Function hodge() in the stokes package

Robin K. S. Hankin

hodge

function (K, n = dovs(K), g, lose = TRUE) 
{
    if (missing(g)) {
        g <- rep(1, n)
    }
    if (is.empty(K)) {
        if (missing(n)) {
            stop("'K' is zero but no value of 'n' is supplied")
        }
        else {
            return(kform(spray(matrix(1, 0, n - arity(K)), 1)))
        }
    }
    else if (is.volume(K, n)) {
        return(scalar(coeffs(K), lose = lose))
    }
    else if (is.scalar(K)) {
        if (missing(n)) {
            stop("'K' is scalar but no value of 'n' is supplied")
        }
        else {
            return(volume(n) * coeffs(K))
        }
    }
    stopifnot(n >= dovs(K))
    f1 <- function(o) {
        seq_len(n)[!seq_len(n) %in% o]
    }
    f2 <- function(x) {
        permutations::sgn(permutations::as.word(x))
    }
    f3 <- function(v) {
        prod(g[v])
    }
    iK <- index(K)
    jj <- apply(iK, 1, f1)
    if (is.matrix(jj)) {
        newindex <- t(jj)
    }
    else {
        newindex <- as.matrix(jj)
    }
    x_coeffs <- elements(coeffs(K))
    x_metric <- apply(iK, 1, f3)
    x_sign <- apply(cbind(iK, newindex), 1, f2)
    as.kform(newindex, x_metric * x_coeffs * x_sign)
}

To cite the stokes package in publications, please use Hankin (2022); this function monograph discusses hodge(). Given a $k$-form $\beta$, function hodge() returns its Hodge dual $\star\beta$. Formally, if $V={\mathbb R}^n$, and $\Lambda^k(V)$ is the space of alternating linear maps from $V^k$ to ${\mathbb R}$, then $\star\colon\Lambda^k(V)\longrightarrow\Lambda^{n-k}(V)$. To define the Hodge dual, we need an inner product $\left\langle\cdot,\cdot\right\rangle$ [function kinner() in the package] and, given this and $\beta\in\Lambda^k(V)$ we define $\star\beta$ to be the (unique) $n-k$-form satisfying the fundamental relation:

$$\alpha\wedge\left(\star\beta\right)=\left\langle\alpha,\beta\right\rangle\omega,$$

for every $\alpha\in\Lambda^k(V)$. Here $\omega=e_1\wedge e_2\wedge\cdots\wedge e_n$ is the unit $n$-vector of $\Lambda^n(V)$. Taking determinants of this relation shows the following. If we use multi-index notation so $e_I=e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $I=\left\lbrace i_1,\cdots,i_k\right\rbrace$, then

$$\star e_I=(-1)^{\sigma(I)}e_J$$

where $J=\left\lbrace j_i,\ldots,j_k\right\rbrace=[n]\setminus\left\lbrace i_1,\ldots,i_k\right\rbrace$ is the complement of $I$, and $(-1)^{\sigma(I)}$ is the sign of the permutation $\sigma(I)=i_1\cdots i_kj_1\cdots j_{n-k}$. We extend to the whole of $\Lambda^k(V)$ using linearity. Package idiom for calculating the Hodge dual is straightforward, being simply hodge().

The Hodge dual on basis elements of $\Lambda^k(V)$

We start by demonstrating hodge() on basis elements of $\Lambda^k(V)$. Recall that if $\left\lbrace e_1,\ldots,e_n\right\rbrace$ is a basis of vector space $V=\mathbb{R}^n$, then $\left\lbrace\omega_1,\ldots,\omega_k\right\rbrace$ is a basis of $\Lambda^1(V)$, where $\omega_i(e_j)=\delta_{ij}$. A basis of $\Lambda^k(V)$ is given by the set

$$\bigcup_{1\leqslant i_1 < \cdots < i_k\leqslant n} \bigwedge_{j=1}^k\omega_{i_j} = \left\lbrace \left.\omega_{i_1}\wedge\cdots\wedge\omega_{i_k} \right|1\leqslant i_1 < \cdots < i_k\leqslant n \right\rbrace.$$

This means that basis elements are things like $\omega_2\wedge\omega_6\wedge\omega_7$. If $V=\mathbb{R}^9$, what is $\star\omega_2\wedge\omega_6\wedge\omega_7$?

(a <- d(2) ^ d(6) ^ d(7))

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

hodge(a,9)

## An alternating linear map from V^6 to R with V=R^9:
##                  val
##  1 3 4 5 8 9  =   -1

See how $\star a$ has index entries 1-9 except $2,6,7$ (from $a$). The (numerical) sign is negative because the permutation has negative (permutational) sign. We can verify this using the permutations package:

p <- c(2,6,7,  1,3,4,5,8,9)
(pw <- as.word(p))

## [1] (1264)(375)
## [coerced from word form]

print_word(pw)

##     1 2 3 4 5 6 7 8 9
## [1] 2 6 7 1 3 4 5 . .

sgn(pw)

## [1] -1

Above we see the sign of the permutation is negative. More succinct idiom would be

hodge(d(c(2,6,7)),9)

## An alternating linear map from V^6 to R with V=R^9:
##                  val
##  1 3 4 5 8 9  =   -1

The second argument to hodge() is needed if the largest index $i_k$ of the first argument is less than $n$; the default value is indeed $n$. In the example above, this is $7$:

hodge(d(c(2,6,7)))

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

Above we see the result if $V=\mathbb{R}^7$.

More complicated examples

The hodge operator is linear and it is interesting to verify this.

(o <- rform())

## An alternating linear map from V^3 to R with V=R^7:
##            val
##  2 6 7  =    6
##  2 5 7  =    5
##  5 6 7  =   -9
##  1 3 7  =    4
##  1 5 7  =    7
##  2 3 5  =   -3
##  1 5 6  =   -8
##  1 2 7  =    2
##  1 4 6  =    1

hodge(o)

## An alternating linear map from V^4 to R with V=R^7:
##              val
##  2 3 5 7  =   -1
##  3 4 5 6  =    2
##  2 3 4 7  =   -8
##  1 4 6 7  =   -3
##  2 3 4 6  =   -7
##  2 4 5 6  =   -4
##  1 2 3 4  =   -9
##  1 3 4 6  =    5
##  1 3 4 5  =   -6

We verify that the fundamental relation holds by direct inspection:

o ^ hodge(o)

## An alternating linear map from V^7 to R with V=R^7:
##                    val
##  1 2 3 4 5 6 7  =  285

kinner(o,o)*volume(dovs(o))

## An alternating linear map from V^7 to R with V=R^7:
##                    val
##  1 2 3 4 5 6 7  =  285

showing agreement (above, we use function volume() in lieu of calculating the permutation’s sign explicitly. See the volume vignette for more details). We may work more formally by defining a function that returns TRUE if the left and right hand sides match

diff <- function(a,b){a^hodge(b) ==  kinner(a,b)*volume(dovs(a))}

and call it with random $k$-forms:

diff(rform(),rform())

## [1] TRUE

Or even

all(replicate(10,diff(rform(),rform())))

## [1] TRUE

Small-dimensional vector spaces

We can work in three dimensions in which case we have three linearly independent $1$-forms: $dx$, $dy$, and $dz$. To work in this system it is better to use dx print method:

options(kform_symbolic_print = "dx")
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.

Vector cross product identities

The three dimensional vector cross product $\mathbf{u}\times\mathbf{v}=\det\begin{pmatrix} i & j & k \\ u_1&u_2&u_3\\ v_1&v_2&v_3 \end{pmatrix}$ is a standard part of elementary vector calculus. In the package the idiom is as follows:

vcp3

## function (u, v) 
## {
##     hodge(as.1form(u)^as.1form(v))
## }

revealing the formal definition of cross product as $\mathbf{u}\times\mathbf{v}=\star{\left(\mathbf{u}\wedge\mathbf{v}\right)}$. There are several elementary identities that are satisfied by the cross product:

$$\begin{aligned} \mathbf{u}\times(\mathbf{v}\times\mathbf{w}) &= \mathbf{v}(\mathbf{w}\cdot\mathbf{u})-\mathbf{w}(\mathbf{u}\cdot\mathbf{v})\\ (\mathbf{u}\times\mathbf{v})\times\mathbf{w} &= \mathbf{v}(\mathbf{w}\cdot\mathbf{u})-\mathbf{u}(\mathbf{v}\cdot\mathbf{w})\\ (\mathbf{u}\times\mathbf{v})\times(\mathbf{u}\times\mathbf{w}) &= (\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}))\mathbf{u} \\ (\mathbf{u}\times\mathbf{v})\cdot(\mathbf{w}\times\mathbf{x}) &= (\mathbf{u}\cdot\mathbf{w})(\mathbf{v}\cdot\mathbf{x}) - (\mathbf{u}\cdot\mathbf{x})(\mathbf{v}\cdot\mathbf{w}) \end{aligned}$$

We may verify all four together:

u <- c(1,4,2)
v <- c(2,1,5)
w <- c(1,-3,2)
x <- c(-6,5,7)
c(
  hodge(as.1form(u) ^ vcp3(v,w))        == as.1form(v*sum(w*u) - w*sum(u*v)),
  hodge(vcp3(u,v) ^ as.1form(w))        == as.1form(v*sum(w*u) - u*sum(v*w)),
  as.1form(as.function(vcp3(v,w))(u)*u) == hodge(vcp3(u,v) ^ vcp3(u,w))     ,
  hodge(hodge(vcp3(u,v)) ^ vcp3(w,x))   == sum(u*w)*sum(v*x) - sum(u*x)*sum(v*w)
)         

## [1] TRUE TRUE TRUE TRUE

Above, note the use of the hodge operator to define triple vector cross products. For example we have $\mathbf{u}\times\left(\mathbf{v}\times\mathbf{w}\right)= \star\left(\mathbf{u}\wedge\star\left(\mathbf{v}\wedge\mathbf{w}\right)\right)$.

Non positive-definite metrics

The inner product $\left\langle\alpha,\beta\right\rangle$ above may be generalized by defining it on decomposable vectors $\alpha=\alpha_1\wedge\cdots\wedge\alpha_k$ and $\beta=\beta_1\wedge\cdots\wedge\beta_k$ as

$$\left\langle\alpha,\beta\right\rangle= \det\left(\left\langle\alpha_i,\beta_j\right\rangle_{i,j}\right)$$

where $\left\langle\alpha_i,\beta_j\right\rangle=\pm\delta_{ij}$ is an inner product on $\Lambda^1(V)$ [the inner product is given by kinner()]. The resulting Hodge star operator is implemented in the package and one can specify the metric. For example, if we consider the Minkowski metric this would be $-1,1,1,1$.

Print methods for the Minkowski metric

The standard print method is not particularly suitable for working with the Minkowski metric:

options(kform_symbolic_print = NULL)  # default print method
(o <- kform_general(4,2,1:6))

## An alternating linear map from V^2 to R with V=R^4:
##          val
##  3 4  =    6
##  2 4  =    5
##  1 4  =    4
##  2 3  =    3
##  1 3  =    2
##  1 2  =    1

Above we see an unhelpful representation. To work with $2$-forms in relativistic physics, it is often preferable to use bespoke print method usetxyz:

options(kform_symbolic_print = "txyz")
o

## An alternating linear map from V^2 to R with V=R^4:
##  +6 dy^dz +5 dx^dz +4 dt^dz +3 dx^dy +2 dt^dy + dt^dx

Specifying the Minkowski metric

Function hodge() takes a g argument to specify the metric:

hodge(o)

## An alternating linear map from V^2 to R with V=R^4:
##  + dy^dz -2 dx^dz +3 dt^dz +4 dx^dy -5 dt^dy +6 dt^dx

hodge(o,g=c(-1,1,1,1))

## An alternating linear map from V^2 to R with V=R^4:
##  - dy^dz +2 dx^dz +3 dt^dz -4 dx^dy -5 dt^dy +6 dt^dx

hodge(o)-hodge(o,g=c(-1,1,1,1))

## An alternating linear map from V^2 to R with V=R^4:
##  +8 dx^dy -4 dx^dz +2 dy^dz

References

Hankin, R. K. S. 2022. “Stokes’s Theorem in R.” arXiv. https://doi.org/10.48550/ARXIV.2210.17008.