Reading Mathematics (2002) [pdf]

#1

Post by kercker » Wed, Dec 27, 2017, 6:51 AM UTC

Reading Mathematics (2002) [pdf]
math.cornell.edu

Re: Reading Mathematics (2002) [pdf]

#2

Post by sergj » Thu, Dec 28, 2017, 2:23 AM UTC

From which book is this?

Re: Reading Mathematics (2002) [pdf]

#3

Post by tw1010 » Thu, Dec 28, 2017, 2:32 AM UTC

Funny how utterly natural and subconsious this stuff becomes after a while. I almost felt like commenting something snyde about how superfluous it is to make it this explicit, but then I realized that it was only a few years ago that it made my consious brain totally overwhelmed.

Re: Reading Mathematics (2002) [pdf]

#4

Post by stablemap » Thu, Dec 28, 2017, 2:52 AM UTC

sergj wrote: post #2

From which book is this?

Looks to be a multivariate calculus text by the Hubbards:

http://matrixeditions.com/5thUnifiedApproach.html

Re: Reading Mathematics (2002) [pdf]

#5

Post by thewayfarer » Thu, Dec 28, 2017, 2:52 AM UTC

sergj wrote: post #2

From which book is this?

Vector Calculus, Linear Algebra, and Differential Forms, A Unified Approach by John H. Hubbard and Barbara Burke Hubbard.

Re: Reading Mathematics (2002) [pdf]

#6

Post by opportune » Thu, Dec 28, 2017, 4:16 AM UTC

tw1010 wrote: post #3

Funny how utterly natural and subconsious this stuff becomes after a while. I almost felt like commenting something snyde about how superfluous it is to make it this explicit, but then I realized that it was only a few years ago that it made my consious brain totally overwhelmed.

Yeah at first I was thinking "is this not the only way of understanding written mathematics?!!"

Then I realized that when I was a freshman undergrad, I understood much less than half of the terminology that is considered basic mathematics. The formalisms behind WOLOG and \forall \epsilon >0 \exists \delta > 0 s.t. yadda yadda were completely lost on me. A lot of mathematics is simply familiarity, and that's easy to forget

Re: Reading Mathematics (2002) [pdf]

#7

Post by johnbender » Thu, Dec 28, 2017, 4:39 AM UTC

Beyond notation and ordering, I have had the best results reading and comprehending complex concepts, including mathematics, by taking the following suggestion to the extreme:

> Read with pencil and paper in hand, making up little examples for yourself as you go on.

I like to find a difficult question that I can answer with an understanding of the material. This acts as a litmus test of my understanding and a forcing function.

The question can be almost anything, but a general approach I use is to write a "compiler" that maps some concept from the material to a concept I already understand (this normally takes the form of a denotational semantics). Then the question would be, "How can I interpret X as Y?" This technique has its limits since the material can't be too far afield from something I already know and the idea isn't novel but it has been effective for me. The critical bit is forcing myself to write down a fairly comprehensive mapping function. This gets me into the dark corners of my understanding very quickly and adds new questions to answer.

Re: Reading Mathematics (2002) [pdf]

#8

Post by aisofteng » Thu, Dec 28, 2017, 4:45 AM UTC

johnbender wrote: post #7

Beyond notation and ordering, I have had the best results reading and comprehending complex concepts, including mathematics, by taking the following suggestion to the extreme: > Read with pencil and paper in hand, making up little examples for yourself as you go on. I like to find a difficult question that I can answer with an understanding of the material. This acts as a litmus test of my understanding and a forcing…

I do the exact opposite - I don't write anything down and try to visualize what I'm reading entirely. I don't start writing anything down when doing exercises until I have the entire solution in mind (unless it's something computational). It's hard to get used to doing at first, but, after a couple years, my abstract reasoning skills developed significantly past most or all of my peers.

Re: Reading Mathematics (2002) [pdf]

#9

Post by du_bing » Thu, Dec 28, 2017, 4:47 AM UTC

thewayfarer wrote: post #5

sergj wrote: post #2

From which book is this?

Vector Calculus, Linear Algebra, and Differential Forms, A Unified Approach by John H. Hubbard and Barbara Burke Hubbard.

Great Thanks!

Re: Reading Mathematics (2002) [pdf]

#10

Post by jostylr » Thu, Dec 28, 2017, 11:08 AM UTC

stablemap wrote: post #4

sergj wrote: post #2

From which book is this?

Looks to be a multivariate calculus text by the Hubbards: http://matrixeditions.com/5thUnifiedApproach.html

That's correct (a later edition than the pdf given the date). It is a beautiful book that blends the practical and abstract together to really give a solid foundation in both linear algebra and multivariable calculus topics. If someone asked me for exactly one book to read after learning single variable calculus, this is the book I would recommend. It is long, but extremely satisfying.

It covers in detail many topics often glossed over while keeping an eye on what is actually applicable, including such things as (taken from their page):

> More big matrices! We included the Perron-Frobenius theorem, and its application to Google's PageRank algorithm More singular values! We included a detailed proof of the singular value decomposition, and show how it applies to facial recognition: "how does Facebook apply names to pictures?"