Peter4075
Reputation
245
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 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