# Overview

The stokes 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 $k$-tensors and $k$-forms. A $k$-tensor is a multilinear map $S\colon V^k\longrightarrow\mathbb{R}$, where $V=\mathbb{R}^n$ is considered as a vector space. Given two $k$-tensors $S,T$ the package can calculate their outer product $S\otimes T$ using natural R idiom (see below and the vignette for details).

A $k$-form is an alternating $k$-tensor, that is a $k$-tensor $\omega$ with the property that linear dependence of $x_1,\ldots,x_n$ implies that $\omega\left(x_1,\ldots,x_n\right)=0$. Given $k$-forms $\omega,\eta$, the package provides R idiom for calculating their wedge product $\omega\wedge\eta$.

# 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. We may define a $k$-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
#> A linear map from V^2 to R with V=R^5:
#>          val
#>  4 3  =    4
#>  3 5  =    3
#>  2 4  =    2
#>  1 3  =    1


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] 11.23556


Cross 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


## Alternating forms

An alternating form (or $k$-form) is an antisymmetric $k$-tensor; the package can convert a general $k$-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 $k$-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


We may coerce KF to functional form:

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


# The wedge product

The wedge product of two $k$-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

• Spivak 1971. Calculus on manifolds, Addison-Wesley.

But an accessible book would be

• Hubbard and Hubbard 2015. Vector calculus, linear algebra, and differential forms: a unified approach. Matrix Editions

# Further information

For more detail, see the package vignette

vignette("stokes")

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stokes documentation built on Jan. 18, 2022, 1:11 a.m.