stokes-package: The Exterior Calculus

In stokes: The Exterior Calculus

The Exterior Calculus

Description

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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

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

• J. H. Hubbard and B. B. Hubbard 2015. Vector calculus, linear algebra and differential forms: a unified approach. Ithaca, NY.

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

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

stokes documentation built on Aug. 19, 2023, 1:07 a.m.