Contractions of \(k\)-forms

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

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.

References

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

Author

Robin K. S. Hankin

See also

wedge,lose

Examples

contract(as.kform(1:5),1:8)
#> An alternating linear map from V^4 to R with V=R^5:
#>              val
#>  1 2 3 4  =    5
#>  1 2 4 5  =    3
#>  2 3 4 5  =    1
#>  1 3 4 5  =   -2
#>  1 2 3 5  =   -4
contract(as.kform(1),3)   # 0-form
#> [1] 3



contract_elementary(c(1,2,5),c(1,2,10,11,71))
#> An alternating linear map from V^2 to R with V=R^5:
#>          val
#>  1 5  =   -2
#>  2 5  =    1
#>  1 2  =   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