Function `phi()` function in the `stokes` package

In stokes: The Exterior Calculus

set.seed(0)
library("stokes")
options(rmarkdown.html_vignette.check_title = FALSE)
knitr::opts_chunk$set(echo = TRUE)
knit_print.function <- function(x, ...){dput(x)}
registerS3method(
  "knit_print", "function", knit_print.function,
  envir = asNamespace("knitr")
)

knitr::include_graphics(system.file("help/figures/stokes.png", package = "stokes"))

phi

To cite the stokes package in publications, please use @hankin2022_stokes. Function phi() returns a tensor dual to the standard basis of $V=\mathbb{R}^n$. Here I discuss phi() but there is some overlap between this vignette and the tensorprod vignette.

In a memorable passage, @spivak1965 states (theorem 4.1):

Integration on chains

Let $v_1,\ldots,v_n$ be a basis for $V$, and let $\phi_1,\ldots,\phi_n$ be the dual basis, $\phi_i(v_j)=\delta_{ij}$. Then the set of all $k$-fold tensor products \[ \phi_{i_1}\otimes\cdots\otimes\phi_{i_k}\qquad 1\leqslant i_1,\ldots,i_k\leqslant n \] is a basis for $\mathcal{J}(V)$, which therefore has dimension $n^k$.

- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 75

Function phi() returns a very simple tensor:

phi(4)

First we will verify the properties of phi(), using $V=\mathbb{R}^5$, specifically

[ \phi_i(e_j) = \delta_{ij} = \begin{cases} 1, & i=j\ 0 & i\neq j. \end{cases} ]

(package idiom is to use e() for basis vectors as opposed to Spivak's $v$). As numerical verification, we will check that $\phi_3(e_2)=0$, $\phi_3(e_3)=1$, $\phi_3(e_4)=0$:

f <- as.function(phi(3))
c(f(as.matrix(e(2,5))), f(as.matrix(e(3,5))), f(as.matrix(e(4,5))))

A more severe test might be

aa <- function(n){
   outer(seq_len(n), seq_len(n),
   Vectorize(function(i, j){as.function(phi(i))(as.matrix(e(j, n)))}))
}

aa(5)

Above, we see that the matrix is $I_5$, as expected. Further:

all(aa(9) == diag(9))

The objects created by phi() may be multiplied together using tensorprod() or its binary operator %X%:

phi(4) %X% phi(3) %X% phi(5)

If we want to create arbitrary tensor products of $\phi$ objects the most natural way would be to use tensorprod() repeatedly:

Reduce(`%X%`,sapply(4:8,phi))

However, function phi() simply takes a vector:

phi(c(2,5,1))

This creates an element of the basis set, in this case $\phi_2\otimes\phi_5\otimes\phi_1$. Verification is straightforward:

(v <- sample(9))
phi(v) == Reduce(`%X%`,sapply(v,phi))

We will consider an element $X$ of $\mathcal{J}^{2}(V)$ where $V=\mathbb{R}^3$ and construct an explicit basis for it along the lines of Spivak's observation above.

(X <- ktensor(spray(matrix(c(1,2,3,2,1,1),3,2),1:3)))

Thus $X=\phi_1\otimes\phi_2 +2\phi_2\otimes\phi_1 +3\phi_3\otimes\phi_1$. Spivak asserts that $\mathcal{J}^{2}(V)$ has dimension $n^k=3^2=9$.

1*phi(c(1,2)) + 2*phi(c(2,1)) + 3*phi(c(3,1))

With a little effort, we can create all $3^2=9$ elements of a basis as follows:

apply(expand.grid(rep(list(seq_len(3)),2)),1,phi)

Or it might be logically better to use ellipsis constructs to pass multiple arguments:

s <- function(...){phi(unlist(list(...)))}
s(3,4,6)

Then we could have

1*s(1,2) + 2*s(2,1) + 3*s(3,1)
1*s(1,2) + 2*s(2,1) + 3*s(3,1) == X

Distributivity

The tensor product is left- and right distributive. To illustrate this we can use the package to calculate, say, $(2\phi_1+3\phi_2)\otimes(5\phi_3+7\phi_4)$:

(2*phi(1) + 3*phi(2)) %X% (5*phi(3) + 7*phi(4) )

Above we see package form for the result which is $10\phi_1\phi_3 + 14\phi_1\phi_4 + 15\phi_2\phi_3 + 21\phi_2\phi_4$ in algebraic notation.

Reconstruction of a given tensor

Consider the following tensor

(b <- ktensor(spray(matrix(c(3,4,7,5,4,3),3,2),7:9)))

We may express $b$ as the sum of its three terms, each with a coefficient:

7*phi(c(3,5)) + 8*phi(c(4,4)) + 9*phi(c(7,3))

Above, observe that the order of the terms may differ between the two methods, as per disordR discipline [@hankin2022_disordR], but they are algebraically identical:

b == 7*phi(c(3,5)) + 8*phi(c(4,4)) + 9*phi(c(7,3))

Function Alt()

Function Alt() returns an alternating tensor as documented in the Alt vignette in the package. It works nicely with phi():

phi(1:3)
Alt(6*phi(1:3))

References

stokes documentation built on April 4, 2025, 1:48 a.m.