Objects ex, ey, and ez in the stokes package

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.