Sec. 2.3 EUCLIDEAN METRIC TENSOR 35 rectangular Cartesian, the length of the element ds is determined by Pythagoras' rule: Here ds2 is the square of ds, not ds with a superscript 2. When a coordinate transformation (1) is effected, we obtain from Eq. (2), according to the ordinary rules of differentiation, (4) Substituting into (3), we have dxa dXi ds2 = -- dOkd8". d6k 88, If we define the functions
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