In RobinHankin/wedge: The Exterior Calculus

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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 $k$-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. In the package, we can create a $k$-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

Above, object jj is equal to $dx_1\otimes dx_2\otimes dx_3 + 2dx_2\otimes dx_3\otimes dx_4$ (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)

Tensor products are implemented:

KT %X% KT

Above we see ${\mathrm KT}\otimes{\mathrm KT}$.

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)

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
KF <- as.kform(M,c(1,5))
KF

Above, we see that KF is equal to $dx_2\wedge dx_3\wedge dx_4 + 5dx_1\wedge dx_2\wedge dx_4$. We may coerce KF to functional form:

f <- as.function(KF)
E <- matrix(rnorm(12),4,3)
f(E)

Above, we evaluate KF at a point in $\left({\mathbb R^4}\right)^3$ [the three columns of matrix E are each interpreted as vectors in ${\mathbb R}^4$].

The wedge product

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

KF2 <- kform_general(6:9,2,1:6)
KF2
KF ^ KF2

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 

A number of useful functions from the exterior calculus are provided, such as the gradient of a scalar function:

grad(1:6)

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

grad(1:4) ^ grad(1:6)

References

The most concise reference is

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

But a more leisurely 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")

RobinHankin/wedge documentation built on June 9, 2025, 2:53 p.m.