Found in 16 comments on Hacker News

rsj_hn · 2021-04-01
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The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Grad...

but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.

The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that

Integral over the boundary = Integral over the interior of the divergence, or more poetically

Int_(dA)A = Int_A dA

Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.

So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a).

The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).

I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.

I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

nabla9 · 2019-10-24
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"Mathematics : Its Content, Methods and Meaning" by A. D. Aleksandrov, A. N. Kolmogorov ,M. A. Lavrent'ev. (3 Volumes)

This is a classic and exactly what you are seeking for. I think it was originally published in 1962.

https://www.goodreads.com/book/show/405880.Mathematics

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

whitepoplar · 2019-01-18
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This book is fantastic and pretty much takes you through an entire undergrad mathematics course: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

synthmeat · 2018-03-11
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I’m going to go with a few assumptions here:

a) You don’t do this full time.

b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.

c) You are skilled with programming/software in general.

In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.

Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I’d recommend getting great and cheap russian recap of mathematics up to 60s [1] and a modern coverage of the field in relatively light essay form [2].

Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.

[1] https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

[2] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

hyporthogon · 2017-03-06
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This is awesome, thanks.

Obligatory 'zoomout' recommendation: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V..., which I learned about from HN (http://hackernewsbooks.com/book/mathematics-its-content-meth...). Wish I had read/pondered this before grad math classes.

whitepoplar · 2016-11-30
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This book is *fantastic* and should get you pretty far: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

dpflan · 2015-09-20
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This is always a good topic with numerous HN submissions and comments:

Just from a search, there are some great results:

1. *Mathematics for Computer Science* - HN Submissions: https://news.ycombinator.com/item?id=9311752, https://news.ycombinator.com/item?id=3694448

2. *How to Read Mathematics* - HN Submissions: https://news.ycombinator.com/item?id=4030812, https://news.ycombinator.com/item?id=1576969

__

Here are some other excellent mathematics books:

1. *Mathematics: Its Content, Methods and Meaning*

Containing the thoughts and direction of numerous mathematicians including Kolmogorov, this is a great survey of the field of mathematics. It touches upon Analysis, Analytic Geometry, Probability, Linear Algebra, Topology, and more. [1.]

2. *Concrete Mathematics: A Foundation for Computer Science*

Containing the thoughts and direction of mathematician and computer scientists such as Donald Knuth, this is a great reference for computer science related mathematical concepts focusing on continuous and discrete concepts. [2.]

__

[1.] http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

[2.] http://www.amazon.com/Concrete-Mathematics-Foundation-Comput...

nabla9 · 2015-02-04
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Usually good books travel from English speaking world to other languages as translations.

One of the masterpieces that has gone the opposite direction is:

Mathematics: Its Content, Methods and Meaning (three volumes bound as one) by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent'ev (18 authors total) http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

This book is really good companion for autodidacts. It's basically overview of mathematics.

wallflower · 2014-12-28
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This isn't a course but this book has been consistently word-of-mouth recommended as a comprehensive survey of undergraduate mathematics.

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

brandonmenc · 2014-12-04
· Original
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Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

Covers something like three years of an undergraduate degree in mathematics. Lots of words - but that text is used to develop an understanding of the concepts and images. Considered a masterpiece. An enjoyable read.

ivan_ah · 2014-10-03
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I've looked at a bunch of these math compendiums while researching what to include in my book, and this one seemed the best so far: http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do... The writing isn't very hand-holdy, but it covers a lot of important topics, and without too much fluff.

For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.

I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.

fensterbrett · 2014-09-02
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http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

oskarth · 2014-04-12
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Not the same type of book, but you could do a lot worse than reading through *Mathematics: Its Content, Methods and Meaning*, by M. A. Lavrent’ev, A. D. Aleksandrov, A. N. Kolmogorov. It's an amazing book which gives a mathematical (but not rigorous in the sense of proofs etc.) overview of most of mathematics.

http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...

nabla9 · 2013-12-28
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May I present one of the greatest math books for general audience: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev.

http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...

Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come even close to this. Reading this book together with the The Princeton Companion to Mathematics was real treat.

programnature · 2008-12-22
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This book looks awesome. Mathematics is so vast that as a student its hard to know what classes you should take, or even what the map of the territory looks like. That is one of the reasons I got frustrated with my pure math BS.

The Russian school put out some pretty massive volumes, like

http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin...

but I found those to be a little too caught up in the proposition/proof cycle to be useful as a guide to the uninitiated.

https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...