� � � Lecture 8 8.251 Spring 2007 In general, A = � dξ1, dξ2√g Note a line moving along in ξ1 direction is not necessarily orthogonal to a line moving in ξ2 direction. d�v1 = (dξ1 , 0) d�v2 = (0, dξ2) 2D: no reference to 3D space, just 2D space param. by ξ1 and ξ2 dA = |d�v1||d�v2| sin θ (d�v1)2 = gij dv1i dv1 j = g11(dξ1)2 d�v1 · d�v2 = gij dv1i dv2 j = g12dξ1dξ2 dA = [g11(dξ)2][g22(dξ2)2] − [g12dξ1dξ2]2 = dξ1dξ22g11g22 − g12 = dξ1dξ2 det(gij ) Works in any number of dimensions (though here proved only for 2) 1Lecture 8 8.251 Spring 2007 Generalization to n dimensions Metric always a square matrix with a determinant. Consider generalized parallelopiped in N dimensions. Volume in terms of corner vectors? (Standard from N-dim Euclidean geometry) Vol = det[Vik] where k is the vector index and i is the vi subscripts [1, . . . , N] Can construct orthogonal vector sets v2�involves adding or subtracting as much of v1 to v2 to get orthogonality. Shifts parallelopiped into rectangle without changing volume. Every operation is determinant-invar.
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