The volume element in n dimensions
Dimension of the space
Object of class
Spivak phrases it well (theorem 4.6, page 82):
If V has dimension n, it follows that . has dimension 1. Thus all alternating n-tensors on V are multiples of any non-zero one. Since the determinant is an example of such a member of . it is not surprising to find it in the following theorem:
Let . be a basis for V and let .. If . then
ommitted; see PDF
(see the examples for numerical verification of this).
Neither the zero k-form, nor scalars, are considered to be a volume element.
Robin K. S. Hankin
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