Alt: Alternating multilinear forms

In stokes: The Exterior Calculus

Alternating multilinear forms

Description

\loadmathjax

Converts a \mjseqnk-tensor to alternating form

Usage

Alt(S,give_kform)

Arguments

S	A multilinear form, an object of class ktensor
give_kform	Boolean, with default FALSE meaning to return an alternating \mjseqnk-tensor [that is, an object of class ktensor that happens to be alternating] and TRUE meaning to return a \mjseqnk-form [that is, an object of class kform]

Details

Given a \mjseqnk-tensor \mjseqnT, we have

\mjdeqn\mathrm

Alt(T)\left(v_1,...,v_k\right)= \frac1k!\sum_\sigma\in S_k\mathrmsgn(\sigma)\cdot T\left(v_\sigma(1),...,v_\sigma(k)\right) omitted; see latex

Thus for example if \mjseqnk=3:

\mjdeqn\mathrm

Alt(T)\left(v_1,v_2,v_3\right)= \frac16\left(\beginarrayc +T\left(v_1,v_2,v_3\right)\quad -T\left(v_1,v_3,v_2\right)
-T\left(v_2,v_1,v_3\right)\quad +T\left(v_2,v_3,v_1\right)
+T\left(v_3,v_1,v_2\right)\quad -T\left(v_3,v_2,v_1\right) \endarray \right) omitted; see latex

and it is reasonably easy to see that \mjeqn\mathrmAlt(T)Alt(T) is alternating, in the sense that

\mjdeqn\mathrm

Alt(T)\left(v_1,...,v_i,...,v_j,...,v_k\right)= -\mathrmAlt(T)\left(v_1,...,v_j,...,v_i,...,v_k\right) omitted; see latex

Function Alt() is intended to take and return an object of class ktensor; but if given a kform object, it just returns its argument unchanged.

A short vignette is provided with the package: type vignette("Alt") at the commandline.

Value

Returns an alternating \mjseqnk-tensor. To work with \mjseqnk-forms, which are a much more efficient representation of alternating tensors, use as.kform().

Author(s)

Robin K. S. Hankin

See Also

kform

Examples

(X <- ktensor(spray(rbind(1:3),6)))
Alt(X)
Alt(X,give_kform=TRUE)

S <- as.ktensor(expand.grid(1:3,1:3),rnorm(9))
S
Alt(S)

issmall(Alt(S) - Alt(Alt(S)))  # should be TRUE; Alt() is idempotent

a <- rtensor()
V <- matrix(rnorm(21),ncol=3)
LHS <- as.function(Alt(a))(V)
RHS <- as.function(Alt(a,give_kform=TRUE))(V)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

stokes documentation built on Aug. 19, 2023, 1:07 a.m.