Most people would agree that doing technical work, whether it be pure or applied, and learning the background knowledge necessary to do this work comprises most of the literature and curriculum in mathematics. What is almost never discussed however is how to communicate mathematics in a way which is as clear as possible to as many people as possible. Writing technical articles is a domain of its own, and one which many mathematicians and applied mathematicians learn from their peers and advisers, as well as here on Math SE (e.g. How to write a good mathematical paper?). There are many templates, good general styles, and even specific styles which should be used for different journals, and over time it is typical that any successful mathematician will become proficient in this area.

A topic on which there appears to be no guidelines from any one of these groups however is textbook writing. In fact, I don't even recall ever having a serious discussion with anyone in any technical field about what even makes a good textbook good! Of course everyone knows the texts they like, and different people prefer different texts for different reasons, but apart from personal opinions, I wonder if is possible to catalog a few strategies and approaches to good textbook writing with which a majority of people might agree.

FYI: Currently I'm trying to think out the best way to write a concise, yet complete, introductory control theory text. Most of the existing introductions are 500-800 pages, mash all sorts of different approaches together, and often leave 3rd year engineering undergraduates thoroughly confused and enraged with the subject. My dream is to write a text which can explain introductory control theory in as lucid and interesting a way as David Griffiths' Introduction to Quantum Mechanics explains the eponymous subject at the undergraduate level.

  • $\begingroup$ Short answer is: It's hard, and not an exact science (not even close). That said, there are certainly things to keep in mind. $\endgroup$ – Justin Benfield Jun 29 '16 at 23:44
  • $\begingroup$ @JustinBenfield Do you mind sharing more thoughts? I sort of meant this 'question' to be more like a discussion topic you might see in a more typical forum type site. $\endgroup$ – Mortified Through Math Jun 29 '16 at 23:47

One of the biggest challenges is organization. Deciding how exactly to organize all of the material you want to cover so that it is both, ordered logically, and also provides motivation for an idea before it is introduced. All the while, you also want to draw attention to the various connections between the various concepts presented in the text. A big part of writing good math is writing good proofs. Good proofs don't just prove the truth of their conclusion from their premise, they elucidate why that conclusion actually follows from those premises. You also need to consider carefully what background knowledge you expect your readers to have, as well as how well they are presumed to know it. For instance, for students being introduced to control theory, I imagine they would have had a least one linear algebra course, but perhaps not much more. Many of the fancier matrix algebra results used in control theory are likely new to those students, and their theoretical justifications should be explained carefully.

  • $\begingroup$ I'm sorry for my tardy reply; I've been in the process of moving cross-country! I think your answer contains a lot of good information for what I might call a text on state variables or some of the more rigorous parts of control theory. The classical control theory I want to focus on is actually far more rooted in complex analysis, and contains several beautiful applications of it, but most people texts introduce state variables far too soon and fail to properly discuss the actual mathematical foundations that Bode, Nyquist, and Black actually relied on! $\endgroup$ – Mortified Through Math Jul 3 '16 at 2:32

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