Function volume() in the stokes package

Robin K. S. Hankin

volume

function (n) 
{
    as.kform(seq_len(n))
}

To cite the stokes package in publications, please use Hankin (2022); this function monograph discusses volume(). 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 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 volume element of $V$, determined by the inner product $T$ and orientation $\mu$. Note that $\operatorname{det}$ is the volume element of $\mathbb{R}^n$ determined by the usual inner product and usual orientation, and that $\left|\operatorname{det}\left(v_1,\ldots,v_n\right)\right|$ is the volume of the parallelepiped spanned by the line segments from $0$ to each of $v_1,\ldots,v_n$.

- 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:

(V <- volume(7))

## 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:

f <- as.function(V)
f(diag(7))

## [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 $A_{ij}=\left(v_i\right)_j$:

A <- matrix(rnorm(49),7,7)
LHS <- f(A)
RHS <- det(A)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

##      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.

References

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.