Two neighbouring
points P and P1 on the surface then correspond to the co-ordinates
P: u,v
P1: 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 P1, as measured
with a little rod, by means of the very small number ds. Then
according to Gauss we have
ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2
where g[11], g[12], g[22], are magnitudes which depend in a perfectly
definite way on u and v. The magnitudes g[11], g[12] and g[22],
determine the behaviour of the rods relative to the u-curves and
v-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the u-curves and v-curves and to attach numbers
to them, in such a manner, that we simply have :
ds2 = du2 + dv2
Under these conditions, the u-curves and v-curves are straight lines
in the sense of Euclidean geometry, and they are perpendicular to each
other. Here the Gaussian coordinates are samply Cartesian ones.
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