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MathPrinceps's profile
Laurens Gunnarsen
Laurens Gunnarsen
Laurens Gunnarsen
@MathPrinceps

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Laurens Gunnarsen

@MathPrinceps

Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.

Joined June 2012

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    1. John Carlos Baez‏ @johncarlosbaez 8 Dec 2019
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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.

      3 replies 2 retweets 30 likes
    2. Steven Strogatz‏Verified account @stevenstrogatz 9 Dec 2019
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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.

      2 replies 0 retweets 18 likes
    3. John Carlos Baez‏ @johncarlosbaez 9 Dec 2019
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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.

      2 replies 4 retweets 13 likes
    4. Laurens Gunnarsen‏ @MathPrinceps 9 Dec 2019
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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}.

      1 reply 0 retweets 1 like
    5. John Carlos Baez‏ @johncarlosbaez 9 Dec 2019
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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

      3 replies 1 retweet 5 likes
    6. John Carlos Baez‏ @johncarlosbaez 9 Dec 2019
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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.

      2 replies 0 retweets 4 likes
    7. John Carlos Baez‏ @johncarlosbaez 9 Dec 2019
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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

      1 reply 8 retweets 22 likes
    8. Robert Low‏ @RobJLow 9 Dec 2019
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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.)

      2 replies 1 retweet 6 likes
    9. John Carlos Baez‏ @johncarlosbaez 9 Dec 2019
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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.

      3 replies 1 retweet 5 likes
    10. Laurens Gunnarsen‏ @MathPrinceps 10 Dec 2019
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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.

      1 reply 0 retweets 1 like
      Laurens Gunnarsen‏ @MathPrinceps 10 Dec 2019
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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?

      9:44 AM - 10 Dec 2019
      1 reply 0 retweets 0 likes
        1. New conversation
        2. John Carlos Baez‏ @johncarlosbaez 10 Dec 2019
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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.

          2 replies 0 retweets 2 likes
        3. Laurens Gunnarsen‏ @MathPrinceps 10 Dec 2019
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          Replying to @johncarlosbaez @RobJLow and

          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.

          1 reply 0 retweets 1 like
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