Proof of euclidean distance

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HallsofIvy
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How do you prove that for a set of coordinates you are supposed to take
[tex]\mathrm{d}s^2=\mathrm{d}x_i\mathrm{d}x^i[/tex]
for the distance? I mean in a very abstract fashion. All I know is that there is some coordinate mesh. Why don't I take other powers for the distance for example?

Or if that bilinear equation is only a special case, then why is our space obeying it?
It finally dawned on me that Gernuk did NOT write [itex]ds^2= dx^idx^i[/itex], he wrote [itex]ds^2= dx_idx^i[/itex]. If the space is NOT Euclidean then [itex]dx_i= g_{ij}dx^j[/itex] so what he really wrote was that [itex]ds^2= g_{ij}dx^idx^j[/itex] the general formula for a metric tensor.
 

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