contract: Contractions of k-forms
In wedge: The Exterior Calculus

Description Usage Arguments Author(s) References See Also Examples

Given a k-form phi and a vector v, the contraction phi_v of phi and v is a k-1-form with

ommitted; see PDF

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

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 v.

1 2	contract(K,v,lose=TRUE) contract_elementary(o,v)

`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

Robin K. S. Hankin

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

wedge,lose

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


## Now some verification:
o <- rform(2,k=5,n=9,coeffs=runif(2))
V <- matrix(rnorm(45),ncol=5)
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]))(V[,-(1:4),drop=FALSE]),
   as.function(contract(o,V[,1:5],lose=FALSE))(V[,-(1:5),drop=FALSE])
)

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