The `kinner()` function in the `stokes` package

In stokes: The Exterior Calculus

knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
library("stokes")
library("emulator")
set.seed(1)

\hfill{width=10%}

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

Slightly slicker:

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

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)

options(kform_symbolic_print = NULL)

stokes documentation built on Aug. 19, 2023, 1:07 a.m.