hodge() function in the stokes package
## 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: 0x55abbc2a2438>
## <environment: namespace:stokes>Given a \(k\)-form \(\omega\), function hodge() returns its Hodge dual \(*\omega\) or \(\star\omega\). 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 \(J=\left\lbrace j_i,\ldots,j_k\right\rbrace=[n]\backslash\left\lbrace i_1,\ldots,i_k\right\rbrace\) 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 is straightforward:
## An alternating linear map from V^3 to R with V=R^7:
## -8 dNA^dNA^dNA +6 dx^dNA^dNA -5 dy^dNA^dNA -4 dx^dz^dNA +3 dy^dz^dNA +9 dx^dy^dNA -7 dx^dNA^dNA -2 dy^dz^dNA + dx^dNA^dNA## An alternating linear map from V^4 to R with V=R^7:
## + dy^dz^dNA^dNA -2 dx^dNA^dNA^dNA -7 dy^dz^dNA^dNA +9 dz^dNA^dNA^dNA -3 dx^dNA^dNA^dNA +4 dy^dNA^dNA^dNA +5 dx^dz^dNA^dNA -6 dy^dz^dNA^dNA -8 dx^dy^dz^dNAWe verify that the fundamental relation holds by direct inspection:
## An alternating linear map from V^7 to R with V=R^7:
## +285 dx^dy^dz^dNA^dNA^dNA^dNA## An alternating linear map from V^7 to R with V=R^7:
## +285 dx^dy^dz^dNA^dNA^dNA^dNAshowing agreement. We may work more formally by defining a function that returns TRUE if the left and right hand sides match
and call it with random \(k\)-forms:
## [1] TRUEOr even
## [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:
## An alternating linear map from V^2 to R with V=R^3:
## + dy^dzThe 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 \(-1,1,1,1\).
To work with \(2\)-forms in relativistic physics, it is often preferable to use bespoke print method usetxyz:
## An alternating linear map from V^2 to R with V=R^4:
## +6 dz^dNA +5 dy^dNA +4 dx^dNA +3 dy^dz +2 dx^dz + dx^dy## 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:
## 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## 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## An alternating linear map from V^2 to R with V=R^4:
## +8 dx^dy -4 dx^dz +2 dy^dz