---
title: "Objects `ex`, `ey`, and `ez` in the `stokes` package"
author: "Robin K. S. Hankin"
bibliography: stokes.bib
output: html_vignette
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{basis}
  %\usepackage[utf8]{inputenc}
---

```{r setup, include=FALSE}
set.seed(0)
library("stokes")
knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
```

```{r out.width='20%', out.extra='style="float:right; padding:10px"',echo=FALSE}
knitr::include_graphics(system.file("help/figures/stokes.png", package = "stokes"))
```


```{r label=defineexeyez}
ex <- e(1,3)
ey <- e(2,3)
ez <- e(3,3)
```

To cite the `stokes` package in publications, please use
@hankin2022_stokes.  Convenience objects `ex`, `ey`, and `ez` are
discussed here (related package functionality is discussed in
`dx.Rmd`).  The dual basis to $(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$
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).


```{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 $\mathrm{d}x^i\frac{\partial}{\partial
x^j}$ is the identity, showing that `ex`, `ey`, `ez` are indeed
conjugate to $\mathrm{d}x,\mathrm{d}y,\mathrm{d}z$.

### Package dataset

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

```{r,label=savedxdydz}
save(ex,ey,ez,file="exeyez.rda")
```

# References