stokes (version 1.2-1)

transform: Linear transforms of \(k\)-forms

Description

Given a \(k\)-form, express it in terms of linear combinations of the \(dx_i\)

Usage

pullback(K,M)
stretch(K,d)

Value

The functions documented here return an object of class

kform.

Arguments

K

Object of class kform

M

Matrix of transformation

d

Numeric vector representing the diagonal elements of a diagonal matrix

Author

Robin K. S. Hankin

Details

Function pullback() calculates the pullback of a function. A vignette is provided at pullback.Rmd.

Suppose we are given a two-form

$$ \omega=\sum_{i < j}a_{ij}\mathrm{d}x_i\wedge\mathrm{d}x_j$$

and relationships

$$\mathrm{d}x_i=\sum_rM_{ir}\mathrm{d}y_r$$

then we would have

$$\omega = \sum_{i < j} a_{ij}\left(\sum_rM_{ir}\mathrm{d}y_r\right)\wedge\left(\sum_rM_{jr}\mathrm{d}y_r\right). $$

The general situation would be a \(k\)-form where we would have $$ \omega=\sum_{i_1 < \cdots < i_k}a_{i_1\ldots i_k}\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}$$

giving

$$\omega = \sum_{i_1 < \cdots < i_k}\left[ a_{i_1,\ldots, i_k}\left(\sum_rM_{i_1r}\mathrm{d}y_r\right)\wedge\cdots\wedge\left(\sum_rM_{i_kr}\mathrm{d}y_r\right)\right]. $$

The transform() function does all this but it is slow. I am not 100% sure that there isn't a much more efficient way to do such a transformation. There are a few tests in tests/testthat and a discussion in the stokes vignette.

Function stretch() carries out the same operation but for \(M\) a diagonal matrix. It is much faster than transform().

References

S. H. Weintraub 2019. Differential forms: theory and practice. Elsevier. (Chapter 3)

See Also

wedge

Examples

# Example in the text:
K <- as.kform(matrix(c(1,1,2,3),2,2),c(1,5))
M <- matrix(1:9,3,3)
pullback(K,M)

# Demonstrate that the result can be complicated:
M <- matrix(rnorm(25),5,5)
pullback(as.kform(1:2),M)

# Numerical verification:
o <- volume(3)

o2 <- pullback(pullback(o,M),solve(M))
max(abs(coeffs(o-o2))) # zero to numerical precision

# Following should be zero:
pullback(as.kform(1),M)-as.kform(matrix(1:5),c(crossprod(M,c(1,rep(0,4)))))

# Following should be TRUE:
issmall(pullback(o,crossprod(matrix(rnorm(10),2,5))))

# Some stretch() use-cases:

p <- rform()
p
stretch(p,seq_len(7))
stretch(p,c(1,0,0,1,1,1,1))   # kills dimensions 2 and 3