contract: Contractions of k-forms

In stokes: The Exterior Calculus

Contractions of k-forms

Description

\loadmathjax

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

Usage

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

Arguments

K	A \mjseqnk-form
o	Integer-valued vector corresponding to one row of an index matrix
lose	Boolean, with default TRUE meaning to coerce a \mjseqn0-form to a scalar and FALSE meaning to return the formal \mjseqn0-form
v	A vector; in function contract(), if a matrix, interpret each column as a vector to contract with

Details

Given a \mjseqnk-form \mjeqn\phiphi and a vector \mjeqn\mathbfvv, the contraction \mjeqn\phi_\mathbfvphi_v of \mjeqn\phiphi and \mjeqn\mathbfvv is a \mjseqnk-1-form with

\mjdeqn \phi

_\mathbfv\left(\mathbfv^1,...,\mathbfv^k-1\right) = \phi\left(\mathbfv,\mathbfv^1,...,\mathbfv^k-1\right) omitted; see PDF

provided \mjseqnk>1; if \mjseqnk=1 we specify \mjeqn\phi_\mathbfv=\phi(\mathbfv)phi_v=phi(v).

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

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 Aug. 19, 2023, 1:07 a.m.