Big title, big subject area, surprisingly small book. If you’re even a little bit tempted, you should pick up a copy.
Title: All the Mathematics You Missed (But Need to Know for Graduate School)
Author: Thomas A. Garrity
Original Publication Date: 2002
Cover Price: Not listed.
You can find the paperback edition ($29.99US = $34.33) at Amazon.com or the hardcover ($54.79Can = $48.84 US) at Amazon.ca
This is a 330 page survey of undergraduate mathematics. The chapter titles are:
- Linear Algebra (page 1)
- Epsilon and Delta Analysis (page 23)
- Calculus for Vector-Valued Functions (page 47)
- Point Set Topology (page 63)
- Classical Stokes’ Theorem (page 81)
- Differential Forms and Stokes’ Theorem (page (111)
- Curvature for Curves and Surfaces (page 145)
- Geometry (page 161)
- Complex Analysis (page 171)
- Countability and the Axiom of Choice (page 201)
- Algebra (page 213)
- Lebesgue Integration (page 231)
- Fourier Analysis (page 243)
- Differential Equations (page 261)
- Combinatorics and Probability (page 285)
- Algorithms (page 307)
- Equivalence Relations (Appendix, pages 327 and 328)
I’ve included the pages numbers to give you a feel for how much time is spent on each topic.
This covers a huge amount of mathematics, explicitly stating the moments when details are skipped for the sake of brevity, and always pointing the reader to materials that include all the details one could want. The best side effect of putting all this material under one cover is that the connections between the different areas of math are made clear. For example, I’ve taken two University level linear algebra courses, and done fairly well in them (I received an 8 and a 9 in the classes, which were marked on the 9-point scale), and yet this book was the first place I learned of a geometrical interpretation of the determinant. (ie. If you use a matrix to transform the n-dimensional unit cube, then the volume of the resulting parallelipiped is the determinant of the matrix. This is mentioned without proof in chapter one, so I immediately set about proving it to myself, only to find afterwards that the author had simply deferred the proof to chapter six.) Every chapter had at least one such revealing moment for me. Keep in mind that my transcript includes 13 university level math courses, as well as two mathematical physics courses, as I tended to choose math courses as options when working on my physics degrees and education degree. This may seem like I’m trying to brag about my background; I’m not. I’m just letting you know how strong my background is so that you’ll understand how unusual it is to find that much material I’d never seen in one book. This is a great resource for the beginning graduate students it was aimed at, as well as a math geek such as myself.
The balance of time spent on particular topics is not quite what I’d hoped. There seems to be a disproportionate amount of time spent on Stokes’ Theorem, and very little time spent on geometry. For example, the axioms of algebra and set theory are explicitly listed, while the axioms of geometry are not. We get no less than 60 pages on Stokes’ Theorem, with only ten pages covering Euclidean, hyperbolic, and elliptic geometries. Perhaps this will bother me more than others, since I’ve seen much of Stokes’ Theorem in class, but haven’t actually taken a class in geometry. (I’ve got textbooks designed to fill that gap; expect more reviews in the coming months.) Still, as always, the geometry chapter ends with recommendations for textbooks to investigate if you’d like to learn more.
The clarity of the text was quite good, often because proofs were glossed over somewhat. (As mentioned above, though, the glossing was noted and described, and came with pointers to the detailed versions.) As mentioned above, the combination of all such topics in one volume made it much easier to tie the various topics together into a single package, which helped illuminate things I thought I understood. (Up until this point, I thought the determinant was a useful quantity that was useful in calculating inverse matrices and eigenvalues, but had no real value and meaning on its own.) However, that same wide variety of topics also means that some are covered in a relatively short span, which can make them hard to understand. (The “P=NP” problem is one that I had a hard time sorting out based solely on this text.) In fact, in retrospect, the “vague” sections were all the ones that I’ve had no previous exposure to, which is a sign that this book may be difficult to delve into without a background in at least some of the topics being discussed. I give it 4 out of 6.
The structure of the text was clear, with the goals and tools of each chapter explicitly stated at the beginning, and the book recommendations and exercises bringing up the rear. When the text is referring to previous (or upcoming) results, that reference is explicitly stated. I give it 6 out of 6.
The examples were sparse, presumably to keep this under the 1000 page count, but they weren’t eliminated. Instead, the examples were carefully chosen to be simple and/or significant, and in some way clarify the concept being discussed. In some cases, such as “P=NP,” the explicit connections between example and theory could have been made more clear. However, as this text is not intended to give someone a mastery of any topic, but rather a starting point to fill in the gaps in one’s knowledge, the small number of examples is not a serious problem. I give it 5 out of 6.
There were few exercises at the end of each chapter, but they were selected with similar care as the examples. Doing the examples will give the reader a good feeling for their level of understanding of the topic, and let the reader decide when it’s time to grab one or more of the recommended books and delve further into the subject. My only complaint about the exercises is that the solutions (or at least answers) were not included, which I feel is a problem in a self-study text such as this. I give it 4 out of 6.
It’s hard to knock the completeness of a text like this. No, you won’t master any given topic using just the material here, but you will certainly see every significant topic in undergraduate mathematics to some degree. This book had lofty goals, and it hit them. I give it 6 out of 6.
The editing was fairly good. I only noticed two typos, and I was following along, doing the proofs, examples and exercises myself as I went. I give it 5 out of 6.
Overall, it’s not a perfect book, but I still recommend it without reservation to anyone who has had some background in university level math courses. It’s a complete and illuminating survey of where undergraduate (and some portions of graduate) mathematics teaching is right now, including some mention of the philosophy behind things without getting overly editorial. I give it 6 out of 6.
In total, All the Mathematics You Missed (But Need to Know for Graduate School) receives 36 out of 42, which is actually a bit lower than I’d have estimated just after putting it down. I think this is mainly due to the fact that the scoring rubric for textbooks was designed around a textbook for use in a particular class. This isn’t well suited to that purpose, so it’s not a good fit for the rubric.
Additional Notes and Comments
I’ve got a number of textbooks that I’ve picked up to continue my studies on my own, now that I’m not a student. Topics include geometry, differential equations, algebraic number theory, algebraic topology, linear algebra, the history of mathematics, and analysis (real, complex, and functional.) Also, I’ve got a large number of physics textbooks and some film studies texts I haven’t reviewed yet. If you have requests, speak now. I’ll read another one after I’ve finished The Babylon 5 Scripts of J. Michael Straczynski Volume 3.