All the Mathematics You Missed But Need to Know For Graduate School

by Thomas A. Garrity

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.

Genres: Mathematics, Nonfiction, Science, Textbooks, Reference, Physics, Engineering

347 pages, Paperback

First published November 12, 2001

Reviews

[Carl · Rating 5 out of 5]

This book is outstanding. I learned about this from the Math Sorcerer youtube channel which also featured an interview with the author.
This book is intended for undergraduate math majors wanting to go to grad school in math. I'm probably not the typical reader. I'm an engineer by training and manage software developers for a living. Reading math is one of (many) hobbies.
That said, this was a great read. Some chapters were fairly easy reads since I have had prior exposure (Linear Algebra, Analysis, Classical Stokes, Elementary number theory, Countability, Curvature, Complex Analysis, Fourier Analysis, Differential Equations, Prob theory, Algorithms) and some topics were more or less new to me (Topology, Differential Forms, some of the Abstract Algebra, Algebraic and Analytical Number theory, Lebesgue integration/measure theory, Category theory). Readers should be warned that you will not master a subject if this is the only book you read on the subject. But as a survey of what you should know after an undergraduate math degree, and a guide to where your gaps are, this is perfect.

[William Schram · Rating 4 out of 5]

Overview of mathematics that one should have been exposed to upon reaching Graduate School. Very wide breadth, but little depth to each subject covered. However, Garrity does have a bibliography for further study and reference. It has problems in it, but to be honest, I never passed Calc II, so it was a bit beyond what I could do, but it did have the proofs of a number of things and it showed Stokes' Equations and other things that you would encounter.

Very fascinating, and the bibliography gives me a bit of new reading material. Also, one of the few books that tried to explain Set Theory notation at least a little, so it gains something for that.

If I would sit down with the book and try the problems and focus, I could probably succeed at doing some of them, but I didn't really feel like spending the time trying to solve any of the problems.

I really didn't have any issue with the depth of coverage either, since he mentions in the book that you should have been exposed to this stuff already, so it's more of a reminder or primer than an in-depth coverage.

Anyway, once I understand more of the notation, I would probably read this again.

[George Marshall · Rating 5 out of 5]

If you are trying to get back a math foundation, or preparing to extend into higher levels of math, this is a great book to start with. The writing has a good flow, and introduces concepts in a way that provides a framework for further growth. It also provides, chapter by chapter, a list of best resources to follow through in the specific branch of math. A friend had suggested this book to me, and it did not disappoint.

[Yasiru · Rating 4 out of 5]

Too light on the abstract algebra and a bit wrongheaded on occasion with some of the general commentary (no, the prevailing belief is not that P=NP is a statement independent of other axioms- see for example Aaronson's paper on the subject), but very useful nevertheless.

[Roger Hallman · Rating 4 out of 5]

This is a well-written, Cliffs Notes summary approach to university level mathematics. The definitions are clearly stated where you may have found that...say an abstract algebra text is written in a different language that only closely resembles English.

I'm using it as a research aid for a paper and presentation on the solubility of the quadratic, cubic, and quartic equation but the insolubility of polynomial equations of degree 5 or higher. In addition to the summaries provided, there are recommended texts for deeper discussions of the covered subjects.

[Darin]

I'm not sure how much math can be learned from a book that covers this many mathematical topics in 300+ pages, but this book does provide a good refresher for those who have studied these subjects in the past. Each section includes a brief discussion of recommended books for further reading if interested.

[Viney Kumar · Rating 5 out of 5]

I've only read through and completed the exercises in 5 of the 20 chapters, but this is an exceptional book. The book is excellent in providing a broad overview that emphasises the essential themes, techniques and theorems, while providing resources to supplement knowledge and refill the holes in our understanding.

The author adds his own creative flavour to the choice of topics in the book. Along with the standard undergrad topics (Multivariable Calculus, Analysis, Abstract Algebra, Point Set Topology, Probability, Algorithms and Combinatorics), the book contains well presented material on everything from analytic number theory to differential geometry that I'll be sure to read when I have some spare time.

The presentation style is clear and succinct. The key objects of study and the 'goal theorem' that each chapter builds up to is stated at the beginning of the chapter, so we know where the author is going with each successive section.

I highly recommend this book as well as some of the additional resources cited in this book (particularly the Berkeley problems in Mathematics).

[Curtis Penner · Rating 3 out of 5]

The writing was excellent (with the exception of Abstract Algebra). He covers the subject matter appropriately. For this the book would be a 5 star. The drawback in the physical book. The binding is so tight that I struggled to hold the pages open. The print was so shinny, it reflected the reading light into my eyes making it difficult to read.

I wanted to cut the spine of the book and make it loose leaf. I won't because I need to keep it on the bookshelf. This is because it was publish by Cambridge Publishing. I have had the same problem with other books for Cambridge Publishing. Their book's contents are good but the physical books are BAD. An example of books that were alway easy to read were MIR books. Not necessarily the contents, but physically the books are easy to read and take notes in.

For these types of books, the structure of the book is important. Being math based, one would take notes in the books. This book you are using whatever to hold the book open, and then trying to write in it is very difficult.

[John Teske · Rating 4 out of 5]

So I did a year of graduate mathematics, and this showed me how much I had yet to go.

Wow.

I thought I might pick up mathematics again as a recreational hobby.

This might be a good reference going forward if I do such an adventure.

[Charlien Tania]

This book turned out written specifically for math students. I didn't continue far more than its' introductory chapter since there were too many technical terms that I don't comprehend.

[C.N. Haugen · Rating 4 out of 5]

Makes sense as a refresher, but it is in no ways an introductory book.

[Anna Sinclair · Rating 5 out of 5]

Good reintroduction to prepare for Grad School but more importantly has great references.

[Lucille Nguyen · Rating 5 out of 5]

Good overview of a lot of different areas in modern mathematics. Excellent presentation and examples, well worth reading.

[Erik · Rating 3 out of 5]

Not a textbook. Useful for structuring a reading list on the various topics. My background varies, generally good on the more abstract conceptual stuff (I know way more about logic and foundations coming out of philosophy than could ever be useful) to very ignorant on actual techniques of integration and problem solving, which I am seeking to fix.

[IQ · Rating 4 out of 5]

A good reference book.

[Ivan · Rating 2 out of 5]

A book teaching you math assuming you already know math well. Sweet sweet Irony. Good if used as reference rather than to learn a missed topic.