All the Mathematics You Missed: But Need to Know for Graduate School
All the Mathematics You Missed: But Need to Know for Graduate School
By Thomas A. Garrity
Publisher: Cambridge University Press
Number Of Pages: 376
Publication Date: 2001-11-12
ISBN-10 / ASIN: 0521797071
ISBN-13 / EAN: 9780521797078
Binding: Paperback
Few beginning graduate students in mathematics and other quantitative subjects possess the daunting breadth of mathematical knowledge expected of them when they begin their studies. This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations.
Summary: Not A Textbook and Far More Enjoyable
Rating: 4
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.
Summary: Bird’s-eye view of the big picture
Rating: 5
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.
Summary: Don’t waste your time or money
Rating: 1
This book sucks. It has lots of useful mathematics in it, if you can decipher it. Most math classes you’ll ever take involve mindless number shuffling and you never truly understand what you’re doing. Colleges just want to produce marketable products (students) instead of good classes so most of us just repeat mathematical processes without understanding the theory behind it. This book assumes you understand those theories. I read of over some of the sections I had already learned about in college and they looked like total jibberish. Why? Because the definitions are so clinical and use notations that most of us have never seen. Not in class. Not in text books. Never! This book will make you appreciate your math teachers. It’s light on examples and the ones given don’t clearly demonstrate the logic involved.
Summary: hatchet job
Rating: 1
You can learn about as much math from this as
you can from reading a course catalogue. There
is no need to put more thought into this review than
the author did into the ‘writing’ of the book – which
is to say not very much. Suffice it to say that:
a) you can learn much more by simply consulting the
relevant topics in wikipedia
b) stylistically the custom of lifting bodily whole
sections from other’s works is both shameful and
lends the work as a whole all the coherence of a
patchwork quilt
c) the only real work the author did appears in chapter 6
but 30 decent pages out of more than 300 total makes for
a pretty lame batting average.
Summary: Logic is conspicuously absent, otherwise a reasonable survey
Rating: 4
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.
link
http://ifile.it/arh7yun/0521797071.rar
