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 .
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
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 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
The most concise reference is
But a more leisurely book would be
For more detail, see the package vignette
vignette("stokes")
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