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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Replying to @johncarlosbaez @RobJLow and
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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I greatly regret any impression I may have given that I was in argumentative mood. I absolutely was not, and meant simply to ask how you prefer to think, and why. I have generally found this a fruitful question with you, and often been impressed and enlightened by your answers.
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Replying to @MathPrinceps @RobJLow and
Thanks - sorry, it seemed to be turning into an argument. I like all the viewpoints you espouse. I also like the diagrammatic approach to classifying reps of compact Lie groups so nicely explained by Predrag here (though I'd use more category theory): http://birdtracks.eu/version9.0/GroupTheory.pdf …
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Replying to @johncarlosbaez @MathPrinceps and
I also think more physicists should learn all the 3d real irreps of GL(3,R). Many of them know the idea informally: vectors are either "axial" or "polar", and they can also have dimensions of length^k for any k, which affects their transformations under rescaling.
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