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Unformatted text preview: L X vol = 1 p | g | i p | g | X i . Proof : 1. Direct calculation. Use L X g i j . The scalar div X is called the divergence of the vector eld X . Remark. Let Y 1 , . . . , Y n be vector elds invariant under the vector eld X . Then div X = ( L X vol )( Y 1 , . . . , Y n ) vol ( Y 1 , . . . , Y n ) = d dt log vol ( Y 1 , . . . , Y n ) t = . Thus, div X is the logarithmic rate of change of the volume along the ow. Remark. Let n-1 be an ( n-1)-form dened by = i X vol . In components i 1 ... i n-1 = X j p | g | ji 1 ... i n-1 Then d = ( div X )vol . To prove this compute in local coordinates ( d ) i 1 ... i n = n [ i 1 X j p | g | ji 2 ... i n di geom.tex; April 12, 2006; 17:59; p. 116...
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