Compendium(s) of Elementary Mathematical Truths

Question

Can anyone suggest a reference (or set of references) that simply list the core definitions/theorems from any (or all) of the listed subject areas:

Precalculus (Elementary Algebra, Trig, Analytic Geometry)
Algebra (Abstract/Linear)
Real Analysis (Single/Multi-variable)
Complex Analysis
Topology (Basic Point-Set and fundamental algebraic concepts)

The depth of the references should be sufficient to at least cover typical undergraduate mathematics and possibly some graduate-level material. I am open to either dead-tree or on-line references. I don't need something that actually develops theory or concepts as I have many, many textbooks that do this and I don't need something that has solved problems such as Schaum's. What I really need is a concise, pithy collection of references that lists definitions and theorems in a logical order.

To help you understand the context of my question, as one of my projects, I'm working through problem sets that cover a broad range of undergraduate and lower-level graduate material. I think it would be really beneficial if I could reference the material I need from a minimal number of sources.

I realize I could make my own reference as I go along and, indeed, this activity in itself could be worthwhile, but I'm hoping something like this already exists.

You could fare worse than looking at the magnificent books of Pólya and Szegő, Problems and Theorems in Analysis, I/II. They cover at least points 1,3,4 of your requirements, and much more. (but maybe I misunderstand your question). — t.b., Jun 21 '11 at 20:24
I don't think it goes as deep as you are requiring, but CRC Standard Mathematical Tables and Formulae includes material in all the subjects you mention (but, for example, topology is only included via the basic definition and a few comments, in half a page). — Arturo Magidin, Jun 21 '11 at 20:24
@Theo Buehler Those are indeed nice books but am really looking for something very spartan that contains only definitions and theorems, not something that develops the material or contains problem/solutions and proofs. — ItsNotObvious, Jun 21 '11 at 20:30
Handbook of Mathematics, by Bronshtein, Semendyayev, Musiol, Mühlig, is pretty comprehensive. For some reason I can't get Amazon to preview this (and Google Books doesn't have a preview either), but here is a table of contents, and there are... other... ways of seeing the rest of it. It doesn't have topology, unfortunately. — Zev Chonoles, Jun 21 '11 at 20:32
Ramanujan found Carr's A synopsis of elementary results in pure mathematics helpful, though it is rather dated now — Henry, Mar 18 '23 at 14:48

score 5 · Answer 1 · answered Jun 21 '11 at 20:36

5

Garrity's All the Mathematics You Missed But Need to Know for Graduate School seems to be what you are looking for. It covers everything you listed and more!

answered Jun 21 '11 at 20:36

Corey

1,766

I bought this book a while back, and recall being very disappointed with the amount of algebra covered. I don't remember how it did with other subjects. – Zev Chonoles Jun 21 '11 at 20:40
I actually have Garrity's book and, unfortunately, find it lacking for my puposes. It covers a fair amount of math but the depth is missing and an cosiderable amount of prose is dedicated to conceptual development which is, again, at odds with what I seeek. – ItsNotObvious Jun 21 '11 at 20:41
@Zev Chonoles My take on this book is that if you want a nice conceptual overview of a broad swath of mathematics then its not a bad read. For any reference/study purposes, however, its not really useful. – ItsNotObvious Jun 21 '11 at 20:45
@3Sphere: I suppose I had just been expecting something else. Though looking at it again, I think you're right about its value as a conceptual overview. – Zev Chonoles Jun 21 '11 at 20:49

Mike Jones · Answer 2 · 2011-06-23T21:21:22.270

3

More wordy that what you're asking for, but maybe very useable until something more spot-on shows up: Math Reference Project http://www.mathreference.com/main.html

edit: Also, somewhat tangential to what you are asking for, but perhaps still germane, is the following website with a long list of (standard? / famous?) problems: http://www.mathproblems.info/

edited Jun 23 '11 at 21:21

answered Jun 21 '11 at 20:36

Mike Jones

4,460

Thanks for the references, I'll look them up – ItsNotObvious Jun 23 '11 at 21:50

Compendium(s) of Elementary Mathematical Truths

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