[1] https://nhigham.com/2021/01/26/what-is-the-nearest-positive-...
I thought the book might be fun to browse around in, so I purchased it. I knew the companion book (…to Mathematics) had a very good reputation.
It didn’t work for me…long story short is, the articles were written at too high a level of sophistication to serve as an introduction for a curious outsider. It was more of what someone in a nearby field might want to get up to speed on what is known, when their background in “that kind of thing” is already quite strong.
I was surprised for various reasons - I know Higham’s technical work, enjoy his blog, and I have a decent math background. (Of course, he’s the editor, not the author - it’s an enormous book.)
Ah, well. Not every shoe has to fit.
“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.
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.
> Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with
In such a case you're not interested in either manifold or sub-manifold or group in and of itself. So a lack of familiarity with theorems isn't an impediment.
Reading mathematical definitions on their own just doesn't give you a whole lot of context about the objects they're describing.
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.
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.
I am interested in visualizations/simulations of physical systems as a way to learn advanced math. Is there any books or resources that take that approach?
https://ocw.mit.edu/courses/res-18-008-calculus-revisited-co...
When looking into simulations of physical systems, you'll run into partial differential equations, but be careful about learning resources that don't put numerical methods front and center. The article on Numerical Weather Prediction in the post has a good description:
> "Analytical solution of the equations is impossible, so approximate methods must be employed. We consider methods of discretizing the spatial domain to reduce the PDEs to an algebraic system and of advancing the solution in time."
Given Python's popularity in scientific computing, a lot of the available materials on the topic are in that language, using libraries like numpy and scipy a lot. I've been playing around with custom ChatGPT here - you can construct a workflow that takes a description of a common equation, generates the LaTex expression for it, translates that to a sympy expression, and then from that generate the numerical method code using numpy and then the code to plot the behavior over a given range in matplotlib. Bonkers, we're living in the future.
I think most differential equations courses are too focused on symbolic solving techniques. To me, understanding a (physical) system is mainly understanding the differential equation (system) itself, not it's solution. MathTheBeautiful really excels at this approach.
Solutions are of course important as well, but i think that's what computers are for.
[0] https://www.youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCV...
Uses Haskell, and teaches it so knowing Haskell is an unnecessary prerequisite, though familiarity with programming in general would be very beneficial, and focuses on 2D and 3D visualisation.
I learned Penn because it's "real" (mainstream) math & lets you read in the same language; I learned GA because it's designed to proselytize to other mathematicians & deliberately presents serous ideas in very approachable ways.
I found the explanation on the development linkages to be intriguing. It described the historical context and the steps of development one key ideas.
Calling curious people posers is off putting.