contract: Contractions of k-forms

In stokes: The Exterior Calculus

Contractions of k-forms

Description

A contraction is a natural linear map from k-forms to k-1-forms.

Usage

contract(K,v,lose=TRUE)
contract_elementary(o,v)

Arguments

K	A k-form
o	Integer-valued vector corresponding to one row of an index matrix
lose	Boolean, with default TRUE meaning to coerce a 0-form to a scalar and FALSE meaning to return the formal 0-form
v	A vector; in function contract(), if a matrix, interpret each column as a vector to contract with

Details

Given a k-form \phi and a vector \mathbf{v}, the contraction \phi_\mathbf{v} of \phi and \mathbf{v} is a k-1-form with

\phi_\mathbf{v}\left(\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) = \phi\left(\mathbf{v},\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right)

provided k>1; if k=1 we specify \phi_\mathbf{v}=\phi(\mathbf{v}).

Function contract_elementary() is a low-level helper function that translates elementary k-forms with coefficient 1 (in the form of an integer vector corresponding to one row of an index matrix) into its contraction with \mathbf{v}.

There is an extensive vignette in the package, vignette("contract").

Value

Returns an object of class kform.

Author(s)

Robin K. S. Hankin

References

Steven H. Weintraub 2014. “Differential forms: theory and practice”, Elsevier (Definition 2.2.23, chapter 2, page 77).

See Also

wedge,lose

Examples

contract(as.kform(1:5),1:8)
contract(as.kform(1),3)   # 0-form



contract_elementary(c(1,2,5),c(1,2,10,11,71))


## Now some verification [takes ~10s to run]:
#o <- kform(spray(t(replicate(2, sample(9,4))), runif(2)))
#V <- matrix(rnorm(36),ncol=4)
#jj <- c(
#   as.function(o)(V),
#   as.function(contract(o,V[,1,drop=TRUE]))(V[,-1]), # scalar
#   as.function(contract(o,V[,1:2]))(V[,-(1:2),drop=FALSE]),
#   as.function(contract(o,V[,1:3]))(V[,-(1:3),drop=FALSE]),
#   as.function(contract(o,V[,1:4],lose=FALSE))(V[,-(1:4),drop=FALSE])
#)

#print(jj)
#max(jj) - min(jj) # zero to numerical precision

stokes documentation built on June 22, 2024, 11:56 a.m.