contract {stokes} | R Documentation |
Contractions of
-forms
Description
A contraction is a natural linear map from -forms to
-forms.
Usage
contract(K,v,lose=TRUE)
contract_elementary(o,v)
Arguments
K |
A |
o |
Integer-valued vector corresponding to one row of an index matrix |
lose |
Boolean, with default |
v |
A vector; in function |
Details
Given a -form
and a vector
,
the contraction
of
and
is a
-form with
provided ; if
we specify
.
Function contract_elementary()
is a low-level helper function
that translates elementary -forms with coefficient 1 (in the form
of an integer vector corresponding to one row of an index matrix) into
its contraction with
.
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
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