Alex Eftimiades
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 Feb 2 comment Hodge decomposition on a manifold with a nontrivial connection Not really--at least not within Cartan's framework. The more I read about these subjects, the more I suspect David Hestenes's Geometric Calculus is an easier framework for me. If I come across an answer, I will be sure to post it. Jan 9 awarded Yearling Aug 20 awarded Revival Jan 27 awarded Tumbleweed Jan 20 revised Cauchy-Schwarz for metrics with arbitrary signatures "negative norm" changed to "negative inner product" Jan 20 revised Cauchy-Schwarz for metrics with arbitrary signatures added some more tags Jan 20 asked Cauchy-Schwarz for metrics with arbitrary signatures Jan 19 awarded Quorum Jan 18 accepted The most general notion of a directional derivative Jan 18 comment The most general notion of a directional derivative I actually tried to (informally) generalize a directional derivative by using the notion of a direction here: pathintegral.org/blog/?p=582 I did not mention it because I did not want to look like I was just advertising myself or something. I was hoping that with some formal definitions, I would be able to prove it was equivalent to some generalized notion of a derivative known to be equivalent to the standard one in $\mathbb R^n$. Thank you for the thorough answer. I learned something about generalizations too--namely that the $C'$ in my question need not be unique! +1 Jan 18 revised The most general notion of a directional derivative minor change--better phrasing Jan 18 asked The most general notion of a directional derivative Jan 17 comment Is there/need there be a mathematical definition of a “direction”? I thought I would mention that on a Minkowski metric, I could say that the product of the "time" direction with itself is -1, so one does not have to equate the "direction product" with the cosine of an angle. I presume in its broadest sense, it is just a mapping from pairs of directions to the interval [-1,1]. Jan 17 awarded Supporter Jan 17 comment Is there/need there be a mathematical definition of a “direction”? I had to read the Wikipedia article on equivalence relations (I have a relatively narrow math background), but once I understood it, I realize your definition fits my notion of a direction quite well. +1 Jan 17 accepted Is there/need there be a mathematical definition of a “direction”? Jan 17 revised Is there/need there be a mathematical definition of a “direction”? added inner product space tag. Jan 17 comment Is there/need there be a mathematical definition of a “direction”? I could say that the reason that a unit vector is still a vector and not a direction is because a vector divided by a magnitude is still a vector. The reals are closed under division, so any time you divide a vector by a magnitude you get a new vector with a different magnitude, not a direction. Jan 17 comment Is there/need there be a mathematical definition of a “direction”? Yes I know. Those however are still technically vectors and can be added and subtracted as such. I was wondering whether there was any point to defining a mathematical object analogous to a unit vector without a well defined notion of addition and subtraction. Jan 17 asked Is there/need there be a mathematical definition of a “direction”?