`volume()`

in the
Stokes package```
function (n)
{
as.kform(seq_len(n))
}
```

To cite the `stokes`

package in publications, please use
Hankin (2022). Spivak (1965), in a memorable passage,
states:

**The volume element**

The fact that \(\operatorname{dim}\Lambda^n\left(\mathbb{R}^n\right)=1\) is probably not new to you, since \(\operatorname{det}\) is often defined as the unique element \(\omega\in\Lambda^n{\left(\mathbb{R}^n\right)}\) such that \(\omega{\left(e_1,\ldots,e_n\right)}=1\). For a general vector space \(V\) there is no extra criterion of this sort to distinguish a particular \(\omega\in\Lambda^n{\left(\mathbb{R}^n\right)}\). Suppose, however, that an inner product \(T\) for \(V\) is given. If \(v_1,\ldots,v_n\) and \(w_1,\ldots, w_n\) are two bases which are orthonormal with respect to \(T\), and the matrix \(A=\left(a_{ij}\right)\) is defined by \(w_i=\sum_{j=1}^n a_{ij}v_j\), then

\[\delta_{ij}=T{\left(w_i,w_j\right)}= \sum_{k,l=1}^n a_{ik}a_{jl}\,T{\left(v_k,v_l\right)}= \sum_{k=1}^n a_{ik}a_{jk}.\]

In other words, if \(A^T\) denotes the transpose of the matrix \(A\), then we have \(A\cdot A^T=I\), so \(\operatorname{det}A=\pm 1\). It follows from Theorem 4-6 [see vignette`det.Rmd`

] that if \(\omega\in\Lambda^n(V)\) satisfies \(\omega{\left(v_1,\ldots,v_n\right)}=\pm
1\), then \(\omega{\left(w_1,\ldots,w_n\right)}=\pm
1\). If an orientation \(\mu\)
for \(V\) has also been given, it
follows that there is a unique \(\omega\in\Lambda^n(V)\) such that \(\omega\left(v_1,\ldots,v_n\right)=1\)
whenever \(v_1,\ldots,v_n\) is an
orthornormal basis such that \(\left[v_1,\ldots,v_n\right]=\mu\). This
unique \(\omega\) is called the
**- Michael Spivak, 1969** *(Calculus on Manifolds, Perseus books).
Page 83*

In the `stokes`

package, function `volume(n)`

returns the volume element on the usual basis, that is, \(\omega{\left(e_1,\ldots,e_n\right)}\). We
will take \(n=7\) as an example:

```
## An alternating linear map from V^7 to R with V=R^7:
## val
## 1 2 3 4 5 6 7 = 1
```

We can verify Spivak’s reasoning as follows:

`## [1] 1`

Above, we see that \(\omega{\left(e_1,\ldots,e_n\right)}=1\). To verify that \(V{\left(v_1,\ldots,v_n\right)}=\operatorname{det}(A)\), where \(A_{ij}=\left(v_i\right)_j\):

```
## LHS RHS diff
## 1.770074 1.770074 0.000000
```

Now we create \(w_1,\ldots,w_n\), another orthonormal set. We may verify by generating a random orthogonal matrix and permuting its rows:

```
M1 <- qr.Q(qr(matrix(rnorm(49),7,7))) # M1: a random orthogonal matrix
M2 <- M1[c(2,1,3,4,5,6,7),] # M2: (odd) permutation of rows of M1
c(f(M1),f(M2))
```

`## [1] 1 -1`

Above we see that the volume element of `M1`

and
`M2`

are \(\pm1\) to within
numerical precision.

Hankin, R. K. S. 2022. “Stokes’s Theorem in
R.” arXiv. https://doi.org/10.48550/ARXIV.2210.17008.

Spivak, M. 1965. *Calculus on Manifolds*. Addison-Wesley.