hodge: Hodge star operator

In stokes: The Exterior Calculus

Hodge star operator

Description

\loadmathjax

Given a \mjseqnk-form, return its Hodge dual

Usage

hodge(K, n=dovs(K), g, lose=TRUE)

Arguments

K	Object of class kform
n	Dimensionality of space, defaulting the the largest element of the index
g	Diagonal of the metric tensor, with missing default being the standard metric of the identity matrix. Currently, only entries of \mjeqn\pm 1+/-1 are accepted
lose	Boolean, with default TRUE meaning to coerce to a scalar if appropriate

Value

Given a \mjseqnk-form, in an \mjseqnn-dimensional space, return a \mjseqn(n-k)-form.

Note

Most authors write the Hodge dual of \mjeqn\psiomitted as \mjeqn*\psiomitted or \mjeqn\star\psiomitted, but Weintraub uses \mjeqn\psi *omitted.

Author(s)

Robin K. S. Hankin

See Also

wedge

Examples

(o <- kform_general(5,2,1:10))
hodge(o)
o == hodge(hodge(o))

Faraday <- kform_general(4,2,runif(6)) # Faraday electromagnetic tensor
mink <- c(-1,1,1,1)  # Minkowski metric
hodge(Faraday,g=mink)

Faraday  == Faraday |>
      hodge(g=mink) |>
      hodge(g=mink) |>
      hodge(g=mink) |>
      hodge(g=mink)

hodge(dx,3) == dy^dz


## Some edge-cases:
hodge(scalar(1),2)
hodge(zeroform(5),9)
hodge(volume(5))
hodge(volume(5),lose=TRUE)
hodge(scalar(7),n=9)

stokes documentation built on Aug. 19, 2023, 1:07 a.m.