The wedge
package provides functionality for working with the exterior
calculus. It includes cross products and wedge products and a variety of
use-cases. The canonical reference would be Spivak (see references). A
detailed vignette is provided in the package.
The package deals with -tensors and -forms. A -tensor is a multilinear map , where is considered as a vector space. Given two -tensors the package can calculate their outer product using natural R idiom (see below and the vignette for details).
A -form is an alternating -tensor, that is a -tensor with the property that linear dependence of implies that . Given -forms , the package provides R idiom for calculating their wedge product .
You can install the released version of wedge from CRAN with:
# install.packages("wedge") # uncomment this to install the package
library("wedge")
wedge
package in useThe package has two main classes of objects, kform
and ktensor
. We
may define a -tensor as
follows
KT <- as.ktensor(cbind(1:4,3:5),1:4)
#> Warning in cbind(1:4, 3:5): number of rows of result is not a multiple of
#> vector length (arg 2)
KT
#> val
#> 1 3 = 1
#> 2 4 = 2
#> 3 5 = 3
#> 4 3 = 4
We can coerce KT
to a function and then evaluate it:
KT <- as.ktensor(cbind(1:4,2:5),1:4)
f <- as.function(KT)
E <- matrix(rnorm(10),5,2)
f(E)
#> [1] -0.1383716
Cross products are implemented:
KT %X% KT
#> val
#> 3 4 3 4 = 9
#> 2 3 1 2 = 2
#> 2 3 2 3 = 4
#> 3 4 1 2 = 3
#> 4 5 1 2 = 4
#> 1 2 1 2 = 1
#> 1 2 2 3 = 2
#> 2 3 3 4 = 6
#> 3 4 2 3 = 6
#> 4 5 4 5 = 16
#> 4 5 2 3 = 8
#> 1 2 3 4 = 3
#> 4 5 3 4 = 12
#> 1 2 4 5 = 4
#> 2 3 4 5 = 8
#> 3 4 4 5 = 12
An alternating form (or -form) is an antisymmetric
-tensor; the package can
convert a general -tensor to alternating form using the Alt()
function:
Alt(KT)
#> val
#> 1 2 = 0.5
#> 2 1 = -0.5
#> 4 3 = -1.5
#> 2 3 = 1.0
#> 3 2 = -1.0
#> 5 4 = -2.0
#> 3 4 = 1.5
#> 4 5 = 2.0
However, the package provides a bespoke and efficient representation for
-forms as objects with
class kform
. Such objects may be created using the as.kform()
function:
M <- matrix(c(4,2,3,1,2,4),2,3,byrow=TRUE)
M
#> [,1] [,2] [,3]
#> [1,] 4 2 3
#> [2,] 1 2 4
KF <- as.kform(M,c(1,5))
KF
#> val
#> 2 3 4 = 1
#> 1 2 4 = 5
We may coerce KF
to functional form:
f <- as.function(KF)
E <- matrix(rnorm(12),4,3)
f(E)
#> [1] -1.895993
The wedge product of two -forms is implemented as %^%
or wedge()
:
KF2 <- kform_general(6:9,2,1:6)
KF2
#> val
#> 6 7 = 1
#> 6 8 = 2
#> 7 9 = 5
#> 7 8 = 3
#> 6 9 = 4
#> 8 9 = 6
KF %^% KF2
#> val
#> 2 3 4 6 7 = 1
#> 1 2 4 6 8 = 10
#> 1 2 4 6 9 = 20
#> 2 3 4 7 9 = 5
#> 1 2 4 7 9 = 25
#> 2 3 4 6 8 = 2
#> 1 2 4 6 7 = 5
#> 2 3 4 8 9 = 6
#> 2 3 4 6 9 = 4
#> 2 3 4 7 8 = 3
#> 1 2 4 7 8 = 15
#> 1 2 4 8 9 = 30
The package can accommodate a number of results from the exterior calculus such as elementary forms:
dx <- as.kform(1)
dy <- as.kform(2)
dz <- as.kform(3)
dx %^% dy %^% dz # element of volume
#> val
#> 1 2 3 = 1
A number of useful functions from the exterior calculus are provided, such as the gradient of a scalar function:
grad(1:6)
#> val
#> 1 = 1
#> 2 = 2
#> 3 = 3
#> 4 = 4
#> 5 = 5
#> 6 = 6
The package takes the leg-work out of the exterior calculus:
grad(1:4) %^% grad(1:6)
#> val
#> 2 5 = 10
#> 3 5 = 15
#> 3 6 = 18
#> 1 5 = 5
#> 2 6 = 12
#> 4 5 = 20
#> 1 6 = 6
#> 4 6 = 24
The most concise reference is
But an accessible book would be
For more detail, see the package vignette
vignette("wedge")
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