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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Replying to @johncarlosbaez @MathPrinceps and
The reason cross product and curl identites seem difficult is that the Levi-Civita symbol is the subtlest tensor most of us have encountered at that point in our development! So this is where diagrammatic methods can do the most good at an early stage.pic.twitter.com/5znqLcPRKM
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Replying to @johncarlosbaez @DynamicsSIAM and
Alternatively, the cross product seems difficult because lurking in there are the notions of oriented inner product space, exterior product, and hodge dual. (Which is why the Levi-Civita symbol is subtle.)
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Replying to @RobJLow @DynamicsSIAM and
One reason the cross product is hard is that to understand it as a map between reps of O(3) (not just SO(3)) we need to understand the vector and pseudovector reps of O(3). For example, the cross product of two vectors is a pseudovector. Well-trained physicists learn this stuff.
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Do you object to viewing the cross-product as simply a bilinear, antisymmetric, Jacobi-identity-obeying Lie bracket of two vectors? Surely this is a simpler, more natural conception of the cross-product than that which involves metric-duals of Hodge duals of bivectors.
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Replying to @MathPrinceps @johncarlosbaez and
I'm merely saying that the Lie bracket tensor, which is 2-covariant and 1-contravariant, ought perhaps to be viewed as the relevant object, if we're talking about cross-products. We can build such a gadget by "raising an index" on epsilon, of course. But why go into all that?
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Replying to @MathPrinceps @RobJLow and
I don't object to any mathematically valid way of working with these things. I like and use them all. I don't know how this turned into an argument.
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