hodge() function in the stokes package
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)
## }
## <bytecode: 0x560d84f40468>
## <environment: namespace:stokes>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().
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 \(\omega_2\wedge\omega_6\wedge\omega_7\). f \(V=\mathbb{R}^9\), 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 = 1hodge(a,9)## An alternating linear map from V^6 to R with V=R^9:
## val
## 1 3 4 5 8 9 = -1See how \(\star a\) has index entries 1-9 except \(2,6,7\) (from \(a\)). The (numerical) sign is negative because the permution 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] -1Above 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 = -1The 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 = -1Above we see the result if \(V=\mathbb{R}^7\).
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
## 5 6 7 = -8
## 1 5 7 = 6
## 2 5 6 = -5
## 1 3 5 = -4
## 2 3 6 = 3
## 1 2 7 = 9
## 1 5 6 = -7
## 2 3 5 = -2
## 1 4 7 = 1hodge(o)## An alternating linear map from V^4 to R with V=R^7:
## val
## 2 3 5 6 = 1
## 1 4 6 7 = -2
## 2 3 4 7 = -7
## 3 4 5 6 = 9
## 1 4 5 7 = -3
## 2 4 6 7 = 4
## 1 3 4 7 = 5
## 2 3 4 6 = -6
## 1 2 3 4 = -8We 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 = 285kinner(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 = 285showing 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] TRUEOr even
all(replicate(10,diff(rform(),rform())))## [1] TRUEWe 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^dzThis is further discussed in the dovs vignette.
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))
## }
## <bytecode: 0x560d84ebd670>
## <environment: namespace:stokes>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 TRUEAbove, note the use of the hodge operator to define triple vector cross products. For example we have \(\mathbf{u}\times\left(\mathbf{v}\times\mathbf{w}\right)= \star\left(\mathbf{u}\wedge\star\left(\mathbf{v}\wedge\mathbf{w}\right)\right)\).
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 \(-1,1,1,1\).
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 = 1Above 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^dxFunction 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^dxhodge(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^dxhodge(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