wedge (version 1.0-3)

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

Description

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

Usage

transform(K,M)
stretch(K,d)

Arguments

K

Object of class kform

M

Matrix of transformation

d

Numeric vector representing the diagonal elements of a diagonal matrix

Value

Returns a \(k\)-form

Details

Suppose we are given a two-form

$$ \omega=\sum_{i<j}a_{ij}dx_i\wedge dx_j$$

and relationships

$$dx_i=\sum_rM_{ir}dy_r$$

then we would have

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

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

giving

$$\omega = \sum_{i_1<\cdots <i_k}\left[ a_{i_1<\cdots < i_k}\left(\sum_rM_{i_1r}dy_r\right)\wedge\cdots\wedge\left(\sum_rM_{i_kr}dy_r\right)\right] $$

So \(\omega\) was given in terms of \(dx_1,\ldots,dx_k\) and we have expressed it in terms of \(dy_1,\ldots,dy_k\). So for example if

$$\omega= dx_1\wedge dx_2 + 5dx_1\wedge dx_3$$

and

$$ \left( \begin{array}{l} dx_1\\ dx_2\\ dx_3 \end{array} \right)= \left( \begin{array}{ccc} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9\\ \end{array} \right) \left( \begin{array}{l} dy_1\\ dy_2\\ dy_3 \end{array} \right) $$

then

$$\begin{array}{ccl} \omega &=& \left(1dy_1+4dy_2+7dy_3\right)\wedge \left(2dy_1+5dy_2+8dy_3\right)+ 5\left(1dy_1+4dy_2+7dy_3\right)\wedge \left(3dy_1+6dy_2+9dy_3\right) \\ &=&2dy_1\wedge dy_1+5dy_1\wedge dy_2+\cdots+ 5\cdot 7\cdot 6dx_3\wedge dx_2+ 5\cdot 7\cdot 9dx_3\wedge dx_3+\\ &=& -33dy_1\wedge dy_2-66dy_1\wedge dy_3-33dy_2\wedge dy_3 \end{array} $$

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.

Function stretch() carries out the same operation but for a matrix with zero off-diagonal elements. It is much faster.

References

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

See Also

wedge

Examples

# NOT RUN {
# Example in the text:
K <- as.kform(matrix(c(1,1,2,3),2,2),c(1,5))
M <- matrix(1:9,3,3)
transform(K,M)

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

# Numerical verification:
o <- rform(terms=2,n=5)

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

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

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

# Some stretch() use-cases:

p <- rform()
p
stretch(p,seq_len(5))
stretch(p,c(1,0,1,1,1))   # kills dimension 2

# Works nicely with pipes:
# }
# NOT RUN {
max(abs(value(o-o %>% transform(M) %>% transform(solve(M)))))
# }
# NOT RUN {
# }