@mkoeris @stevenstrogatz @AIsakovic1 @johncarlosbaez @techreview
A thread on diagrammatic calculations:
Re. Nov 13, 2019 MIT Technology Review
"How to turn the complex mathematics of vector calculus into simple pictures"https://www.technologyreview.com/s/614704/how-to-turn-the-complex-mathematics-of-vector-calculus-into-simple-pictures/ …
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Replying to @chaaosbook @mkoeris and
Thanks for the thread. Interesting to hear that you are sold on this, Predrag.
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Replying to @stevenstrogatz @chaaosbook and
Sold on it? He practically wrote the book! And the book is here: http://birdtracks.eu/version9.0/GroupTheory.pdf …
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Replying to @johncarlosbaez @chaaosbook and
I wasn’t talking about bird tracks! I know Predrag wrote the book – I have it, and love it! I was talking about the newly proposed diagrammatic approach to vector calculus (The topic of Predrag’s thread; I asked him about it because I know he likes diagrammatic methods.)
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Replying to @stevenstrogatz @chaaosbook and
The diagrammatic methods for vector calculus and the diagrammatic methods for group representation theory are two aspects of the same game. Penrose invented these methods for tensor calculus, which combines vector calculus and group rep theory. So Predrag should like both.
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Replying to @johncarlosbaez @chaaosbook and
Ok, thanks. I had not appreciated that the vector calculus diagrammatic methods were essentially the same thing as the methods for group theory.
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Replying to @stevenstrogatz @chaaosbook and
Yeah, the cool thing is that all the usual operations in 3d vector calculus are covariant under 3d rotations, so they're all about "things you can do with representations of SO(3)". The all-important Levi-Civita symbol epsilon_{ijk} is the star of the show here.
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Replying to @johncarlosbaez @stevenstrogatz and
Actually, I would argue that the real star of the show is the inner product tensor, g_{ij}.
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Replying to @MathPrinceps @stevenstrogatz and
Yes, I wanted to mention that one too. I think epsilon_{ijk} is more "tricky" and thus commands more attention, especially in the rules below, which look much more fun as diagrams. So it's like the temperamental star of a soap opera, while g_{ij} is the sturdy sidekick.pic.twitter.com/jUL2papPOk
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Replying to @johncarlosbaez @MathPrinceps and
Of course conceptually it's more fundamental that SO(3) preserves the metric than that it preserves the volume 3-form. But when we learn vector calculus the intricacies of the cross product and curl eat up more brain cells than the dot product and divergence.
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The interesting thing is that we can identify SO(3) without mentioning epsilon: it simply consists of those O(3) elements that preserve orientation. And an orientation is a choice not of an epsilon tensor, but rather of a ray in the 1-d vector space of such tensors.
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