212 reputation
18
bio website pathintegral.org/blog
location
age
visits member for 1 year, 6 months
seen Aug 14 '13 at 17:49

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”?
Jan
15
awarded  Scholar