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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Replying to @johncarlosbaez @RobJLow and
Off topic & high school. I try to persuade people that force isn't a vector: a force has a vector property. You can't add 2 forces to get a force, but you can add their vector properties to get the resultant, which is a vector but not a force: it no position. What d'you think?
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Replying to @pippinsboss @RobJLow and
As a mathematical physicist, the difference between "being a force" and "having a vector property" is too philosophical for me to bother with unless it's encoded in math in some way.
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I think he may have been aiming at the distinction between a vector and a translation -- that is, between a group and a space on which a group acts simply-transitively.
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