Objects dx, dy, and dz in the stokes package

Robin K. S. Hankin

dx <- d(1)
dy <- d(2)
dz <- d(3)

To cite the stokes package in publications, please use Hankin (2022). Convenience objects dx, dy, and dz, corresponding to elementary differential forms, are discussed here (basis vectors \(e_1\), \(e_2\), \(e_2\) are discussed in ex.Rmd). Spivak (1965), in a memorable passage, states:

Fields and forms

If \(f\colon\mathbb{R}^n\longrightarrow\mathbb{R}\) is differentiable, then \(Df(p)\in\Lambda^1(\mathbb{R}^n)\). By a minor modification we therefore obtain a \(1\)-form \(\mathrm{d}f\), defined by

\[\mathrm{d}f(p)(v_p)=Df(p)(v).\]

Let us consider in particular the \(1\)-forms \(\mathrm{d}\pi^i\) 1. It is customary to let \(x^i\) denote the function \(\pi^i\) (on \(\mathbb{R}^3\) we often denote \(x^1\), \(x^2\), and \(x^3\) by \(x\), \(y\), and \(z\)) \(\ldots\) Since \(\mathrm{d}x^i(p)(v_p)=\mathrm{d}\pi^i(p)(v_p)=D\pi^i(p)(v)=v^i\), we see that \(\mathrm{d}x^1(p),\ldots,\mathrm{d}x^n(p)\) is just the dual basis to \((e_1)_p,\ldots, (e_n)_p\).

- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 89

Spivak goes on to observe that every \(k\)-form \(\omega\) can be written \(\omega=\sum_{i_1 < \cdots < i_k}\omega_{i_1,\ldots i_k}\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}\). If working in \(\mathbb{R}^3\), we have three elementary forms \(\mathrm{d}x\), \(\mathrm{d}y\), and \(\mathrm{d}z\); in the package we have the pre-defined objects dx, dy, and dz. These are convenient for reproducing textbook results. We start with some illustrations of the package print method.

dx
## An alternating linear map from V^1 to R with V=R^1:
##        val
##  1  =    1

This is somewhat opaque and difficult to understand. It is easier to start with a more complicated example: take \(\mathrm{d}x\wedge\mathrm{d}y -7\mathrm{d}x\wedge\mathrm{d}z + 3\mathrm{d}y\wedge\mathrm{d}z\):

dx^dy -7*dx^dz + 3*dy^dz
## An alternating linear map from V^2 to R with V=R^3:
##          val
##  1 3  =   -7
##  2 3  =    3
##  1 2  =    1

We see three rows for the three elementary components. Taking the row with coefficient \(-7\) [which would be \(-7\mathrm{d}x\wedge\mathrm{d}z\)], this maps \(\left(\mathbb{R}^3\right)^2\) to \(\mathbb{R}\) and we have

\[(-7\mathrm{d}x\wedge\mathrm{d}z)\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix}, \begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\right)= -7\det\begin{pmatrix}u_1&v_1\\u_3&v_3\end{pmatrix}\]

Armed with this familiar fact, we can interpret \(dx\) as a map from \(\left(\mathbb{R}^3\right)^1\) to \(\mathbb{R}\) with

\[\mathrm{d}x\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix} \right)= \det\begin{pmatrix}u_1\end{pmatrix}=u_1\]

or, in other words, \(\mathrm{d}x\) picks out the first component of its vector (as the print method gives, albeit obscurely). This is easily shown in the package:

as.function(dx)(c(113,3,6))
## [1] 113

We might want to verify that \(\mathrm{d}x\wedge\mathrm{d}y=-\mathrm{d}y\wedge\mathrm{d}x\):

dx ^ dy == -dy ^ dx
## [1] TRUE

Elementary forms and the print method

The print method is configurable and can display kforms in symbolic form. For working with dx dy dz we may set option kform_symbolic_print to dx:

options(kform_symbolic_print = 'dx')

Then the results of calculations are more natural:

dx
## An alternating linear map from V^1 to R with V=R^1:
##  + dx
dx^dy + 56*dy^dz
## An alternating linear map from V^2 to R with V=R^3:
##  + dx^dy +56 dy^dz

However, this setting can be confusing if we work with \(\mathrm{d}x^i,i>3\), for the print method runs out of alphabet:

rform()
## An alternating linear map from V^3 to R with V=R^7:
##  +6 dy^dNA^dNA +5 dy^dNA^dNA -9 dNA^dNA^dNA +4 dx^dz^dNA +7 dx^dNA^dNA -3 dy^dz^dNA -8 dx^dNA^dNA +2 dx^dy^dNA + dx^dNA^dNA

Above, we see the use of NA because there is no defined symbol.

The Hodge dual

Function hodge() returns the Hodge dual:

hodge(dx^dy + 13*dy^dz)
## An alternating linear map from V^1 to R with V=R^3:
##  +13 dx + dz

Note that calling hodge(dx) can be confusing:

hodge(dx)
## [1] 1

This returns a scalar because dx is interpreted as a one-form on one-dimensional space, which is a scalar form. One usually wants the result in three dimensions:

hodge(dx,3)
## An alternating linear map from V^2 to R with V=R^3:
##  + dy^dz

This is further discussed in the dovs vignette.

Other ways to create the elementary one-forms

It is possible to create these objects using package idiom:

d(1) == dx
## [1] TRUE

Basis vectors

Package dataset

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

save(dx,dy,dz,file="dx.rda")

References

Hankin, R. K. S. 2022. “Stokes’s Theorem in R.” arXiv. https://doi.org/10.48550/ARXIV.2210.17008.
Spivak, M. 1965. Calculus on Manifolds. Addison-Wesley.

  1. Spivak introduces the \(\pi^i\) notation on page 11: “if \(\pi\colon\mathbb{R}^n\longrightarrow\mathbb{R}^n\) is the identity function, \(\pi(x)=x\), then [its components are] \(\pi^i(x)=x^i\); the function \(\pi^i\) is called the \(i^\mathrm{th}\) projection function”↩︎