wedge (version 1.0-3)

wedge-package: wedge

Description

wedge

Details

The DESCRIPTION file: wedge wedge

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.

References

• 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.

See Also

spray

Examples

# NOT RUN {
## 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
# }