Lecture 8 In general A 8 251 Spring 2007 d 1 d 2 g Note a line moving along in 1 direction is not necessarily orthogonal to a line moving in 2 direction d v1 d 1 0 d v2 0 d 2 2D no reference to 3D space just 2D space param by 1 and 2 dA d v1 d v2 sin d v1 gij dv1i dv1j g11 d 1 2 2 d v1 d v2 gij dv1i dv2j g12 d 1 d 2 g11 d 2 g22 d 2 2 g12 d 1 d 2 2 2 d 1 d 2 g11 g22 g12 d 1 d 2 det gij dA Works in any number of dimensions though here proved only for 2 1 Lecture 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 2
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