k-forms

Functionality for dealing with $k$ -forms

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

Arguments

n: Dimension of the vector space $V=\mathbb{R}^n$
i: Integer
k: A $k$ -form maps $V^k$ to $\mathbb{R}$
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

Details

A $k$ -form is an alternating $k$ -tensor. In the package, $k$ -forms are represented as sparse arrays (spray objects), but with a class of c("kform", "spray"). The constructor function kform() takes a spray object and returns a kform object: it ensures that rows of the index matrix are strictly nonnegative integers, have no repeated entries, and are strictly increasing. Function as.kform() is more user-friendly.

kform() is the constructor function. It takes a spray object and returns a kform.
as.kform() also returns a kform but is a bit more user-friendly than kform().
kform_basis() is a low-level helper function that returns a matrix whose rows constitute a basis for the vector space $\Lambda^k(\mathbb{R}^n)$ of $k$ -forms.
kform_general() returns a kform object with terms that span the space of alternating tensors.
is.kform() returns TRUE if its argument is a kform object.
d() is an easily-typed synonym for as.kform(). The idea is that d(1) = dx, d(2)=dy, d(5)=dx^5, etc. Also note that, for example, d(1:3)=dx^dy^dz, the volume form.

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

$T\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)= -T\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right)$

In the package, an object of class kform is an efficient representation of an alternating tensor.

Function kform_basis() is a low-level helper function that returns a matrix whose rows constitute a basis for the vector space $\Lambda^k(\mathbb{R}^n)$ of $k$ -forms:

$\phi=\sum_{1\leq i_1 < \cdots < i_k\leq n} a_{i_1\ldots i_k}\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}$

and indeed we have:

$a_{i_1\ldots i_k}=\phi\left(\mathbf{e}_{i_1},\ldots,\mathbf{e}_{i_k}\right)$

where $\mathbf{e}_j,1\leq j\leq k$ is a basis for $V$ .

Value

All functions documented here return a kform object except as.function.kform(), which returns a function, and is.kform(), which returns a Boolean, and e(), which returns a conjugate basis to that of d().

References

Hubbard and Hubbard; Spivak

Author

Robin K. S. Hankin

Note

Hubbard and Hubbard use the term “ $k$ -form”, but Spivak does not.

Examples


as.kform(cbind(1:5,2:6),rnorm(5))
#> An alternating linear map from V^2 to R with V=R^6:
#>                  val
#>  5 6  =   0.54448610
#>  4 5  =  -1.08907888
#>  3 4  =   1.21604011
#>  2 3  =   0.32836490
#>  1 2  =   0.09588097
kform_general(1:4,2,coeffs=1:6)  # used in electromagnetism
#> An alternating linear map from V^2 to R with V=R^4:
#>          val
#>  3 4  =    6
#>  2 4  =    5
#>  1 4  =    4
#>  2 3  =    3
#>  1 3  =    2
#>  1 2  =    1

K1 <- as.kform(cbind(1:5,2:6),rnorm(5))
K2 <- kform_general(5:8,2,1:6)
K1^K2  # or wedge(K1,K2)
#> An alternating linear map from V^4 to R with V=R^8:
#>                      val
#>  3 4 5 6  =  -0.69195268
#>  1 2 6 8  =   2.30636367
#>  3 4 6 7  =  -2.07585805
#>  4 5 7 8  =   0.04131579
#>  2 3 7 8  =  -8.92659065
#>  1 2 7 8  =   2.76763641
#>  3 4 6 8  =  -3.45976341
#>  1 2 6 7  =   1.38381820
#>  3 4 5 8  =  -2.76781073
#>  2 3 5 6  =  -1.48776511
#>  3 4 5 7  =  -1.38390536
#>  2 3 5 7  =  -2.97553022
#>  2 3 6 7  =  -4.46329533
#>  1 2 5 6  =   0.46127273
#>  1 2 5 8  =   1.84509094
#>  4 5 6 7  =   0.02065789
#>  5 6 7 8  =  -0.92407972
#>  1 2 5 7  =   0.92254547
#>  3 4 7 8  =  -4.15171609
#>  4 5 6 8  =   0.03442982
#>  2 3 6 8  =  -7.43882554
#>  2 3 5 8  =  -5.95106044

d(1:3)
#> An alternating linear map from V^3 to R with V=R^3:
#>            val
#>  1 2 3  =    1
dx^dy^dz   # same thing
#> An alternating linear map from V^3 to R with V=R^3:
#>            val
#>  1 2 3  =    1

d(sample(9)) # coeff is +/-1 depending on even/odd permutation of 1:9
#> An alternating linear map from V^9 to R with V=R^9:
#>                        val
#>  1 2 3 4 5 6 7 8 9  =   -1

f <- as.function(wedge(K1,K2))
E <- matrix(rnorm(32),8,4)
f(E) + f(E[,c(1,3,2,4)])  # should be zero by alternating property
#> [1] 0

options(kform_symbolic_print = 'd')
(d(5)+d(7)) ^ (d(2)^d(5) + 6*d(4)^d(7))
#> An alternating linear map from V^3 to R with V=R^7:
#>  -6 dx4^dx5^dx7 + dx2^dx5^dx7 
options(kform_symbolic_print = NULL)  # revert to default