Objects ex, ey, and ez in the stokes package
Author: Robin K. S. Hankin

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

#### Robin K. S. Hankin

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

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

    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)

    ##      [,1] [,2] [,3]
    ## [1,]    1    0    0
    ## [2,]    0    1    0
    ## [3,]    0    0    1

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.

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

# References

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