Function `tensorprod()` function in the `stokes` package
In stokes: The Exterior Calculus

set.seed(0)
library("spray")
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"))

tensorprod
tensorprod2

To cite the stokes package in publications, please use @hankin2022_stokes. Function tensorprod() returns the tensor cross product of any number of ktensor objects; tensorprod2() is a lower-level helper function that returns the product of two such objects. These functions use spraycross() from the spray package [@hankin2022_spray].

The tensor cross product

In a memorable passage, @spivak1965 states:

Integration on chains

If $V$ is a vector space over $\mathbb{R}$, we denote the $k$-fold product $V\times\cdots\times V$ by $V^k$. A function $T\colon V^k\longrightarrow\mathbb{R}$ is called *multilinear* if for each $i$ with $1\leqslant i\leqslant k$ we have $$ T\left(v_1,\ldots, v_i + {v'}_i,\ldots, v_k\right)= T\left(v_1,\ldots,v_i,\ldots,v_k\right)+ T\left(v_1,\ldots,{v'}_i,\ldots,v_k\right),\\ T\left(v_1,\ldots,av_i,\ldots,v_k\right)=aT\left(v_1,\ldots,v_i,\ldots,v_k\right). $$ A multilinear function $T\colon V^k\longrightarrow\mathbb{R}$ is called a *$k$-tensor* on $V$ and the set of all $k$-tensors, denoted by $\mathcal{J}^k(V)$, becomes a vector space (over $\mathbb{R}$) if for $S,T\in\mathcal{J}^k(V)$ and $a\in\mathbb{R}$ we define $$ (S+T)(v_1,\ldots,v_k) = S(v_1,\ldots,v_k) + T(v_1,\ldots,v_k)\\ (aS)(v_1,\ldots,v_k) = a\cdot S(v_1,\ldots,v_k). $$ There is also an operation connecting the various spaces $\mathcal{J}(V)$. If $S\in\mathcal{J}^k(V)$ and $T\in\mathcal{J}^l(V)$, we define the *tensor product* $S\otimes T\in\mathcal{J}^{k+l}(V)$ by $$ S\otimes T(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l})= S(v_1,\ldots,v_k)\cdot T(v_{k+1},\ldots,v_{k+l}). $$

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

Spivak goes on to observe that the tensor product is distributive and associative but not commutative. He then proves that the set of all $k$-fold tensor products

$$ \phi_{i_1}\otimes\cdots\otimes\phi_{i_k},\qquad 1\leqslant i_1,\ldots,i_k\leqslant n $$

[where $\phi_i(v_j)=\delta_{ij}$,$v_1,\ldots,v_k$ being a basis for $V$] is a basis for $\mathcal{J}^k(V)$, which therefore has dimension $n^k$. Function tensorprod() evaluates the tensor product and I give examples here.

(a <- ktensor(spray(matrix(c(1,1,2,1),2,2),3:4)))
(b <- ktensor(spray(matrix(c(3,4,7,5,4,3),3,2),7:9)))

Thus $a=4\phi_1\otimes\phi_1+3\phi_1\otimes\phi_2$ and $b=7\phi_3\otimes\phi_5+8\phi_4\otimes\phi_4+9\phi_7\otimes\phi_3$. Now the cross product $a\otimes b$ is given by tensorprod():

tensorprod(a,b)

We can see that the product includes the term $21\phi_1\otimes\phi_2\otimes\phi_3\otimes\phi_5$ and five others.

Verification

Spivak proves that the tensor product is associative and distributive, which are demonstrated here.

S <- rtensor()
T <- rtensor()
U <- rtensor()
c( left_distributive = S %X% (T+U) == S*T + S*U,
  right_distributive = (S+T) %X% U == S %X% U + T %X% U,
  associative        = S %X% (T %X% U) == (S %X% T) %X% U
  )

Note on associativity

It is interesting to note that, while the tensor product is associative, disord discipline obscures this fact. Consider the following:

x <- ktensor(spray(matrix(c(1,1,2,1),2,2),1:2))
y <- ktensor(spray(matrix(c(3,4,7,5,4,3),3,2),1:3))
z <- ktensor(spray(matrix(c(1,1,2,1),2,2),1:2))
tensorprod(x, tensorprod(y, z))
tensorprod(tensorprod(x, y), z)