stokes (version 1.0-8)

volume: The volume element

Description

The volume element in \(n\) dimensions

Usage

volume(n)
is.volume(K)

Arguments

n

Dimension of the space

K

Object of class kform

Value

Function volume() returns an object of class kform; function is.volume() returns a Boolean.

Details

Spivak phrases it well (theorem 4.6, page 82):

If \(V\) has dimension \(n\), it follows that ^n(V). 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 ^n(V). it is not surprising to find it in the following theorem:

Let v_1,…,v_nv_1,...,v_n be a basis for \(V\) and let ^n(V).. If w_i=_j=1^n a_ijv_j. then

(w_1,…,w_n)=(a_ij)(v_1,… v_n)omitted; see PDF

(see the examples for numerical verification of this).

Neither the zero \(k\)-form, nor scalars, are considered to be a volume element.

References

Spivak

See Also

zeroform,as.1form

Examples

# NOT RUN {
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


# }