volume: The volume element
In wedge: The Exterior Calculus

Description Usage Arguments Details Author(s) References See Also Examples

The volume element in n dimensions

1 2	volume(n) is.volume(K)

`n`	Dimension of the space
`K`	Object of class `kform`

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

Spivak

zeroform,as.1form

as.kform(1) %^% as.kform(2) %^% as.kform(3)  == volume(3)  # should be TRUE

o <- volume(5)
M <- matrix(runif(25),5,5)
det(M) - as.function(o)(M)   # should be zero