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\section{Gaussian Coordinates}  Two neighbouring points $P$ and $P'$ on the surface then correspond to the co-ordinates  $$   P: (u, v) \\   P': (u + du, v + dv),   $$ where $du$ and $dv$ signify very small numbers. In a similar manner we may indicate the distance (line-interval) between $P$ and $P'$, as measured with a little rod, by means of the very small number ds. Then according to Gauss we have   $$   ds^2 = g_{11} du^2 +2g_{12}\,du \,dv + g_{22} dv^2   $$ have.