Roughly speaking, the metric tensor g_(i j) is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements d x_i in a generalized Pythagorean theorem: d s^2 = g_11 d x_1^2 + g_12 d x_1 d x_2 + g_22 d x_2^2 + .... In Euclidean space, g_(i j) = δ_(i j) where δ is the Kronecker delta (which is 0 for i!=j and 1 for i = j), reproducing the usual form of the Pythagorean theorem d s^2 = d x_1^2 + d x_2^2 + .... In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be "arithmetized" by way of introducing a sort of generalized coordinate system.

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