The kinner() function in the stokes package

Robin K. S. Hankin

kinner

## 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)
## }
## <bytecode: 0x5587b1b36a40>
## <environment: namespace:stokes>

To cite the stokes package in publications, please use Hankin (2022). 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     1

Slightly slicker:

outer(p,p,Vectorize(function(i,j){kinner(do(i),do(j),M=mink)}))

##      [,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    1

Tidyup

It is important to remove the dt, dx, dt, dx as created above because they will interfere with the other vignettes:

rm(dt,dx,dy,dz)

References

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