifics. I would highly recommend this book for that purpose. I don't know about other purposes, but for that it has been great for me.

The author gives many insights that nobody bothers to tell you in textbooks or in any specific class. For example, in the preface he explains that mathematics on the whole is about sets of certain types of objects and certain types of functions between those objects. This is a major simplification- but that's the point! I applaud Garrity for having the guts to say this, though he makes himself a target for ridicule by making such a gross simplification. Students like me need to hear it. The rest of the book begins each chapter by telling the reader what types of objects are studied in that field of math, and what the functions are that map between said objects.

It's a blurry, bird's-eye view of the big picture. But it motivates me. I have an idea about what to look forward to in a given class. I love this book. I had it out from my university's library for almost an entire year, and then realized I wanted my own copy so I could keep it.

While there is some truth to both segments of the title, in my experience there is also a great deal that can be disputed. Many of the topics that Garrity discusses in the book are standard fare in an undergraduate mathematics major. Chapter one is a recapitulation and summary of a basic course in linear algebra, certainly not something that any math major would have missed. The topic of chapter two is epsilon and delta real analysis, the mainstay of first year calculus. Chapter three covers calculus of vector-values functions, a primary topic of third semester calculus. Finally, the basics of abstract algebra, groups, rings and fields, are covered in chapter eleven. Therefore, four of the sixteen chapters describe topics that no math major could have missed.
Some of the other chapters cover topics that may or may not be requirements for completion of a major:

*) Chapter 4 point set topology
*) Chapter 8 geometry
*) Chapter 14 differential equations
*) Chapter 15 combinatorics and probability

However, it is most unlikely that anyone could receive a math major without taking at least two of these courses.
My disputes with the second part of the title are twofold. The first is that the topic may not be needed in graduate school. Chapter thirteen covers Fourier analysis and chapter sixteen algorithms. I am not convinced that graduate students really need to know either of these topics. My second point of dispute is that some of these topics are the basic topics that you study in graduate school. Stokes' Theorem, differential forms, curvature for curves and surfaces, complex analysis, countability and the axiom of choice and Lebesgue integration are all described at a level that I consider to be above the undergraduate.
Putting these criticisms aside, this book is a good survey of most of the topics that you would be expected to master in graduate school. The one conspicuous absence is any mention of logic. The word proposition or even the word logic does not appear in the index, and this is a topic that is needed in graduate school. To me, this is a glaring and unfortunate oversight.

Published in Journal of Recreational Mathematics, reprinted with permission.

This book is not comprehensive and doesn't explain things well at all. It should have been titled, "All the Mathematics I, the Writer, Missed But Needed to Know For Graduate School," because that's exactly what it is. Do not misunderstand me: it is a good book and covers some interesting topics. However, it's not a book that will prepare you for graduate school. It's more of a collection of mathematical topics that the writer found interesting. It's similar to having dinner with a mathematician who can't stop talking about the topics they love.

This is a great book to read, and read-again if you are entering a graduate program in mathematics, computational science, physics, or any mathematics-related graduate program. Includes many topics often found on qualifying exams.

If you need to learn the math this is not for you. If you know the math and need a refresher this is an excellent reference. If you travel a lot an need a reference but don't want to carry around a 10 lbs. book, this is a life saver.

This book is more like a popular account of mathematics than a mathematics text. Granted, there are considerably more equations here than one would typically find in a popular work, but the level of knowledge conveyed is similar, probably something along the lines of Penrose's Road to Reality but with a few more proofs thrown in. On the other hand, to be able to make sense of the text and the accompanying exercises, one would have to have more than a layman's knowlege of mathematics, say at least undergraduate analysis/algebra.

Most are of the proofs are of the "easy to prove" variety. The more difficult proofs - that one would be truly most have likely missed as an undergraduate - are argued heuristically, if at all. For example, the author states, but does not prove even in the simplest case, the Inverse Function Theorem. He follows this with a skeletal proof of the Implicit Function Theorem assuming the IFT. The IFT is a core result in undergraduate analysis but how many student's "miss" the proof? I would say most and the book does little to alleviate this particular gap.

The absolutely fatal flaw of this work however is that no answers are provided for the exercises. In a book of this sort, specifically targeted to the self-studier, there is no exuse for the author not providing answers to his problems. In order to learn mathematics, one must do exercises. Unless you are a reasearch-level mathematician (which is decidedly not the target audience for this book) you need to be able to compare your answer with the known answer to verify that you have completed the exercise correctly. Since there are no answers in this book, it is impossible for the self-studier to get anything out of doing the exercises since he will have know way of knowing whether his work is correct. Hence, the value of this book to anyone attempting to fill gaps in their mathematical knowledge is highly questionable.

To end on a less pessimistic note, there is some value in this book as the explanations are relatively good and the mathematics disussed by the author is genuinely interesting. There is also a good discussion of differential forms Stoke's Theorem (proved for the case where the manifold over which integration will occur is a unit cube.)