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The `hodge()` function in the `stokes` package
In stokes: The Exterior Calculus
knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
library("permutations")
library("stokes")
options(polyform = FALSE)
set.seed(1)
![](`r system.file("help/figures/stokes.png", package = "stokes")`){width=10%}
hodge
Given a $k$-form $\beta$, function hodge() returns its Hodge dual
$\star\beta$. Formally, if $V={\mathbb R}^n$, and $\Lambda^k(V)$ is
the space of alternating linear maps from $V^k$ to ${\mathbb R}$, then
$\star\colon\Lambda^k(V)\longrightarrow\Lambda^{n-k}(V)$.
To define the Hodge dual, we need an inner product
$\left\langle\cdot,\cdot\right\rangle$ [function kinner() in the
package] and, given this and $\beta\in\Lambda^k(V)$ we define
$\star\beta$ to be the (unique) $n-k$-form satisfying the fundamental
relation:
$$
\alpha\wedge\left(\star\beta\right)=\left\langle\alpha,\beta\right\rangle\omega,$$
for every $\alpha\in\Lambda^k(V)$. Here $\omega=e_1\wedge
e_2\wedge\cdots\wedge e_n$ is the unit $n$-vector of $\Lambda^n(V)$.
Taking determinants of this relation shows the following.
If we use multi-index notation so $e_I=e_{i_1}\wedge\cdots\wedge
e_{i_k}$ with $I=\left\lbrace i_1,\cdots,i_k\right\rbrace$, then
$$\star e_I=(-1)^{\sigma(I)}e_J$$
where $J=\left\lbrace
j_i,\ldots,j_k\right\rbrace=[n]\setminus\left\lbrace
i_1,\ldots,i_k\right\rbrace$ is the complement of $I$, and
$(-1)^{\sigma(I)}$ is the sign of the permutation $\sigma(I)=i_1\cdots
i_kj_1\cdots j_{n-k}$. We extend to the whole of $\Lambda^k(V)$ using
linearity. Package idiom for calculating the Hodge dual is
straightforward, being simply hodge().
options(kform_symbolic_print = NULL)
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))
hodge(a,9)
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))
print_word(pw)
sgn(pw)
Above we see the sign of the permutation is negative. More succinct
idiom would be
hodge(d(c(2,6,7)),9)
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)))
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())
hodge(o)
We verify that the fundamental relation holds by direct inspection:
o ^ hodge(o)
kinner(o,o)*volume(dovs(o))
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())
Or even
all(replicate(10,diff(rform(),rform())))
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)
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
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)
)
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
$-1,1,1,1$.
Print methods for the Minkowski metric
The standard print method is not particularly suitable for working
with the Minkowski metric:
options(kform_symbolic_print = NULL) # default print method
(o <- kform_general(4,2,1:6))
Above we see an unhelpful representation. To work with $2$-forms in
relativistic physics, it is often preferable to use bespoke print
method usetxyz:
options(kform_symbolic_print = "txyz")
o
Specifying the Minkowski metric
Function hodge() takes a g argument to specify the metric:
hodge(o)
hodge(o,g=c(-1,1,1,1))
hodge(o)-hodge(o,g=c(-1,1,1,1))
options(kform_symbolic_print = NULL)
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knitr::opts_chunk$set(echo = TRUE) options(rmarkdown.html_vignette.check_title = FALSE) library("permutations") library("stokes") options(polyform = FALSE) set.seed(1)
![](`r system.file("help/figures/stokes.png", package = "stokes")`){width=10%}
hodge
Given a $k$-form $\beta$, function hodge() returns its Hodge dual
$\star\beta$. Formally, if $V={\mathbb R}^n$, and $\Lambda^k(V)$ is
the space of alternating linear maps from $V^k$ to ${\mathbb R}$, then
$\star\colon\Lambda^k(V)\longrightarrow\Lambda^{n-k}(V)$.
To define the Hodge dual, we need an inner product
$\left\langle\cdot,\cdot\right\rangle$ [function kinner() in the
package] and, given this and $\beta\in\Lambda^k(V)$ we define
$\star\beta$ to be the (unique) $n-k$-form satisfying the fundamental
relation:
$$ \alpha\wedge\left(\star\beta\right)=\left\langle\alpha,\beta\right\rangle\omega,$$
for every $\alpha\in\Lambda^k(V)$. Here $\omega=e_1\wedge e_2\wedge\cdots\wedge e_n$ is the unit $n$-vector of $\Lambda^n(V)$. Taking determinants of this relation shows the following.
If we use multi-index notation so $e_I=e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $I=\left\lbrace i_1,\cdots,i_k\right\rbrace$, then
$$\star e_I=(-1)^{\sigma(I)}e_J$$
where $J=\left\lbrace
j_i,\ldots,j_k\right\rbrace=[n]\setminus\left\lbrace
i_1,\ldots,i_k\right\rbrace$ is the complement of $I$, and
$(-1)^{\sigma(I)}$ is the sign of the permutation $\sigma(I)=i_1\cdots
i_kj_1\cdots j_{n-k}$. We extend to the whole of $\Lambda^k(V)$ using
linearity. Package idiom for calculating the Hodge dual is
straightforward, being simply hodge().
options(kform_symbolic_print = NULL)
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)) hodge(a,9)
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)) print_word(pw) sgn(pw)
Above we see the sign of the permutation is negative. More succinct idiom would be
hodge(d(c(2,6,7)),9)
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)))
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()) hodge(o)
We verify that the fundamental relation holds by direct inspection:
o ^ hodge(o) kinner(o,o)*volume(dovs(o))
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())
Or even
all(replicate(10,diff(rform(),rform())))
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)
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
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) )
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
$-1,1,1,1$.
Print methods for the Minkowski metric
The standard print method is not particularly suitable for working with the Minkowski metric:
options(kform_symbolic_print = NULL) # default print method (o <- kform_general(4,2,1:6))
Above we see an unhelpful representation. To work with $2$-forms in
relativistic physics, it is often preferable to use bespoke print
method usetxyz:
options(kform_symbolic_print = "txyz") o
Specifying the Minkowski metric
Function hodge() takes a g argument to specify the metric:
hodge(o) hodge(o,g=c(-1,1,1,1)) hodge(o)-hodge(o,g=c(-1,1,1,1))
options(kform_symbolic_print = NULL)
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