How do you use reference books?

Question

Reference books at the research levels often does not include any problem or exercise. While you can't read these books like novels(you normally need to work on other sheet of paper), I'm just wondering if you actually use theorems on books like problems and try to completely prove them on your own on other sheet of paper, or you just read until you understand. I've just started using these sort of books so I'm just curious how do you make sure that you really understand and able to use the materials, or you just use them as references, not as textbooks.

Answer 1

I'm not a mathematician so my experience may be different from others.

I've dealt with two kinds of technical references. The first is a sort of expository overview of a field/subfield with many proofs/derivations and detailed explanations. The other is more like a gigantic list of identities, facts, data, etc.

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Expository References

With these I treat them the same way I would read a research paper. On my first way through a particular section I don't worry too much about the details, but do common sense checks on the formulas or identities involved. My goal is to understand what is generally accepted/purported to be true. If the mathematics involved is very new to me then I may work through a few choice proofs on a first passing to make sure I understand the definitions and notation.

What I do next depends on the relevance of the material to my field. If this is a work on which I am expected to be an expert then I don't just stop at one reading. On the other hand if it is work in a field I need to communicate with, but not necessarily be proficient in I may stop there and worry about other things.

On subsequent readings I try to prove and develop the material in my own way. I generally avoid reading the details of the authors proof of a statement until I have tried it myself for at least a couple of hours. If the proof is beyond me I try to at least construct a few nontrivial examples.

Often the methods of proving the theorems are very different from the methods needed to solve problems with the theorems. In the absence of good problem sets I try to invent my own problems/applications. When I read a theorem I ask myself $``$ What is this theorem good for?$"$.

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Big List References:

Two examples of what I am going to call the "Big List" reference material are listed below. These are massive references and it is just about impossible to work through every identity yourself. The great thing about these works is that they allow you to quickly get a top level view/perspective of what is true and what isn't true.

When I find myself stuck working on a problem which involves some special funciton I am unfamiliar with I open up my copy of A&S and flip to the appropriate section. Then I write down every identity which could possibly be relevant to my problem and shuffle these around until they are useful.

• $``$Quantum Theory of Angular Momemtum $"$ by D. A. Varshalovich , A. N. Moskalev, V. K. Khersonskii

• $``$Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables$"$ by Milton Abramowitz (Editor), Irene A. Stegun (Editor)