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 \(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z\).

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