stokes (version 1.2-0)

contract: 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)

Value

Returns an object of class kform.

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

Author

Robin K. S. Hankin

Details

Given a k-form phi and a vector vv, the contraction _vphi_v of phi and vv is a k-1-form with

_v(v^1,...,v^k-1) = (v,v^1,...,v^k-1) omitted; see PDF

provided k>1; if k=1 we specify _v=(v)phi_v=phi(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 vv.

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

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