[tedheath123]

How should one use a book like this? Is it to get an overview of a topic before diving in? I don’t think I’ve ever learnt any mathematics from reference works, so I’m curious as to their intended audience.

[lemonwaterlime]

You use it like a conceptual dictionary. Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with (e.g. a submanifold). You’d look up “submanifold” and see that it is “ subset of a manifold that is itself a manifold, but has smaller dimension.” Okay, that seems to fit the intuition of a “sub”-something. But I don’t know what a “manifold” is. So I’d look that up.

“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”

At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.

Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.

[Tainnor]

> At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done.

I'm not sure. I only have a rather rudimentary understanding of topology, so I do understand the definition of a manifold on a technical level, but I don't know any interesting examples or theorems about them so it wouldn't be immediately clear to me why something being a submanifold is worth mentioning.

Similarly, I don't think that just reading the definition really gives you a good understanding of groups. You probably want to work through some examples of groups, and arguably, the importance of groups doesn't really become clear until you've encountered group actions.

[bluenose69]

The historical notes are a great strength of this book. As for learning the material, from what you've written, you would likely be better off with the sort of books used in first and second year university.

A way to find good ones is to look at some university webpages, to see what books they use in 1-level and 2-level classes. (Of course, start with 1-level.). Those textbooks will be more expansive, with interesting diagrams, problem sets, and so forth. And they will use fancy typesetting patterns, like insets in boxes for subtopics, etc.

I suspect quite a few purchasers will be university teachers who want to have this on their shelves, for when students come by and ask for a book to borrow overnight to brush up on a topic.

[kxyvr]

Outside of the other suggestions in this thread, this book may also be helpful to someone interested in studying applied mathematics in college, but unsure of what that means either in terms of topics or career. I've only flipped through the book, but it seems to do a good job at giving a high level overview of various topics and applications. If one were to like what they see, then perhaps one should investigate further.

In a similar topic, if someone is considering a career in mathematics, I like the book, "A Mathematician's Survival Guide: Graduate School and Early Career Development." It applies to both pure and applied mathematicians, but it does a good job of walking through undergraduate studies all of the way to being a professor. Not all mathematicians end up in the professoriate, but the graduate school information is still valuable.

[simonjgreen]

I’d say appreciation of background and inspiration. The preface linked on the page does a good job of positioning it too.

http://assets.press.princeton.edu/chapters/p10592.pdf

[kayo_20211030]

I wouldn't use a book like this for foundational learning. It's more a precis of existing information on a topic. Looking at one of the entries for Numerical Weather Forecasting, it presupposes at least a solidly-established understanding in Applied Math or Math Physics. If you're approaching that topic without a basic knowledge of what a divergence is, what vorticity is, what a gravity wave is, or the difference between implicit and explicit FD equations, etc. it's probably not going to teach you much. But, if you do have the background it's a great resource - a really super resource. It's a bit like Wikipedia, I suppose. Super helpful at some level, but not at others.

[epgui]

It’s a great way of getting to know what the landscape of mathematics looks like.

[detourdog]

I would read the table of contents and pick the most intriguing/relevant topic and see if it's comprehensible and relevant.

[Mathnerd314]

Wait until it is out of copyright and then import it wholesale into Wikipedia. But honestly Wikipedia already has most of the content.