stokes (version 1.2-0)

stokes-package: tools:::Rd_package_title("stokes")

Description

tools:::Rd_package_description("stokes")

Author

tools:::Rd_package_author("stokes")

Maintainer: tools:::Rd_package_maintainer("stokes")

Details

The DESCRIPTION file: tools:::Rd_package_DESCRIPTION("stokes") tools:::Rd_package_indices("stokes")

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 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