[HN Gopher] The Princeton Companion to Applied Mathematics
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The Princeton Companion to Applied Mathematics
Author : teleforce
Score : 180 points
Date : 2024-04-28 05:05 UTC (17 hours ago)
(HTM) web link (nhigham.com)
(TXT) w3m dump (nhigham.com)
| tedheath123 wrote:
| 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.
| simonjgreen wrote:
| 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
| bluenose69 wrote:
| 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.
| kayo_20211030 wrote:
| 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.
| detourdog wrote:
| I would read the table of contents and pick the most
| intriguing/relevant topic and see if it's comprehensible and
| relevant.
| epgui wrote:
| It's a great way of getting to know what the landscape of
| mathematics looks like.
| lemonwaterlime wrote:
| 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 wrote:
| > 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.
| kxyvr wrote:
| 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.
| Mathnerd314 wrote:
| Wait until it is out of copyright and then import it wholesale
| into Wikipedia. But honestly Wikipedia already has most of the
| content.
| dtquad wrote:
| The most advanced math I had were the 1st and 2nd year
| multivariable calculus and linear algebra courses in college.
|
| 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?
| bloqs wrote:
| Brilliant.com
| ericdfoley wrote:
| Computational Science and Engineering by Gilbert Strang
| kubielid wrote:
| Absolutely love the book "Learn Physics with Functional
| Programming".
|
| 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.
| thechao wrote:
| I recently bootstrapped myself from your level to algebraic
| geometry -- enough to read an annotated version of Einstein's
| paper. If I had to do it again, I'd just skip to the parts that
| worked: (1) hit up Michael Penn on YT, but work all the
| problems he works; and, (2) learn GA, I like MacDonald's
| "Linear and Geometric Algebra". I'm going to be honest: unless
| you _do_ the homework, you can 't learn the math.
|
| 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.
| lagrange77 wrote:
| Take a look at this playlist [0] by MathTheBeautiful. It says
| it's about PDEs, but it starts with ODEs.
|
| 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...
| photochemsyn wrote:
| The next two steps beyond that are probably vector calculus and
| complex analysis. Check out Herb Gross's classic chalkboard
| lectures on the topic:
|
| 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.
| gautamcgoel wrote:
| For those of you who don't know: this is a companion to the
| Princeton Companion to Mathematics, which is excellent but
| focuses mainly on pure math. Also, the editor of this new volume
| was a highly respected applied mathematician named Nick Higham,
| who sadly died a few months ago.
| pvitz wrote:
| Oh no... I used his work on nearest positive semidefinite
| matrices and liked to read his blog. In my opinion, he was very
| good in explaining his research.
| abhgh wrote:
| Thanks for mentioning that. It's a problem I occasionally
| need to solve and based on your comment, I found this helpful
| write up by Nick Higham [1].
|
| [1] https://nhigham.com/2021/01/26/what-is-the-nearest-
| positive-...
| pvitz wrote:
| In my case, I am applying this in the calibration of
| instantaneous correlation matrices. His relevant papers can
| be found at [0]. It was really helpful that he also
| published MATLAB code of most of his algorithms.
|
| [0] https://nhigham.com/correlation-matrices/
| jimbokun wrote:
| This seems like the perfect book to put on your bookshelf to make
| people think you are smart.
| kubielid wrote:
| I joked at an estate sale once that everyone has 3 bookshelves:
| the books they read, the books they want to read, and the books
| they want other people to think they read.
| DeepSeaTortoise wrote:
| IMO very few things can beat rows upon rows of fancy second
| hand hard cover folders.
| flatline wrote:
| In fairness this one's predecessor has some pretty accessible
| articles that I've actually read. It's more like an
| encyclopedia. Some things I have the background for, others not
| so much.
| detourdog wrote:
| This is a really bad take. I went straight to the algebraic
| geometry. I subject I'm very interested but have no math
| training.
|
| 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.
| kubielid wrote:
| I read it as a joke, fellow MIDI enthusiast.
| mturmon wrote:
| I'm an encyclopedia lover who works on a variety of applied math
| problems as part of $dayjob at a national lab.
|
| 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.
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