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The stokes package: exterior calculus in R

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Overview

The stokes package provides functionality for working with the exterior calculus. It includes tensor 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 .

Installation

You can install the released version of stokes from CRAN with:

# install.packages("stokes")  # uncomment this to install the package
library("stokes")
set.seed(0)

The stokes package in use

The package has two main classes of objects, kform and ktensor. In the package, we can create a -tensor by supplying function as.ktensor() a matrix of indices and a vector of coefficients, for example:

jj <- as.ktensor(rbind(1:3,2:4),1:2)
jj
#> A linear map from V^3 to R with V=R^4:
#>            val
#>  2 3 4  =    2
#>  1 2 3  =    1

Above, object jj is equal to (see Spivak, p76 for details).

We can coerce tensors 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] 11.23556

Tensor products are implemented:

KT %X% KT
#> A linear map from V^4 to R with V=R^5:
#>              val
#>  1 2 1 2  =    1
#>  2 3 1 2  =    2
#>  3 4 3 4  =    9
#>  2 3 4 5  =    8
#>  1 2 2 3  =    2
#>  1 2 4 5  =    4
#>  4 5 4 5  =   16
#>  2 3 3 4  =    6
#>  4 5 3 4  =   12
#>  1 2 3 4  =    3
#>  3 4 4 5  =   12
#>  3 4 2 3  =    6
#>  4 5 2 3  =    8
#>  3 4 1 2  =    3
#>  2 3 2 3  =    4
#>  4 5 1 2  =    4

Above we see .

Alternating forms

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)
#> A linear map from V^2 to R with V=R^5:
#>           val
#>  5 4  =  -2.0
#>  4 5  =   2.0
#>  4 3  =  -1.5
#>  3 2  =  -1.0
#>  2 3  =   1.0
#>  3 4  =   1.5
#>  2 1  =  -0.5
#>  1 2  =   0.5

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
#> An alternating linear map from V^3 to R with V=R^4:
#>            val
#>  1 2 4  =    5
#>  2 3 4  =    1

Above, we see that KF is equal to . We may coerce KF to functional form:

f <- as.function(KF)
E <- matrix(rnorm(12),4,3)
f(E)
#> [1] -5.979544

Above, we evaluate KF at a point in [the three columns of matrix E are each interpreted as vectors in ].

The wedge product

The wedge product of two -forms is implemented as ^ or wedge():

KF2 <- kform_general(6:9,2,1:6)
KF2
#> An alternating linear map from V^2 to R with V=R^9:
#>          val
#>  8 9  =    6
#>  7 9  =    5
#>  6 9  =    4
#>  7 8  =    3
#>  6 8  =    2
#>  6 7  =    1
KF ^ KF2
#> An alternating linear map from V^5 to R with V=R^9:
#>                val
#>  1 2 4 6 7  =    5
#>  1 2 4 6 8  =   10
#>  2 3 4 6 8  =    2
#>  2 3 4 7 8  =    3
#>  2 3 4 6 9  =    4
#>  1 2 4 6 9  =   20
#>  2 3 4 6 7  =    1
#>  2 3 4 7 9  =    5
#>  1 2 4 7 8  =   15
#>  2 3 4 8 9  =    6
#>  1 2 4 7 9  =   25
#>  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 
#> An alternating linear map from V^3 to R with V=R^3:
#>            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)
#> An alternating linear map from V^1 to R with V=R^6:
#>        val
#>  6  =    6
#>  5  =    5
#>  4  =    4
#>  3  =    3
#>  2  =    2
#>  1  =    1

The package takes the leg-work out of the exterior calculus:

grad(1:4) ^ grad(1:6)
#> An alternating linear map from V^2 to R with V=R^6:
#>          val
#>  4 5  =   20
#>  1 5  =    5
#>  2 5  =   10
#>  3 5  =   15
#>  2 6  =   12
#>  4 6  =   24
#>  3 6  =   18
#>  1 6  =    6

References

The most concise reference is

But a more leisurely book would be

Further information

For more detail, see the package vignette

vignette("stokes")

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.