stokes-package {stokes}

The Exterior Calculus

Description

Provides functionality for working with tensors, alternating forms, wedge products, Stokes's theorem, and related concepts from the exterior calculus. Uses 'disordR' discipline (Hankin, 2022, <doi:10.48550/ARXIV.2210.03856>). The canonical reference would be M. Spivak (1965, ISBN:0-8053-9021-9) "Calculus on Manifolds". To cite the package in publications please use Hankin (2022) <doi:10.48550/ARXIV.2210.17008>.

Details

The DESCRIPTION file:

Package:	stokes
Type:	Package
Title:	The Exterior Calculus
Version:	1.2-1
Depends:	R (>= 3.5.0)
Suggests:	knitr, Deriv, testthat, markdown, rmarkdown, quadform, magrittr, covr
VignetteBuilder:	knitr
Imports:	permutations (>= 1.1-2), partitions, methods, disordR (>= 0.9-7), spray (>= 1.0-25)
Authors@R:	person( given=c("Robin", "K. S."), family="Hankin", role = c("aut","cre"), email="[e-mail]", comment = c(ORCID = "0000-0001-5982-0415"))
Maintainer:	Robin K. S. Hankin <[e-mail]>
Description:	Provides functionality for working with tensors, alternating forms, wedge products, Stokes's theorem, and related concepts from the exterior calculus. Uses 'disordR' discipline (Hankin, 2022, <doi:10.48550/ARXIV.2210.03856>). The canonical reference would be M. Spivak (1965, ISBN:0-8053-9021-9) "Calculus on Manifolds". To cite the package in publications please use Hankin (2022) <doi:10.48550/ARXIV.2210.17008>.
License:	GPL-2
LazyData:	yes
URL:	https://github.com/RobinHankin/stokes
BugReports:	https://github.com/RobinHankin/stokes/issues
Author:	Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Index of help topics:

Alt                     Alternating multilinear forms
Ops.kform               Arithmetic Ops Group Methods for 'kform' and
                        'ktensor' objects
as.1form                Coerce vectors to 1-forms
coeffs                  Extract and manipulate coefficients
consolidate             Various low-level helper functions
contract                Contractions of k-forms
dovs                    Dimension of the underlying vector space
dx                      Elementary forms in three-dimensional space
ex                      Basis vectors in three-dimensional space
hodge                   Hodge star operator
inner                   Inner product operator
issmall                 Is a form zero to within numerical precision?
keep                    Keep or drop variables
kform                   k-forms
kinner                  Inner product of two kforms
ktensor                 k-tensors
print.stokes            Print methods for k-tensors and k-forms
rform                   Random kforms and ktensors
scalar                  Scalars and losing attributes
stokes-package          The Exterior Calculus
summary.stokes          Summaries of tensors and alternating forms
symbolic                Symbolic form
tensorprod              Tensor products of k-tensors
transform               Linear transforms of k-forms
vector_cross_product    The Vector cross product
volume                  The volume element
wedge                   Wedge products
zap                     Zap small values in k-forms and k-tensors
zero                    Zero tensors and zero forms

Generally in the package, arguments that are k-forms are denoted K, k-tensors by U, and spray objects by S. Multilinear maps (which may be either k-forms or k-tensors) are denoted by M.

Author(s)

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <[e-mail]>

References

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

• R. K. S. Hankin 2022. “Disordered vectors in R: introducing the disordR package.” https://arxiv.org/abs/2210.03856.

• R. K. S. Hankin 2022. “Sparse arrays in R: the spray package. https://arxiv.org/abs/2210.03856.”

See Also

spray

Examples

## Some k-tensors:
U1 <- as.ktensor(matrix(1:15,5,3))
U2 <- as.ktensor(cbind(1:3,2:4),1:3)

## Coerce a tensor to functional form, here mapping V^3  -> R (here V=R^15):
as.function(U1)(matrix(rnorm(45),15,3))

## Tensor product is tensorprod() or %X%:
U1 %X% U2


## A k-form is an alternating k-tensor:
K1 <- as.kform(cbind(1:5,2:6),rnorm(5))
K2 <- kform_general(3:6,2,1:6)
K3 <- rform(9,3,9,runif(9))

## The distributive law is true

(K1 + K2) ^ K3 == K1 ^ K3 + K2 ^ K3 # TRUE to numerical precision

## Wedge product is associative (non-trivial):
(K1 ^ K2) ^ K3
K1 ^ (K2 ^ K3)


## k-forms can be coerced to a function and wedge product:
f <- as.function(K1 ^ K2 ^ K3)

## E is a a random point in V^k:
E <- matrix(rnorm(63),9,7)

## f() is alternating:
f(E)
f(E[,7:1])



## The package blurs the distinction between symbolic and numeric computing:
dx <- as.kform(1)
dy <- as.kform(2)
dz <- as.kform(3)

dx ^ dy ^ dz

K3 ^ dx ^ dy ^ dz