Relativity
The Special and General Theory
in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of u and a value of v belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point P in the diagram has the Gaussian co-ordinates u = 3, v = 1. Two neighboring 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
| ds2 = g11du2 + 2g12dudv + g22dv2 |
where g11, g12, g22, are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g11, g12 and g22, determine the behavior 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:
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 co-ordinates are simply Cartesian ones. It is clear
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