The Hodge dual on basis elements of \(\Lambda^k(V)\)

We start by demonstrating hodge() on basis elements of \(\Lambda^k(V)\). Recall that if \(\left\lbrace e_1,\ldots,e_n\right\rbrace\) is a basis of vector space \(V=\mathbb{R}^n\), then \(\left\lbrace\omega_1,\ldots,\omega_k\right\rbrace\) is a basis of \(\Lambda^1(V)\), where \(\omega_i(e_j)=\delta_{ij}\). A basis of \(\Lambda^k(V)\) is given by the set

\[ \bigcup_{1\leqslant i_1 < \cdots < i_k\leqslant n} \bigwedge_{j=1}^k\omega_{i_j} = \left\lbrace \left.\omega_{i_1}\wedge\cdots\wedge\omega_{i_k} \right|1\leqslant i_1 < \cdots < i_k\leqslant n \right\rbrace. \]

This means that basis elements are things like \(\omega_2\wedge\omega_6\wedge\omega_7\). f \(V=\mathbb{R}^9\), what is \(\star\omega_2\wedge\omega_6\wedge\omega_7\)?

(a <- d(2) ^ d(6) ^ d(7))

## An alternating linear map from V^3 to R with V=R^7:
##            val
##  2 6 7  =    1

hodge(a,9)

## An alternating linear map from V^6 to R with V=R^9:
##                  val
##  1 3 4 5 8 9  =   -1

See how \(\star a\) has index entries 1-9 except \(2,6,7\) (from \(a\)). The (numerical) sign is negative because the permution has negative (permutational) sign. We can verify this using the permutations package:

p <- c(2,6,7,  1,3,4,5,8,9)
(pw <- as.word(p))

## [1] (1264)(375)
## [coerced from word form]

print_word(pw)

##     {1} {2} {3} {4} {5} {6} {7} {8} {9}
## [1] 2   6   7   1   3   4   5   .   .

sgn(pw)

## [1] -1

Above we see the sign of the permutation is negative. More succinct idiom would be

hodge(d(c(2,6,7)),9)

## An alternating linear map from V^6 to R with V=R^9:
##                  val
##  1 3 4 5 8 9  =   -1

The second argument to hodge() is needed if the largest index \(i_k\) of the first argument is less than \(n\); the default value is indeed \(n\). In the example above, this is \(7\):

hodge(d(c(2,6,7)))

## An alternating linear map from V^4 to R with V=R^5:
##              val
##  1 3 4 5  =   -1

Above we see the result if \(V=\mathbb{R}^7\).

More complicated examples

The hodge operator is linear and it is interesting to verify this.

(o <- rform())

## An alternating linear map from V^3 to R with V=R^7:
##            val
##  2 6 7  =    6
##  2 5 7  =    5
##  5 6 7  =   -9
##  1 3 7  =    4
##  1 5 7  =    7
##  2 3 5  =   -3
##  1 5 6  =   -8
##  1 2 7  =    2
##  1 4 6  =    1

hodge(o)

## An alternating linear map from V^4 to R with V=R^7:
##              val
##  2 3 5 7  =   -1
##  3 4 5 6  =    2
##  2 3 4 7  =   -8
##  1 4 6 7  =   -3
##  2 3 4 6  =   -7
##  2 4 5 6  =   -4
##  1 2 3 4  =   -9
##  1 3 4 6  =    5
##  1 3 4 5  =   -6

We verify that the fundamental relation holds by direct inspection:

o ^ hodge(o)

## An alternating linear map from V^7 to R with V=R^7:
##                    val
##  1 2 3 4 5 6 7  =  285

kinner(o,o)*volume(dovs(o))

## An alternating linear map from V^7 to R with V=R^7:
##                    val
##  1 2 3 4 5 6 7  =  285

showing agreement (above, we use function volume() in lieu of calculating the permutation’s sign explicitly. See the volume vignette for more details). We may work more formally by defining a function that returns TRUE if the left and right hand sides match

diff <- function(a,b){a^hodge(b) ==  kinner(a,b)*volume(dovs(a))}

and call it with random \(k\)-forms:

diff(rform(),rform())

## [1] TRUE

Or even

all(replicate(10,diff(rform(),rform())))

## [1] TRUE

Small-dimensional vector spaces

We can work in three dimensions in which case we have three linearly independent \(1\)-forms: \(dx\), \(dy\), and \(dz\). To work in this system it is better to use dx print method:

options(kform_symbolic_print = "dx")
hodge(dx,3)

## An alternating linear map from V^2 to R with V=R^3:
##  + dy^dz

This is further discussed in the dovs vignette.

Vector cross product identities

The three dimensional vector cross product \(\mathbf{u}\times\mathbf{v}=\det\begin{pmatrix} i & j & k \\ u_1&u_2&u_3\\ v_1&v_2&v_3 \end{pmatrix}\) is a standard part of elementary vector calculus. In the package the idiom is as follows:

vcp3

## function (u, v) 
## {
##     hodge(as.1form(u)^as.1form(v))
## }

revealing the formal definition of cross product as \(\mathbf{u}\times\mathbf{v}=\star{\left(\mathbf{u}\wedge\mathbf{v}\right)}\). There are several elementary identities that are satisfied by the cross product:

\[ \begin{aligned} \mathbf{u}\times(\mathbf{v}\times\mathbf{w}) &= \mathbf{v}(\mathbf{w}\cdot\mathbf{u})-\mathbf{w}(\mathbf{u}\cdot\mathbf{v})\\ (\mathbf{u}\times\mathbf{v})\times\mathbf{w} &= \mathbf{v}(\mathbf{w}\cdot\mathbf{u})-\mathbf{u}(\mathbf{v}\cdot\mathbf{w})\\ (\mathbf{u}\times\mathbf{v})\times(\mathbf{u}\times\mathbf{w}) &= (\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}))\mathbf{u} \\ (\mathbf{u}\times\mathbf{v})\cdot(\mathbf{w}\times\mathbf{x}) &= (\mathbf{u}\cdot\mathbf{w})(\mathbf{v}\cdot\mathbf{x}) - (\mathbf{u}\cdot\mathbf{x})(\mathbf{v}\cdot\mathbf{w}) \end{aligned} \]

We may verify all four together:

u <- c(1,4,2)
v <- c(2,1,5)
w <- c(1,-3,2)
x <- c(-6,5,7)
c(
  hodge(as.1form(u) ^ vcp3(v,w))        == as.1form(v*sum(w*u) - w*sum(u*v)),
  hodge(vcp3(u,v) ^ as.1form(w))        == as.1form(v*sum(w*u) - u*sum(v*w)),
  as.1form(as.function(vcp3(v,w))(u)*u) == hodge(vcp3(u,v) ^ vcp3(u,w))     ,
  hodge(hodge(vcp3(u,v)) ^ vcp3(w,x))   == sum(u*w)*sum(v*x) - sum(u*x)*sum(v*w)
)         

## [1] TRUE TRUE TRUE TRUE

Above, note the use of the hodge operator to define triple vector cross products. For example we have \(\mathbf{u}\times\left(\mathbf{v}\times\mathbf{w}\right)= \star\left(\mathbf{u}\wedge\star\left(\mathbf{v}\wedge\mathbf{w}\right)\right)\).

Non positive-definite metrics

The inner product \(\left\langle\alpha,\beta\right\rangle\) above may be generalized by defining it on decomposable vectors \(\alpha=\alpha_1\wedge\cdots\wedge\alpha_k\) and \(\beta=\beta_1\wedge\cdots\wedge\beta_k\) as

\[\left\langle\alpha,\beta\right\rangle= \det\left(\left\langle\alpha_i,\beta_j\right\rangle_{i,j}\right)\]

where \(\left\langle\alpha_i,\beta_j\right\rangle=\pm\delta_{ij}\) is an inner product on \(\Lambda^1(V)\) [the inner product is given by kinner()]. The resulting Hodge star operator is implemented in the package and one can specify the metric. For example, if we consider the Minkowski metric this would be \(-1,1,1,1\).

options(kform_symbolic_print = NULL)  # default print method
(o <- kform_general(4,2,1:6))

## 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

options(kform_symbolic_print = "txyz")
o

## An alternating linear map from V^2 to R with V=R^4:
##  +6 dy^dz +5 dx^dz +4 dt^dz +3 dx^dy +2 dt^dy + dt^dx

hodge(o)

## An alternating linear map from V^2 to R with V=R^4:
##  + dy^dz -2 dx^dz +3 dt^dz +4 dx^dy -5 dt^dy +6 dt^dx

hodge(o,g=c(-1,1,1,1))

## An alternating linear map from V^2 to R with V=R^4:
##  - dy^dz +2 dx^dz +3 dt^dz -4 dx^dy -5 dt^dy +6 dt^dx

hodge(o)-hodge(o,g=c(-1,1,1,1))

## An alternating linear map from V^2 to R with V=R^4:
##  +8 dx^dy -4 dx^dz +2 dy^dz