taping this to every freshman classroom in my degree
postato in Uncategorized
La nuova BBS è in fase Alpha. I post precedenti al 22 luglio 2024 potrebbero non essere trasferibili, ma rimarranno disponibili per la lettura su /old/.
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taping this to every freshman classroom in my degree
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RE: @schappi@brotka.st do u know anything at all about the lie derivative vs covariant derivative discourse ?@fiore The downside of the Lie derivative is that yeah, it is *just* the directional derivative. Think of how the directional derivative of f from Calc 2 doesn't involve the dot product at all, you just kind of take the 1D derivative of f along the line x+tv...postato in Uncategorized
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RE: @schappi@brotka.st do u know anything at all about the lie derivative vs covariant derivative discourse ?@fiore oh ok since you are doing GR then I can tell you the more general version lol one nice thing about the lie derivative is that it extends very nicely to the more general case of tensor fields (because the Lie derivative is basically taking the pullback of a flow and you can write that in coordinates), though the covariant derivative has a bit of a nasty expression where you need to take the action of the vector field and then do the covariant derivative fo each component in the input...postato in Uncategorized
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RE: @schappi@brotka.st do u know anything at all about the lie derivative vs covariant derivative discourse ?@fiore yes, in fact you have a formula that relates the covariant derivative and the metric:postato in Uncategorized
D_v g(X,Y)=g(CD_v X,Y)+g(X, CD_v Y)
where CD is the covariant derivative and D_v is the "regular" derivative (which only really works for functions).
afaik there is no formula like this for the lie derivative -
RE: @schappi@brotka.st do u know anything at all about the lie derivative vs covariant derivative discourse ?@fiore In a very crude way, the covariant derivative is the one that ignores what is happening outside of the space you are in, whereas the Lie derivative takes everything into account.postato in Uncategorized
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RE: @schappi@brotka.st do u know anything at all about the lie derivative vs covariant derivative discourse ?@fiore I don't really know a source for it that I can recommend since this is a complex topic but I think the fundamental different that is worth to keep in mind is that the Lie derivative only cares about the topology of the space (so, the overall shape), whereas the covariant derivative/connection also cares about the geometry of the space (the distances between points, the angles between lines, the areas...).postato in Uncategorized
For example, you can define a torus as a piece of papers with 1D portals on the sides. The Lie derivative will not change regardless of whether you consider a donut or a square with portals. But if you want to consider a connection, it will be different in both cases.
Another way to see it is with physics and circular paths. When you are moving in circles with a certain speed, then you have both a "linear" acceleration (the one that pushes you forward) and a "centripetal" acceleration (the one that makes you turn, preventing you from escaping the circular path). The "linear" accel. is the covariant derivative of your speed. -
i love when people write "spoiler" as cwi love when people write "spoiler" as cwpostato in Uncategorized
spoiler for what
SPOILER FOR WHAT??? -
I must cringe.I must cringe.postato in Uncategorized
Cringe is the mind-foster.
Cringe is the little-life that prevents total obliteration.
I will not face my cringe.
I will not permit it to pass over me and through me.
And when it has filled me, I will turn the inner eye to see its path.
Where the cringe has fostered there will be nothing.
Only cringe will remain. -
why do my students say i make things look hard fedi???why do my students say i make things look hard fedi???postato in Uncategorized