Objects `dx`, `dy`, and `dz` in the `stokes` package

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.

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”↩︎