wedge-package: The Exterior Calculus
In wedge: The Exterior Calculus

Description Details Author(s) References See Also Examples

Provides functionality for working with differentials, k-forms, wedge products, Stokes's theorem, and related concepts from the exterior calculus. The canonical reference would be: M. Spivak (1965, ISBN:0-8053-9021-9). "Calculus on Manifolds", Benjamin Cummings.

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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.

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

Maintainer: Robin K. S. Hankin <hankin.robin@gmail.com>

J. H. Hubbard and B. B. Hubbard 2015. Vector calculus, linear algebra and differential forms: a unified aproach. Ithaca, NY.
M. Spivak 1971. Calculus on manifolds. Addison-Wesley.

spray

## 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 cross-product is cross() 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