Here we list a number of general identities for the completely antisymmetric volume tensor in dimensions. The most useful are those involving the tensor product (including, as needed, contractions over indices), of the volume tensor with itself :
Start by writing Equation (360) as
The last thing we need is the variation of the determinant of the metric, since it enters directly in the integrals of the actions. Treating the metric as a matrix, and “normalizing” the by dividing by its one independent component, the determinant is given by
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