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Functional Differential Geometry (2012) [pdf]

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Re: Functional Differential Geometry (2012) [pdf]

#11

Post by adolgert »

leephillips wrote: post #7
adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

What do you think of Burke's Applied Differential Geometry ?

Haven't read it unfortunately. The table of contents shows a much more rapid treatment. When working outside of a university course, examples can be very useful. Anyone else read Burke?

Re: Functional Differential Geometry (2012) [pdf]

#12

Post by wuliwong »

adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

I thought the same thing when I saw this on the front page! I have a Ph.D. in physics and I still have only had one or two courses which delved much into differential geometry. :)

Re: Functional Differential Geometry (2012) [pdf]

#13

Post by [deleted] »

ajtulloch wrote: post #6

I think a functional (or CS) perspective is an interesting approach, but I think you can lose some nice results in introductory differential geometry by following just this course. For example, the Gauss-Bonet theorem doesn't appear to be covered, which is an incredibly beautiful result linking the geometry and topology of manifolds. For a more classical introduction to differential geometry requiring only multivaria…

[deleted]

Re: Functional Differential Geometry (2012) [pdf]

#14

Post by andyclymer »

What makes this book exciting to me is that "computer programming is used to explain the mathematical formulas".

As someone who struggles with some areas of mathematics (the ones I wish I paid more attention to in high school because I could be using them now) but who finds more ease with understanding the code examples used in this book, I would love to read more books like this. What other books on mathematics could you recommend to a programmer such as myself?

Are there any books on Linear Algebra or Calculus who take the same approach, to teach using a programming language?

Re: Functional Differential Geometry (2012) [pdf]

#15

Post by esom »

adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

Maybe the reason people are upvoting this is because the book uses software and programming to make the mathematics precise.

This is not a typical math book - just like Sussman's classes in which you program up physics stuff as you go along is not like a typical physics class.

Re: Functional Differential Geometry (2012) [pdf]

#16

Post by notthemessiah »

throwaway283719 wrote: post #9

I started implementing the 'scmutils' library that this book uses in Haskell. I stopped working on it a year or so ago (because life) but it's here if anyone is interested - https://github.com/chris-taylor/Classical-Mechanics

Glad you posted this, had tried to do the same earlier this year. Also related, this POPL paper shows how Noether's Theorem (the idea that every symmetry yields a conservation law e.g. rotational symmetry => angular momentum conservation) can be derived from applying parametric polymorphism to Lagrangian mechanics: http://bentnib.org/conservation-laws.html

This summary of the above paper is much more comprehensible: http://ezyang.tumblr.com/post/74405137526/robert-atkey-from-...

Re: Functional Differential Geometry (2012) [pdf]

#17

Post by snake_plissken »

adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

Where else other than HackerNews am I supposed to find random and interesting PDFs to print out and leave in my bathroom for people to read?

Re: Functional Differential Geometry (2012) [pdf]

#18

Post by micro_cam »

adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

Hirani's thesis on Dec is also worth a read. Useful stuff indeed:

http://www.cs.jhu.edu/~misha/Fall09/Hirani03.pdf

Re: Functional Differential Geometry (2012) [pdf]

#19

Post by cowpig »

adolgert wrote: post #2

Enough of you care about this to vote it to the front page? Who are you people? In that case, I have two favorite books on this topic. Bamberg and Sternberg, A Course in Mathematics for Physics Students. It's a redo of calculus using differential geometry from the start. A very pretty way to do E&M, or calculations on the surface of the Earth, or vector flows. Grady and Polimeni, Discrete Calculus. This is how you do…

Treating data as a distribution in high-dimensional space is at the core of machine learning, and differentiating across those dimensions is typically how learning is done.

I don't know if this is true, but at a glance it looks to me like differential geometry could be useful to people in machine learning. Am I right about that? I am probably going to look closer tonight...

Re: Functional Differential Geometry (2012) [pdf]

#20

Post by jordigh »

I have a big problem when people describe mathematics as just "a language". The people who usually do this say this because they are not interested in the mathematics itself, but in some other subject that happens to require mathematics. In this case, the actual subject of interest appears to be relativity, not mathematics. It seems to me like a disrespect to say that mathematics is a language, because it seems to suggest that mathematics on its own has no substance, nothing intrinsic to itself.

To be sure, there is a mathematical language, a notation with much ambiguity that is typically ok when conveyed to other humans, because humans are great with ambiguity. This book is attempting to get rid of that ambiguity by rewriting it in notation that does not allow for ambiguity and present it to computers, who are still very bad at ambiguity. But this mathematical language is not mathematics, and there is real substance behind the language itself, just like poems can be translated into other natural languages or algorithms can be written in several programming languages.

Calling mathematics a language confuses form with substance or declares that it's all form and no substance.

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