Function kinner() in the
stokes package
## function (o1, o2, M)
## {
## stopifnot(arity(o1) == arity(o2))
## k <- arity(o1)
## if (missing(M)) {
## M <- diag(nrow = max(c(index(o1), index(o2))))
## }
## out <- 0
## k <- arity(o1)
## c1 <- elements(coeffs(o1))
## c2 <- elements(coeffs(o2))
## for (no1 in seq_len(nterms(o1))) {
## for (no2 in seq_len(nterms(o2))) {
## MM <- matrix(0, k, k)
## for (i in seq_len(k)) {
## for (j in seq_len(k)) {
## MM[i, j] <- MM[i, j] + M[index(o1)[no1, i],
## index(o2)[no2, j]]
## }
## }
## out <- out + det(MM) * c1[no1] * c2[no2]
## }
## }
## return(out)
## }To cite the stokes package in publications, please use
Hankin (2022); this function monograph
discusses function kinner(). Given two \(k\)-forms \(\alpha,\beta\), function
kinner() returns an inner product \(\left\langle\cdot,\cdot\right\rangle\) of
\(\alpha\) and \(\beta\). If \(\alpha=\alpha_1\wedge\cdots\wedge\alpha_k\)
and \(\beta=\beta_1\wedge\cdots\wedge\beta_k\),
and we have an inner product \(\left\langle\alpha_i,\beta_j\right\rangle\)
then
\[\left\langle\cdot,\cdot\right\rangle= \det\left(\left\langle\alpha_i,\beta_j\right\rangle_{ij}\right)\]
We extend this inner product by bilinearity to the whole of \(\Lambda^k(V)\).
Some simple examples
Michael Penn uses a metric of
\[ \begin{blockarray}{crrrr} & dt & dx & dy & dz\\ \begin{block}{c[rrrr]} dt & 1 & 0 & 0 & 0 \bigstrut[t] \\ dx & 0 & -1 & 0 & 0 \\ dy & 0 & 0 & -1 & 0 \bigstrut[b]\\ dz & 0 & 0 & 0 &-1 \bigstrut[b]\\ \end{block} \end{blockarray} \]
and shows that
\[ \begin{blockarray}{crrrrrrr} & dt\wedge dx & dt\wedge dy & dt\wedge dz & dx\wedge dy & dx\wedge dz& dy\wedge dz\\ \begin{block}{c[rrrrrrr]} dt\wedge dx & -1 & 0 & 0 &0&0&0&\bigstrut[t] \\ dt\wedge dy & 0 & -1 & 0 &0&0&0& \\ dt\wedge dz & 0 & 0 & -1 &0&0&0&\bigstrut[b]\\ dx\wedge dy & 0 & 0 & 0 &1&0&0&\bigstrut[b]\\ dx\wedge dz & 0 & 0 & 0 &0&1&0&\bigstrut[b]\\ dy\wedge dz & 0 & 0 & 0 &0&0&1&\bigstrut[b]\\ \end{block} \end{blockarray} \]
so, for example, \(\left\langle dt\wedge
dx,dt\wedge
dx\right\rangle=-1\) and \(\left\langle
dt\wedge dx,dt\wedge
dy\right\rangle=0\). We can reproduce this relatively easily in
the package as follows. First we need to over-write the default values
of dx, dy, and dz (which are
defined in three dimensions) and define dt dx dy dz:
dt <- d(1)
dx <- d(2)
dy <- d(3)
dz <- d(4)
p <- c("dt^dx","dt^dy","dt^dz","dx^dy","dx^dz","dy^dz")
mink <- diag(c(1,-1,-1,-1)) # Minkowski metric
M <- matrix(NA,6,6)
rownames(M) <- p
colnames(M) <- p
do <- function(x){eval(parse(text=x))}
for(i in seq_len(6)){
for(j in seq_len(6)){
M[i,j] <- kinner(do(p[i]),do(p[j]),M=mink)
}
}
M## dt^dx dt^dy dt^dz dx^dy dx^dz dy^dz
## dt^dx -1 0 0 0 0 0
## dt^dy 0 -1 0 0 0 0
## dt^dz 0 0 -1 0 0 0
## dx^dy 0 0 0 1 0 0
## dx^dz 0 0 0 0 1 0
## dy^dz 0 0 0 0 0 1Slightly slicker:
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] -1 0 0 0 0 0
## [2,] 0 -1 0 0 0 0
## [3,] 0 0 -1 0 0 0
## [4,] 0 0 0 1 0 0
## [5,] 0 0 0 0 1 0
## [6,] 0 0 0 0 0 1Tidyup
It is important to remove the dt, dx,
dt, dx as created above because they will
interfere with the other vignettes: