stokes (version 1.0-8)

kform: k-forms

Description

Functionality for dealing with \(k\)-forms

Usage

kform(S)
as.kform(M,coeffs,lose=TRUE)
kform_basis(n, k)
kform_general(W,k,coeffs,lose=TRUE)
is.kform(x)
# S3 method for kform
as.function(x,...)

Arguments

n

Dimension of the vector space V=R^nV=R^n

k

A \(k\)-form maps V^kV^k to \(R\)

W

Integer vector of dimensions

M,coeffs

Index matrix and coefficients for a \(k\)-form

S

Object of class spray

lose

Boolean, with default TRUE meaning to coerce a \(0\)-form to a scalar and FALSE meaning to return the formal \(0\)-form

x

Object of class kform

...

Further arguments, currently ignored

Value

All functions documented here return a kform object except as.function.kform(), which returns a function.

Details

A \(k\)-form is an alternating \(k\)-tensor.

Recall that a \(k\)-tensor is a multilinear map from \(V^k\) to the reals, where \(V=R^n\) is a vector space. A multilinear \(k\)-tensor \(T\) is alternating if it satisfies

T(v_1,…,v_i,…,v_j,…,v_k)= T(v_1,…,v_j,…,v_i,…,v_k) omitted; see latex

Function kform_basis() is a low-level helper function that returns a matrix whose rows constitute a basis for the vector space ^k(R^n)L^k(R^n) of \(k\)-tensors:

=_1 i_1 < < i_k n a_i_1… i_kdx_i_1 dx_i_komitted; see latex

and indeed we have:

a_i_1… i_k=(e_i_1,…,e_i_k) omitted; see latex

where e_j,1 j ke_j,1<=j<=k is a basis for \(V\).

In the stokes package, \(k\)-forms are represented as sparse arrays (spray objects), but with a class of c("kform", "spray"). The constructor function (kform()) ensures that rows of the index matrix are strictly nonnegative integers, have no repeated entries, and are strictly increasing.

References

Hubbard and Hubbard; Spivak

See Also

ktensor,lose

Examples

# NOT RUN {
as.kform(cbind(1:5,2:6),rnorm(5))
kform_general(1:4,2,coeffs=1:6)  # used in electromagnetism

K1 <- as.kform(cbind(1:5,2:6),rnorm(5))
K2 <- kform_general(5:8,2,1:6)
wedge(K1,K2)


f <- as.function(wedge(K1,K2))
E <- matrix(rnorm(32),8,4)

f(E) + f(E[,c(1,3,2,4)])  # should be zero


# }