Gaussian Co-ordinates-background and derivation

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Gaussian Co-ordinates are introduced in Section XXV of Relativity, focusing on their derivation and background. The formula for the distance between two points in this system is expressed as ds² = g₁₁du² + 2g₁₂dudv + g₂₂dv², which represents a quadratic relationship essential for maintaining non-negative distances. The discussion highlights the importance of understanding this quadratic form as the most general representation in two-dimensional space.

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In Section XXV of Relativity we are introduced to Gaussian Co-ordinates. I can't seem to find much about this on the internet, perhaps there is another name? Can anyone link some background behind why this system was initially derived, and most importantly does anyone know the derivation of ds^{2} =g11du^{2}+2g12dudv+g22dv^{2} (formula for distance between two points on gaussian system)
Thanks in advance.
 
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That's pretty much just saying that the distance between two points must be a
quadratic
since distance cannot be negative- and that's the most general possible quadratic formula in two dimensions.
 
Thanks for that.
 

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