hodge.RdGiven a \(k\)-form, return its Hodge dual
hodge(K, n=dovs(K), g, lose=TRUE)Object of class kform
Dimensionality of space, defaulting the the largest element of the index
Diagonal of the metric tensor, with missing default being the standard metric of the identity matrix. Currently, only entries of \(\pm 1\) are accepted
Boolean, with default TRUE meaning to coerce to a
scalar if appropriate
Given a \(k\)-form, in an \(n\)-dimensional space, return a \((n-k)\)-form.
Most authors write the Hodge dual of \(\psi\) as \(*\psi\) or \(\star\psi\), but Weintraub uses \(\psi *\).
(o <- kform_general(5,2,1:10))
#> An alternating linear map from V^2 to R with V=R^5:
#> val
#> 2 5 = 8
#> 1 5 = 7
#> 3 4 = 6
#> 2 4 = 5
#> 4 5 = 10
#> 1 4 = 4
#> 2 3 = 3
#> 1 3 = 2
#> 3 5 = 9
#> 1 2 = 1
hodge(o)
#> An alternating linear map from V^3 to R with V=R^5:
#> val
#> 3 4 5 = 1
#> 1 2 4 = -9
#> 2 4 5 = -2
#> 2 3 5 = 4
#> 1 2 3 = 10
#> 1 4 5 = 3
#> 1 3 5 = -5
#> 1 2 5 = 6
#> 2 3 4 = -7
#> 1 3 4 = 8
o == hodge(hodge(o))
#> [1] TRUE
Faraday <- kform_general(4,2,runif(6)) # Faraday electromagnetic tensor
mink <- c(-1,1,1,1) # Minkowski metric
hodge(Faraday,g=mink)
#> An alternating linear map from V^2 to R with V=R^4:
#> val
#> 3 4 = -0.6140056
#> 2 4 = 0.4414992
#> 1 4 = 0.3155947
#> 2 3 = -0.1013138
#> 1 3 = -0.2726462
#> 1 2 = 0.6536994
Faraday == Faraday |>
hodge(g=mink) |>
hodge(g=mink) |>
hodge(g=mink) |>
hodge(g=mink)
#> [1] TRUE
hodge(dx,3) == dy^dz
#> [1] TRUE
## Some edge-cases:
hodge(scalar(1),2)
#> An alternating linear map from V^2 to R with V=R^2:
#> val
#> 1 2 = 1
hodge(zeroform(5),9)
#> The zero alternating linear map from V^4 to R with V=R^n:
#> empty sparse array with 4 columns
hodge(volume(5))
#> [1] 1
hodge(volume(5),lose=TRUE)
#> [1] 1
hodge(scalar(7),n=9)
#> An alternating linear map from V^9 to R with V=R^9:
#> val
#> 1 2 3 4 5 6 7 8 9 = 7