# volume: The volume element In wedge: The Exterior Calculus

## Description

The volume element in n dimensions

## Usage

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

## Arguments

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

## Details

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.

## Author(s)

Robin K. S. Hankin

## References

Spivak

`zeroform`,`as.1form`
 ```1 2 3 4 5``` ```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 ```