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 calculus on graphs, which is how you do scoring algorithms on graphs. You know, for big data.
Discrete exterior calculus in Python! (almost forgot) http://arxiv.org/abs/1103.3076
They have demos that show how extraordinarily powerful it is.
That ought to keep you busy for a few months, you odd ducks.
Wait, Bamberg wrote a book on differential forms and didn't tell me when I asked him for self-study references (he recommended Hubbard&Hubbard IIRC)? Crazy. (Context: I took a Hilbert Space class from him and only then realized what I had missed by skipping his vector calculus class for the standard intro to analysis.) Maybe he made the generous assumption that I'd already thoroughly reviewed the available books and had found his wanting?
Oh well. Either way, I'll have to check it out. I never did get far in Flanders (which left too much implicit, making it difficult to read) and H&H didn't seem to emphasize the E&M applications I was looking for (admittedly I only skimmed it). It appears that said applications are the focus of Bamberg's book, so it would likely be a better fit.
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...
It is -- there's a whole area called Information Geometry which treats the parameter spaces of statistical models as Riemannian manifolds under the Fisher information metric.
As an example application, when sampling from the posterior of a Bayesian model it can help to take this natural geometry of the parameter space into account, e.g. via Riemannian Manifold Hamiltonian Monte Carlo [1]
Then again, from what I've seen most Manifold Learning only uses the word manifold in a loose handwavey sense to motivate what they're doing -- the added abstraction level of differential geometry doesn't always add very much you're just interested in learning smooth functions from R^m -> R^n.
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. :)
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.
I second the Bamberg and Sternberg recommendation. Those books are great introductions to differential geometry and E&M. You can go directly to part 2 if you already have a strong calculus foundation, but part 1 will give that to you and make a gentle introduction to part 2 where differential geometry is explored in more detail.
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?
I've read Burke. ADG is excellent, though it would be even better if it had more concrete examples. It's also very much aimed at physicists (assuming advanced undergraduate knowledge of mechanics, E&M, etc.) and has less emphasis on mathematical rigor than most differential geometry texts.
I actually helped Burke out with a follow-up to ADG called Div, Grad, Curl are Dead, which looked very promising with lots of examples, but tragically he died from injuries sustained in a car accident before it could be polished for publication.
Thanks for the reference to p. It looks like a good addition to the Thorpe I've done and the do Carmo and Lee that I am currently working through. I appreciate it. This is reasonable?
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 multivariate calculus and some real analysis/point set topology, Do Carmo's "Differential Geometry of Curves and Surfaces" is a great textbook.
I put together a summary (key definitions/theorems) from an undergraduate course following Do Carmo at [2].
Lee's books are great as well, very different perspective, audience, motivation and development, but great. I am working through his first on topological manifolds now.
Also, at about the same level of difficulty but a different perspective:
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 -
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
In [Structure and Interpretation of Classical Mechanics](http://mitpress.mit.edu/sites/default/files/titles/content/s...), the same authors mention that a computational approach to calculus revealed errors in their own understanding of classical mechanics equations (such as Lagrange's equations), and they introduced new notation to address the problem. It appears that Functional Differential Geometry picks up that idea and runs with it.
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?
This reminds me of a concept I have been toying with for a while now that I would love to see implemented.
What I would absolutely adore is a wiki styled over the idea of covering mathematical topics in the style of a programming API, and teach it in the same manner. Meaning that someone could use the math as a concept to kludge some code together, or teach it to themselves in this style using the material.
I've been too lazy to start the wiki myself, and not nearly knowledgeable enough to throw together to a few topics to get the ball rolling.
If anyone likes this idea, please steal it, it's something I would love to see.
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.
I followed the installation instructions exactly and it works fine on my Debian machine. I had to modify the script `/usr/local/bin/mechanics` since I don't have a binary named `xterm` on my path. While I was at it, I made it skip that and just `exec $CMD` as it would do if `$DISPLAY` weren't set.
This `edwin` that it comes with is a weird (faithful) emacs clone that runs under mit-scheme. Emacs18, that is.
I carefully typed in the first few code examples and they ran flawlessly. There even is some gui tool which pops up a rendering of mathematical expressions.
This is amazing! I'd read Lagrange's eq in classical mechanics texts many times but my brain is just so tuned in thinking in terms of computations and programming that this approach in this PDF just feels natural. If lot of other things (how about General Relativity?) were written like this in terms of code, I would eat it up like buttered bread.
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 calculus on graphs, which is how you do scoring algorithms on graphs. You know, for big data.
Discrete exterior calculus in Python! (almost forgot) http://arxiv.org/abs/1103.3076 They have demos that show how extraordinarily powerful it is.
That ought to keep you busy for a few months, you odd ducks.