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 e1, e2, e2 are discussed in ex.Rmd). Spivak (1965), in a memorable passage, states:

Fields and forms

If f: ℝn → ℝ is differentiable, then Df(p) ∈ Λ1(ℝn). By a minor modification we therefore obtain a 1-form df, defined by

df(p)(vp) = Df(p)(v).

Let us consider in particular the 1-forms dπi 1. It is customary to let xi denote the function πi (on ℝ3 we often denote x1, x2, and x3 by x, y, and z) … Since dxi(p)(vp) = dπi(p)(vp) = Dπi(p)(v) = vi, we see that dx1(p), …, dxn(p) is just the dual basis to (e1)p, …, (en)p.

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

Spivak goes on to observe that every k-form ω can be written ω = ∑i1 < ⋯ < ikωi1, …ikdxi1 ∧ ⋯ ∧ dxik. If working in ℝ3, we have three elementary forms dx, dy, and dz; 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 dx ∧ dy − 7dx ∧ dz + 3dy ∧ dz:

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 −7dx ∧ dz], this maps (ℝ3)2 to 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 (ℝ3)1 to 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, dx 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 dx ∧ dy = −dy ∧ dx:

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 dxi, 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 πi notation on page 11: “if π: ℝn → ℝn is the identity function, π(x) = x, then [its components are] πi(x) = xi; the function πi is called the ith projection function”↩︎