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  • 32 votes cast
Mar
30
accepted Position or rank of an arbitrary rational number
Mar
28
comment Position or rank of an arbitrary rational number
I was just looking for a way to rank the rational numbers, so I don't mind including repeated rationals. Pedantically, the sequence you quote only follows a northeast direction, not zig-zagging northeast, southwest, northeast as it does in the table.
Mar
28
revised Position or rank of an arbitrary rational number
added 12 characters in body
Mar
28
asked Position or rank of an arbitrary rational number
Feb
3
awarded  Tumbleweed
Jan
27
revised How can two parameters define a point?
added 44 characters in body
Jan
27
asked How can two parameters define a point?
Jan
1
comment Total confusion about differential one-forms and non-coordinate bases
Thanks very much.
Jan
1
accepted Total confusion about differential one-forms and non-coordinate bases
Dec
31
comment Total confusion about differential one-forms and non-coordinate bases
But my question was asking for examples of non-coordinate bases.
Dec
31
comment Total confusion about differential one-forms and non-coordinate bases
Thanks for that. I was aware of the summation convention. I've come thus far via self-studying general relativity, where the usual bases are the coordinate basis vectors $e_{\mu}=\frac{\partial}{\partial x^{\mu}}$ and basis one-forms $\omega^{\nu}=dx^{\nu}$. That's why I was confused by janmarqz's answer, which seemed to me to imply $e_{k}=\frac{\partial}{\partial x^{k}}$, which is a coordinate basis? Also, as I asked in my previous comment, does $df=\frac{\partial f}{\partial x^{a}}dx^{a}$ for both coordinate and non-coordinate bases? If yes, what are the basis one-forms for $df$?
Dec
16
comment Total confusion about differential one-forms and non-coordinate bases
@janmarqz - But doesn't $e_{k}(f)={\rm grad}f\cdot e_{k}=\frac{\partial f}{\partial x^{k}}$ imply $e_{k}=\frac{\partial}{\partial x^{k}}$, which is a coordinate vector basis?
Dec
15
comment Total confusion about differential one-forms and non-coordinate bases
Afraid I'm still confused as to how there are any basis vectors on the rhs. Given the definition $\omega^{a}e_{k}=\delta_{k}^{a}$ I would have thought both the basis one-forms and basis vectors on the lhs would have all disappeared in a puff of 1's and 0's.
Dec
14
comment Total confusion about differential one-forms and non-coordinate bases
So is there a relationship between a coordinate vector basis $\frac{\partial}{\partial x^{j}}$ and a non-coordinate vector basis $e_{k}$?
Dec
14
asked Total confusion about differential one-forms and non-coordinate bases
Dec
10
comment Basis of differential one-form confusion
@Phoenix87 - thanks very much. That's much clearer.
Dec
10
comment Basis of differential one-form confusion
Profuse apologies. I stand corrected.
Dec
10
comment Basis of differential one-form confusion
I didn't know that (obviously, otherwise I wouldn't have asked the question - seems a tad harsh to downvote after six minutes). How would I write the first equation, with indices, to show it referred, for example, only to coordinate bases?
Dec
10
asked Basis of differential one-form confusion
Oct
18
awarded  Benefactor