Functions wedge() and wedge2() in the stokes package
Author: Robin K. S. Hankin

# Functions `wedge()` and `wedge2()` in the `stokes` package

#### Robin K. S. Hankin

    wedge

    function (x, ...)
    {
        if (nargs() < 3) {
            wedge2(x, ...)
        }
        else {
            wedge2(x, Recall(...))
        }
    }

    wedge2

    function (K1, K2)
    {
        if (missing(K2)) {
            return(K1)
        }
        if (is.ktensor(K1) | is.ktensor(K2)) {
            stop("wedge product only defined for kforms")
        }
        if (!is.kform(K1) | !is.kform(K2)) {
            return(K1 * K2)
        }
        if (is.empty(K1) | is.empty(K2)) {
            return(zeroform(arity(K1) + arity(K2)))
        }
        kform(spraycross(K1, K2))
    }

To cite the `stokes` package in publications, please use Hankin (2022). In a memorable passage, Spivak (1965) states:

\(\ldots\) we would like a theorem analogous to 4.1 [the dimensionality of \(k\)-fold tensor products is \(n^k\)]. Of course, if \(\omega\in\Lambda^k(V)\) and \(\eta\in\Lambda^l(V)\), then \(\omega\otimes\eta\) is usually not in \(\Lambda^{k+l}(V)\). We will therefore define a new product, the wedge product \(\omega\wedge\eta\in\Lambda^{k+l}(V)\) by

\[ \omega\wedge\eta=\frac{\left(k+l\right)!}{k!l!}\operatorname{Alt}(\omega\otimes\eta),\qquad\omega\in\Lambda^k(V),\eta\in\Lambda^l(V) \]

(The reason for the strange coefficient will appear later).

- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 79

Function `wedge()` returns the wedge product of any number of \(k\)-forms; function `wedge2()` returns the wedge product of two \(k\)-forms. The idiom of `wedge2()` is somewhat opaque, especially the “strange” combinatorial coefficient \((k+l)!/(k!l!)\), which is discussed in detail below.

## Digression: function `spraycross()`

Function `wedge()` is essentially a convenience wrapper for `spraycross()`; the meat of `wedge2()` is the last line: `kform(spraycross(K1, K2))`. Function `spraycross()` is part of the `spray` package and gives a tensor product of sparse arrays, interpreted as multivariate polynomials:

    (a <- spray(matrix(1:4,2,2),c(2,5)))

    ##          val
    ##  2 4  =    5
    ##  1 3  =    2

    (b <- spray(matrix(c(10,11,12,13),2,2),c(7,11)))

    ##            val
    ##  11 13  =   11
    ##  10 12  =    7

    spraycross(a,b)

    ##                val
    ##  1 3 10 12  =   14
    ##  1 3 11 13  =   22
    ##  2 4 10 12  =   35
    ##  2 4 11 13  =   55

    spraycross(b,a)

    ##                val
    ##  10 12 1 3  =   14
    ##  11 13 1 3  =   22
    ##  10 12 2 4  =   35
    ##  11 13 2 4  =   55

Observe that `spraycross()` (and by association `wedge()`) is associative and distributive but not commutative.

### Cut to the chase: `wedge2()`

Function `wedge2()` takes two kforms and we will start with a very simple example:

    (x <- as.kform(cbind(1,2),5))

    ## An alternating linear map from V^2 to R with V=R^2:
    ##          val
    ##  1 2  =    5

    (y <- as.kform(cbind(3,4,7),7))

    ## An alternating linear map from V^3 to R with V=R^7:
    ##            val
    ##  3 4 7  =    7

    wedge2(x,y)

    ## An alternating linear map from V^5 to R with V=R^7:
    ##                val
    ##  1 2 3 4 7  =   35

It looks like the combinatorial term has not been included but it has. We will express `x` and `y` as tensors (objects of class `ktensor`) and show how the combinatorial term arises.

     tx <- as.ktensor(x)    # "tx" = tensor 'x'
    (ty <- as.ktensor(y))   # "ty" = tensor 'y'

    ## A linear map from V^3 to R with V=R^7:
    ##            val
    ##  7 4 3  =   -7
    ##  7 3 4  =    7
    ##  4 7 3  =    7
    ##  4 3 7  =   -7
    ##  3 7 4  =   -7
    ##  3 4 7  =    7

As functions, `y` and `ty` are identical:

    M <- matrix(round(rnorm(21),2),7,3) # member of (R^7)^3
    c(as.function(y)(M),as.function(ty)(M))

    ## [1] 15.23211 15.23211

Both are equivalent to

    7*(
     +M[3,1]*M[4,2]*M[7,3]
     -M[3,1]*M[4,3]*M[7,2]
     -M[3,2]*M[4,1]*M[7,3]
     +M[3,2]*M[4,3]*M[7,1]
     +M[3,3]*M[4,1]*M[7,2]
     -M[3,3]*M[4,2]*M[7,1]
     )

    ## [1] 15.23211

We can see that `y` is a more compact and efficient representation of `ty`: both are alternating tensors but `y` has alternatingness built in to its evaluation, while `ty` is alternating by virtue of including all permutations of its arguments, with the sign of the permutation.

We can evaluate Spivak’s formula (but without the combinatorial term) for \(x\wedge y\) by coercing to ktensors and using `tensorprod()`:

    (z <- tensorprod(as.ktensor(x),as.ktensor(y)))

    ## A linear map from V^5 to R with V=R^7:
    ##                val
    ##  1 2 3 4 7  =   35
    ##  2 1 3 7 4  =   35
    ##  1 2 4 3 7  =  -35
    ##  2 1 3 4 7  =  -35
    ##  1 2 4 7 3  =   35
    ##  2 1 4 3 7  =   35
    ##  2 1 4 7 3  =  -35
    ##  1 2 7 3 4  =   35
    ##  2 1 7 3 4  =  -35
    ##  1 2 3 7 4  =  -35
    ##  1 2 7 4 3  =  -35
    ##  2 1 7 4 3  =   35

Above, each coefficient is equal to \(\pm 35\) (the sign coming from the sign of the permutation), and we have \(2!3!=12\) rows. We can now calculate \(\operatorname{Alt}(z)\), which would have \(5!=120\) rows, one per permutation of \([5]\), each with coefficient \(\pm\frac{12\times 35}{5!}=\pm 3.5\).

We define \(x\wedge y\) to be \(\frac{5!}{3!2!}\operatorname{Alt}(z)\), so each coefficient would be \(\pm\frac{5!}{3!2!}\cdot\frac{12\times 35}{5!}=35\). We know that \(x\wedge y\) is an alternating form. So to represent it as an object of class `kform`, we need a `kform` object with single index entry `1 2 3 4 7`. This would need coefficient 35, on the grounds that it is linear, alternating, and maps \(\begin{pmatrix} 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&1 \end{pmatrix}\) to \(35\); and indeed this is what we see:

    wedge(x,y)

    ## An alternating linear map from V^5 to R with V=R^7:
    ##                val
    ##  1 2 3 4 7  =   35

So to conclude, the combinatorial term is present in the R idiom, it is just difficult to see at first glance.

# Algebraic properties

First of all we should note that \(\Lambda^k(V)\) is a vector space (this is considered in the `kform` vignette). If \(\omega,\omega_i\in\Lambda^k(V)\) and \(\eta,\eta_i\in\Lambda^l(V)\) then

\[\begin{eqnarray} (\omega_1+\omega_2)\wedge\eta &=& \omega_1\wedge\eta+\omega_2\wedge\eta\\ \omega\wedge(\eta_1+\eta_2) &=&\omega\wedge\eta_1 + \omega\wedge\eta_2\\ \end{eqnarray}\]

(that is, the wedge product is left- and right- distributive); if \(a\in\mathbb{R}\) then

\[\begin{equation} a\omega\wedge\eta = \omega\wedge a\eta=a(\omega\wedge\eta) \end{equation}\]

and \[\begin{equation} \omega\wedge\eta = (-1)^{kl}\eta\wedge\omega. \end{equation}\]

These rules make expansion of wedge products possible by expressing a general kform in terms of a basis for \(\Lambda^k(V)\). Spivak (1965) tells us that, if \(v_1,\ldots,v_k\) is a basis for \(V\), then the set of all

\[\begin{equation} \phi_{i_1}\wedge\phi_{i_2}\wedge\cdots\wedge\phi_{i_k}\qquad 1\leq i_1 < \cdots < i_k\leq n \end{equation}\]

is a basis for \(\Lambda^k(V)\) where \(\phi_i(v_j)=\delta_{ij}\). The package expresses a \(k\)-form in terms of this basis as in the following example:

    (omega <- as.kform(rbind(c(1,2,8),c(1,3,7)),5:6))

    ## An alternating linear map from V^3 to R with V=R^8:
    ##            val
    ##  1 3 7  =    6
    ##  1 2 8  =    5

In algebraic notation, `omega` (or \(\omega\)) would be \(5\phi_1\wedge\phi_2\wedge\phi_8+6\phi_1\wedge\phi_3\wedge\phi_7\) and we may write this as \(\omega=5\phi_{128}+6\phi_{137}\). To take a wedge product of this with \(\eta=2\phi_{235}+3\phi_{356}\) we would write

\[\begin{eqnarray} \omega\wedge\eta &=& (5\phi_{128}+6\phi_{137})\wedge (2\phi_{235}+3\phi_{356})\\ &=& 10\phi_{128}\wedge\phi_{235} + 15\phi_{128}\wedge\phi_{356} + 12\phi_{137}\wedge\phi_{235} + 18\phi_{137}\wedge\phi_{356}\\ &=& 10\phi_1\wedge\phi_2\wedge\phi_8\wedge\phi_2\wedge\phi_3\wedge\phi_5 + 15\phi_1\wedge\phi_2\wedge\phi_8\wedge\phi_3\wedge\phi_5\wedge\phi_6\\&{}&\qquad + 12\phi_1\wedge\phi_3\wedge\phi_7\wedge\phi_2\wedge\phi_3\wedge\phi_5 + 18\phi_1\wedge\phi_3\wedge\phi_7\wedge\phi_3\wedge\phi_5\wedge\phi_6\\ &=& 0+ 15\phi_1\wedge\phi_2\wedge\phi_8\wedge\phi_3\wedge\phi_5\wedge\phi_6+0+0\\ &=& -15\phi_1\wedge\phi_2\wedge\phi_3\wedge\phi_5\wedge\phi_6\wedge\phi_8 \end{eqnarray}\]

where we have used the rules repeatedly (especially the fact that \(\omega\wedge\omega=0\) for any alternating form). Package idiom would be:

    eta <- as.kform(rbind(c(2,3,5),c(3,5,6)),2:3)
    wedge(omega,eta)

    ## An alternating linear map from V^6 to R with V=R^8:
    ##                  val
    ##  1 2 3 5 6 8  =  -15

See how function `wedge()` does the legwork.

# 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.