Objects `ex`, `ey`, and `ez` in the `stokes` package

In stokes: The Exterior Calculus

knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
library("stokes")
set.seed(0)

![](`r system.file("help/figures/stokes.png", package = "stokes")`){width=10%}

ex <- e(1,3)
ey <- e(2,3)
ez <- e(3,3)

Convenience objects ex, ey, and ez are discussed here. Elementary forms dx, dy and dz are discussed in dx.Rmd.

The dual basis to $(dx,dy,dz)$ is, depending on context, written $(e_x,e_y,e_z)$, or $(i,j,k)$ or sometimes $\left(\frac{\partial}{\partial x},\frac{\partial}{\partial x},\frac{\partial}{\partial x}\right)$. Here they are denoted ex, ey, and ez (rather than i,j,k which cause problems in the context of R).

fdx <- as.function(dx)
fdy <- as.function(dy)
fdz <- as.function(dz)
matrix(c(
      fdx(ex),fdx(ey),fdx(ez),
      fdy(ex),fdy(ey),fdy(ez),
      fdz(ex),fdz(ey),fdz(ez)
      ),3,3)

Above we see that the matrix $dx^i\frac{\partial}{\partial x^j}$ is the identity, showing that ex, ey, ez are indeed conjugate to $dx,dy,dz$.

Package dataset

Following lines create exeyez.rda, residing in the data/ directory of the package.

save(ex,ey,ez,file="exeyez.rda")

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