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Exterior calculus with R
In wedge: The Exterior Calculus
The exterior calculus
Ordinary differential calculus may be generalized to
arbitrary-dimensional oriented manifolds using the exterior calculus.
Here I show how the wedge package furnishes functionality for working
with the exterior calculus, and provide numerical verification of a
number of theorems. Notation follows that of Spivak, and Hubbard and
Hubbard.
Recall that a $k$-tensor is a multilinear map $S\colon
V^k\longrightarrow\mathbb{R}$, where $V=\mathbb{R}^n$ is considered as
a vector space; Spivak denotes the space of multilinear maps as
$\mathcal{J}^k(V)$. Formally, multilinearity means
[
S\left(v_1,\ldots,av_i,\ldots,v_k\right)=a\cdot S\left(v_1,\ldots,v_i,\ldots,v_k\right)
]
and
[S\left(v_1,\ldots,v_i+{v_i}',\ldots,v_k\right)=S\left(v_1,\ldots,v_i,\ldots,x_v\right)+
S\left(v_1,\ldots,{v_i}',\ldots,v_k\right).
]
where $v_i\in V$. If $S\in\mathcal{J}^k(V)$ and
$T\in\mathcal{J}^l(V)$, then we may define $S\otimes
T\in\mathcal{J}^{k+l}(V)$ as
[
S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l}\right)=
S\left(v_1,\ldots,v_k\right)\cdot
T\left(v_1,\ldots,v_l\right)
]
Spivak observes that $\mathcal{J}^k(V)$ is spanned by the $n^k$
products of the form
[
\phi_{i_1}\otimes\phi_{i_2}\otimes\ldots\otimes\phi_{i_k}\qquad
1\leq i_i,i_2,\ldots,i_k\leq n
]
where $v_1,\ldots,v_k$ is a basis for $V$ and
$\phi_i\left(v_j\right)=\delta_{ij}$; we can therefore write
[
S=\sum_{1\leq i_1,\ldots,i_k\leq n} a_{i_1\ldots i_k}
\phi_{i_1}\otimes\ldots\otimes\phi_{i_k}.
]
The space spanned by such products has a natural representation in R
as an array of dimensions $n\times\cdots\times n$. If A is such
an array, then the element A[i_1,i_2,...,i_k] is the coefficient
of $\phi_{i_1}\otimes\ldots\otimes\phi_{i_k}$. However, it is more
efficient and conceptually cleaner to consider a sparse array, as
implemented by the spray package. We will consider the case
$n=5,k=4$, so we have multilinear maps from
$\left(\mathbb{R}^5\right)^4$ to $\mathbb{R}$. Below, we will test
algebraic identities in R using the idiom furnished by the wedge
package. For our example we will define
$S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2$
using a matrix with three rows, one per term, and whose rows
correspond to each terms's cross products of the $\phi$'s. We first
have to load the wedge library:
library("wedge")
Then the idiom is straightforward:
k <- 4
n <- 5
M <- matrix(c(5,1,1,1, 1,1,2,3, 1,3,4,2),3,4,byrow=TRUE)
M
S <- as.ktensor(M,coeffs= 0.5 + 1:3)
S
Observe that, if stored as an array of size $n^k$, $S$ would have
$5^4=625$ elements, all but three of which are zero. So $S$ is a
4-tensor, mapping $V^4$ to $\mathbb{R}$, where $V=\mathbb{R}^5$. Here
we have
$S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2$.
Note that in some implementations the row order of object S will
differ from that of M. We will define $E$ to be a random point
in $V^k$ in terms of a matrix:
E <- matrix(rnorm(n*k),n,k) # A random point in V^k
Then
f <- as.function(S)
f(E)
Testing multilinearity is straightforward in the package:
E1 <- E
E2 <- E
E3 <- E
x1 <- rnorm(n)
x2 <- rnorm(n)
r1 <- rnorm(1)
r2 <- rnorm(1)
E1[,2] <- x1
E2[,2] <- x2
E3[,2] <- r1*x1 + r2*x2
Then we can verify the multilinearity of $S$ by coercing to a function
which is applied to E1, E2, E3:
f <- as.function(S)
way1 <- r1*f(E1) + r2*f(E2)
way2 <- f(E3)
c(way1,way2)
(that is, identical up to numerical precision). Note that this is
not equivalent to linearity over $V^{nk}$:
E1 <- matrix(rnorm(n*k),n,k)
E2 <- matrix(rnorm(n*k),n,k)
way1 <- f(r1*E1+r2*E2)
way2 <- r1*f(E1)+r2*f(E2)
c(way1,way2)
Given two k-tensor objects $S,T$ we can form the cross product
$S\otimes T$, defined as
[
S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)=
S\left(v_1,\ldots v_k\right)\cdot T\left(v_{k+1},\ldots
v_{k+l}\right)
]
S1 <- ktensor(spray(cbind(1:3,2:4),1:3))
S2 <- as.ktensor(matrix(1:6,2,3))
S1
S2
The R idiom for $S1\otimes S2$ would be cross(), or %X%:
cross(S1,S2)
Then, for example:
as.function(cross(S1,S2))(matrix(rnorm(30),6,5))
Alternating forms
An alternating form is a multilinear map $T$ satisfying
[
\mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)=
-\mathrm{T}\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right)
]
(or, equivalently,
$\mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_i,\ldots,v_k\right)= 0$). We
write $\Lambda^k(V)$ for the space of all alternating multilinear maps
from $V^k$ to $\mathbb{R}$. Spivak gives
$\operatorname{Alt}\colon\mathcal{J}^k(V)\longrightarrow\Lambda^k(V)$
defined by
[\operatorname{Alt}(T)\left(v_1,\ldots,v_k\right)=
\frac{1}{k!}\sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\cdot
T\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right)
]
where the sum ranges over all permutations of
$\left[n\right]=\left{1,2,\ldots,n\right}$ and
$\operatorname{sgn}(\sigma)\in\pm 1$ is the sign of the permutation.
If $T\in\mathcal{J}^k(V)$ and $\omega\in\Lambda^k(V)$, it is
straightforward to prove that $\operatorname{Alt}(T)\in\Lambda^k(V)$,
$\operatorname{Alt}\left(\operatorname{Alt}\left(T\right)\right)=\operatorname{Alt}\left(T\right)$,
and $\operatorname{Alt}\left(\omega\right)=\omega$.
In the wedge package, this is effected by the Alt() function:
S1
Alt(S1)
Verifying that S1 is in fact alternating is straightforward:
E <- matrix(rnorm(8),4,2)
Erev <- E[,2:1]
as.function(Alt(S1))(E) + as.function(Alt(S1))(Erev) # should be zero
However, we can see that this form for alternating tensors (here
called $k$-forms) is inefficient and highly redundant: in this example
there is a 1 2 term and a 2 1 term (the coefficients are
equal and opposite). In this example we have $k=2$ but in general
there would be potentially $k!$ essentially repeated terms which
collectively require only a single coefficient. The package provides
kform objects which are inherently alternating using a more
efficient representation; they are described using wedge products
which are discussed next.
Wedge products and the exterior calculus
This section follows the exposition of Hubbard and Hubbard, who
introduce the exterior calculus starting with a discussion of
elementary forms, which are alternating forms with a particularly
simple structure. An elementary form is an alternating multilinear
map with a particularly simple structure. An example of an elementary
form would be $dx_1\wedge dx_3$ [treated as an indivisible entity],
which is an alternating multilinear map from
$\mathbb{R}^n\times\mathbb{R}^n$ to $\mathbb{R}$ with
[
\left(
dx_1\wedge dx_3
\right)\left(
\begin{pmatrix}a_1\a_2\a_3\ \vdots\ a_n\end{pmatrix},
\begin{pmatrix}b_1\b_3\b_3\ \vdots\ b_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix} a_1 & b_1 \ a_3 & b_3\end{pmatrix}
=a_1b_3-a_3b_1
]
That this is alternating follows from the properties of the
determinant. In general of course, $dx_i\wedge dx_j\left(
\begin{pmatrix}a_1\ \vdots\ a_n\end{pmatrix}, \begin{pmatrix}b_1\
\vdots\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_i &
b_i \ a_j & b_j\end{pmatrix}$. Because such objects are linear, it
is possible to consider sums of elementary forms, such as $dx_1\wedge
dx_2 + 3 dx_2\wedge dx_3$ with
[
\left(
dx_1\wedge dx_2 + 3dx_2\wedge dx_3
\right)\left(
\begin{pmatrix}a_1\a_2\ \vdots\ a_n\end{pmatrix},
\begin{pmatrix}b_1\b_2\ \vdots\ b_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix} a_1 & b_1\ a_2 & b_2\end{pmatrix}
+3\mathrm{det}
\begin{pmatrix} a_2 & b_2\ a_3 & b_3\end{pmatrix}
]
or even $K=dx_1\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4$
which would be a linear map from $\left(\mathbb{R}^n\right)^3$ to
$\mathbb{R}$ with
[
\left(
dx_4\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4
\right)\left(
\begin{pmatrix}a_1\a_2\ \vdots\ a_n\end{pmatrix},
\begin{pmatrix}b_1\b_2\ \vdots\ b_n\end{pmatrix},
\begin{pmatrix}c_1\c_2\ \vdots\ c_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix}
a_4 & b_4 & c_4\
a_2 & b_2 & c_2\
a_3 & b_3 & c_3
\end{pmatrix}
+5\mathrm{det}
\begin{pmatrix}
a_1 & b_1 & c_1\
a_2 & b_2 & c_2\
a_4 & b_4 & c_4
\end{pmatrix}.
]
Defining $K$ has ready R idiom in which we define a matrix whose rows
correspond to the differentials in each term:
M <- matrix(c(4,2,3,1,2,4),2,3,byrow=TRUE)
M
K <- as.kform(M,c(1,5))
K
Note that the order of the rows in K is immaterial and indeed in
some implementations will appear in a different order: the wedge
package uses the spray package, which in turn utilises the STL
map class of C++.
Formal definition of wedge product
In the previous section we defined objects such as "$dx_1\wedge dx_6$"
as a single entity. Here I define the elementary form $dx_i$ and the
wedge product $\wedge$ formally. The elementary form $dx_i$ is simply
a map from $\mathbb{R}^n$ to $\mathbb{R}$ with
$dx_i\left(x_1,x_2,\ldots,x_n\right)=x_i$. The wedge product maps two
alternating forms to another alternating form; given
$\omega\in\Lambda^k(V)$ and $\eta\in\Lambda^l(V)$, Spivak defines the
wedge product $\omega\wedge\eta\in\Lambda^{k+l}(V)$ as
[
\omega\wedge\eta={k+l\choose k\,l}\operatorname{Alt}(\omega\otimes\eta)
]
and this is implemented in the package by function wedge(), or,
more idiomatically, %^%:
M1 <- matrix(c(3,4,5, 4,6,1),2,3,byrow=TRUE)
K1 <- as.kform(M1,c(2,7))
K1
M2 <- cbind(1:5,3:7)
K2 <- as.kform(M2,1:5)
K2
K1 %^% K2
(we might write the product as $-35dx_1\wedge dx_4\wedge dx_5\wedge
dx_6\wedge dx_7 -21dx_1\wedge dx_3\wedge dx_4\wedge dx_5\wedge dx_6$).
See how the wedge product eliminates rows with repeated entries,
gathers permuted rows together (respecting the sign of the
permutation), and expresses the result in terms of elementary forms.
The product is a linear combination of two elementary forms but only
two coefficients are nonzero out of a possible ${7\choose 5}=21$
terms. Note again that the order of the rows in the product is
arbitrary.
The wedge product has formal properties such as distributivity but by
far the most interesting one is associativity, which I will
demonstrate below:
F1 <- as.kform(matrix(c(3,4,5, 4,6,1,3,2,1),3,3,byrow=TRUE))
F2 <- as.kform(cbind(1:6,3:8),1:6)
F3 <- kform_general(1:8,2)
(F1 %^% F2) %^% F3
F1 %^% (F2 %^% F3)
Note carefully in the above that the terms in (F1 %^% F2) %^% F3
and F1 %^% (F2 %^% F3) appear in a different order. They are
nevertheless algebraically identical:
(F1 %^% F2) %^% F3 - F1 %^% (F2 %^% F3)
Spivak observes that $\Lambda^k(V)$ is spanned by
the $n\choose k$ wedge products of the form
[
dx_{i_1}\wedge dx_{i_2}\wedge\ldots\wedge dx_{i_k}\qquad
1\leq i_i<i_2<\cdots <i_k\leq n
]
where these products are the elementary forms: every element of the
space $\Lambda^k(V)$ is a linear combination of elementary forms, as
illustrated in the package by function kform_general().
Consider the following idiom:
Krel <- kform_general(4,2,1:6)
Krel
Object Krel is a two-form, specifically a map from
$\left(\mathbb{R}^4\right)^2$ to $\mathbb{R}$. Observe that
Krel has ${4\choose 2}=6$ components, which do not appear in any
particular order. Addition of such $k$-forms is straightforward in R
idiom but algebraically nontrivial:
K1 <- as.kform(matrix(1:4,2,2),1:2)
K2 <- as.kform(matrix(c(1,3,7,8,2,4),ncol=2,byrow=TRUE),c(-1,5,4))
K1
K2
K1+K2
In the above, note how the $dx_2\wedge dx_4$ terms combine [to give
2 4 = 6] and the $dx_1\wedge dx_3$ term vanishes.
Print methods
Although the spray form used above is probably the most direct and natural
representation of differential forms in numerical work, sometimes we need
a more algebraic print method.
U <- ktensor(spray(cbind(1:4,2:5),1:4))
U
we can represent this more algebraically using the as.symbolic() function:
as.symbolic(U)
In the above, U is a multilinear map from
$\left(\mathbb{R}^5\right)^2$ to $\mathbb{R}$. Symbolically, a
represents the map that takes $(a,b,c,d,e)$ to $a$, b the map that
takes $(a,b,c,d,e)$ to b, and so on. The asterisk * represents
the tensor product $\otimes$. Alternating forms work similarly but
$k$-forms have different defaults:
K <- kform_general(3,2,1:3)
K
as.symbolic(K,d="d",symbols=letters[23:26])
Note that the wedge product $\wedge$, although implemented in package
idiom as %^%, appears in the symbolic representation as an ascii
caret, ^.
Contractions
Given a $k$-form $\phi\colon V^k\longrightarrow\mathbb{R}$ and a
vector $\mathbf{v}\in V$, the contraction $\phi_\mathbf{v}$ of
$\phi$ and $\mathbf{v}$ is a $k-1$-form with
[
\phi_\mathbf{v}\left(\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) =
\phi\left(\mathbf{v},\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right)
]
if $k>1$; we specify $\phi_\mathbf{v}=\phi(\mathbf{v})$ if $k=1$.
Verification is straightforward:
(o <- rform()) # a random 3-form
V <- matrix(runif(21),ncol=3)
LHS <- as.function(o)(V)
RHS <- as.function(contract(o,V[,1]))(V[,-1])
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
It is possible to iterate the contraction process; if we pass a matrix
$V$ to contract() then this is interpreted as repeated contraction
with the columns of $V$:
as.function(contract(o,V[,1:2]))(V[,-(1:2),drop=FALSE])
If we pass three columns to contract() the result is a $0$-form:
contract(o,V)
In the above, the result is coerced to a scalar; in order to work with
a formal $0$-form (which is represented in the package as a spray
with a zero-column index matrix) we can use the lose=FALSE argument:
contract(o,V,lose=FALSE)
Exterior derivatives
Given a $k$-form $\omega$, Spivak defines the differential of $\omega$
to be a $(k+1)$-form $d\omega$ as follows. If
[
\omega =
\sum_{
i_1 < i_2 <\cdots<i_k}
\omega_{i_1i_2\ldots i_k}
dx^{i_1}\wedge
dx^{i_2}\wedge\cdots\wedge dx^{i_k}
]
then
[
d\omega =
\sum_{
i_1 < i_2 <\cdots<i_k}
\sum_{\alpha=1}^n D_\alpha\left(\omega_{i_1i_2\ldots i_k}\right)
\cdot
dx^{i_1}\wedge
dx^{i_2}\wedge\cdots\wedge dx^{i_k}
]
Hubbard and Hubbard define the exterior derivative $d\phi$ (they use a
bold font, $\mathbf{d}\phi$) of the $k$-form $\phi$ as the
$(k+1)$-form given by
[ {d}\phi
\left({v}i,\ldots,{v}{k+1}\right)
=
\lim_{h\longrightarrow 0}\frac{1}{h^{k+1}}\int_{\partial
P_{x}\left(h{v}1,\ldots,h{v}{k+1}\right)}\phi
]
which, by their own account, is a rather opaque mathematical idiom.
However, the definition makes sense and it is consistent with Spivak's
definition above. The definition allows one to express the
fundamental theorem of calculus in an arbitrary number of dimensions
without modification.
It can be shown that
[
{d}\left(f\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\right)=
{d}f\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}
]
where $f\colon\mathbb{R}^n\longrightarrow\mathbb{R}$ is a scalar
function of position. The package provides grad(), which when
given a vector $x_1,\ldots,x_n$ returns the one-form
[
\sum_{i=1}^n x_idx_i
]
This is useful because $df=\sum_{j=1}^n\left(D_j f\right)\,dx_j$. Thus
grad(c(0.4,0.1,-3.2,1.5))
We will use the grad() function to verify that, in
$\mathbb{R}^n$, a certain $(k-1)$-form has zero work function.
Motivated by the fact that
[
F_3=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}
\begin{pmatrix}x\y\z\end{pmatrix}
]
is a divergenceless velocity field in $\mathbb{R}^3$, H\&H go on to define
[
\omega_n=
d\frac{1}{\left(x_1^2+\ldots +x_n^2\right){^n/2}}\sum_{i=1}^{n}(-1)^{i-1}
x_idx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n
]
(where a hat indicates the absence of a term), and show analyically
that $d\omega=0$. Here I show this using R idiom. The first thing is
to define a function that implements the hat:
f <- function(x){
n <- length(x)
as.kform(t(apply(diag(n)<1,2,which)))
}
So, for example:
f(1:5)
Then we can use the grad() function to calculate $d\omega$,
using the quotient law to express the derivatives analytically:
df <- function(x){
n <- length(x)
S <- sum(x^2)
grad(rep(c(1,-1),length=n)*(S^(n/2) - n*x^2*S^(n/2-1))/S^n
)
}
Thus
df(1:5)
Now we can use the wedge product of the two parts to show that the exterior derivative is zero:
x <- rnorm(9)
print(df(x) %^% f(x)) # should be zero
Differential of the differential, $d^2=0$
We can use the package to verify the celebrated fact that, for any
$k$-form $\phi$, $d\left(d\phi\right)=0$. The first step is to define
scalar functions f1(), f2(), f3(), all $0$-forms:
f1 <- function(w,x,y,z){x + y^3 + x*y*w*z}
f2 <- function(w,x,y,z){w^2*x*y*z + sin(w) + w+z}
f3 <- function(w,x,y,z){w*x*y*z + sin(x) + cos(w)}
Now we need to define elementary $1$-forms:
dw <- as.kform(1)
dx <- as.kform(2)
dy <- as.kform(3)
dz <- as.kform(4)
I will demonstrate the theorem by defining a $2$-form which is the sum
of three elementary two-forms, evaluated at a particular point in
$\mathbb{R}^4$:
phi <-
(
+f1(1,2,3,4) %^% dw %^% dx
+f2(1,2,3,4) %^% dw %^% dy
+f3(1,2,3,4) %^% dy %^% dz
)
We can use slightly slicker R idiom by defining elementary forms
e1,e2,e3 and then defining phi to be a linear sum, weighted with
$0$-forms given by the (scalar) functions:
e1 <- dw %^% dx
e2 <- dw %^% dy
e3 <- dy %^% dz
phi <-
(
+f1(1,2,3,4) %^% e1
+f2(1,2,3,4) %^% e2
+f3(1,2,3,4) %^% e3
)
phi
Now to evaluate first derivatives of f1() etc at point
$(1,2,3,4)$, using Deriv() from the Deriv package:
library("Deriv")
Df1 <- Deriv(f1)(1,2,3,4)
Df2 <- Deriv(f2)(1,2,3,4)
Df3 <- Deriv(f3)(1,2,3,4)
So Df1 etc are numeric vectors of length 4, for example:
Df1
Calculating dphi, or $d\phi$, is easy:
dphi <-
(
+grad(Df1) %^% e1
+grad(Df2) %^% e2
+grad(Df3) %^% e3
)
Now work on the differential of the differential. First evaluate
the Hessians (4x4 numeric matrices) at the same point:
Hf1 <- matrix(Deriv(f1,nderiv=2)(1,2,3,4),4,4)
Hf2 <- matrix(Deriv(f2,nderiv=2)(1,2,3,4),4,4)
Hf3 <- matrix(Deriv(f3,nderiv=2)(1,2,3,4),4,4)
rownames(Hf1) <- c("w","x","y","z")
colnames(Hf1) <- c("w","x","y","z")
For example
Hf1
(note the nonzero diagonal term). But $dd\phi$ is clearly zero as the
Hessians are symmetrical:
ddphi <- # should be zero
(
+as.kform(which(!is.na(Hf1),arr.ind=TRUE),c(Hf1))
+as.kform(which(!is.na(Hf2),arr.ind=TRUE),c(Hf2))
+as.kform(which(!is.na(Hf3),arr.ind=TRUE),c(Hf3))
)
ddphi
as expected.
Stokes's theorem
In its most general form, Stokes's theorem states
[
\int_{\partial X}\phi=\int_Xd\phi
]
where $X\subset\mathbb{R}^n$ is a compact oriented $(k+1)$-dimensional
manifold with boundary $\partial X$ and $\phi$ is a $k$-form defined
on a neighborhood of $X$.
We will verify Stokes, following 6.9.5 of Hubbard in which
[
\phi=
\left(x_1-x_2^2+x_3^3-\cdots\pm x_n^n\right)
\left(
\sum_{i=1}^n
dx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n
\right)
]
(a hat indicates that a term is absent), and we wish to evaluate
$\int_{\partial C_a}\phi$ where $C_a$ is the cube $0\leq x_j\leq a,
1\leq j\leq n$. Stokes tells us that this is equal to $\int_{C_a}
d\phi$, which is given by
[
d\phi = \left(
1+2x_2+\cdots + nx_n^{n-1}\right)
dx_1\wedge\cdots\wedge dx_n
]
and so the volume integral is just
[
\sum_{j=1}^n
\int_{x_1=0}^a
\int_{x_2=0}^a
\cdots
\int_{x_i=0}^a
jx_j^{j-1}
dx_1 dx_2\ldots dx_n=
a^{n-1}\left(a+a^2+\cdots+a^n\right).
]
Stokes's theorem, being trivial, is not amenable to direct numerical
verification but the package does allow slick creation of $\phi$:
phi <- function(x){
n <- length(x)
sum(x^seq_len(n)*rep_len(c(1,-1),n)) * as.kform(t(apply(diag(n)<1,2,which)))
}
phi(1:9)
Further, $d\phi$ is given by
dphi <- function(x){
nn <- seq_along(x)
sum(nn*x^(nn-1)) * as.kform(seq_along(x))
}
dphi(1:9)
References
- J. H. Hubbard and B. B. Hubbard 2015. Vector
calculus, linear algebra and differential forms: a unified
aproach, Ithaca, NY.
- M. Spivak 1971. Calculus on manifolds, Addison-Wesley.
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The exterior calculus
Ordinary differential calculus may be generalized to arbitrary-dimensional oriented manifolds using the exterior calculus. Here I show how the wedge package furnishes functionality for working with the exterior calculus, and provide numerical verification of a number of theorems. Notation follows that of Spivak, and Hubbard and Hubbard.
Recall that a $k$-tensor is a multilinear map $S\colon V^k\longrightarrow\mathbb{R}$, where $V=\mathbb{R}^n$ is considered as a vector space; Spivak denotes the space of multilinear maps as $\mathcal{J}^k(V)$. Formally, multilinearity means
[ S\left(v_1,\ldots,av_i,\ldots,v_k\right)=a\cdot S\left(v_1,\ldots,v_i,\ldots,v_k\right) ]
and
[S\left(v_1,\ldots,v_i+{v_i}',\ldots,v_k\right)=S\left(v_1,\ldots,v_i,\ldots,x_v\right)+ S\left(v_1,\ldots,{v_i}',\ldots,v_k\right). ]
where $v_i\in V$. If $S\in\mathcal{J}^k(V)$ and $T\in\mathcal{J}^l(V)$, then we may define $S\otimes T\in\mathcal{J}^{k+l}(V)$ as
[ S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l}\right)= S\left(v_1,\ldots,v_k\right)\cdot T\left(v_1,\ldots,v_l\right) ]
Spivak observes that $\mathcal{J}^k(V)$ is spanned by the $n^k$ products of the form
[ \phi_{i_1}\otimes\phi_{i_2}\otimes\ldots\otimes\phi_{i_k}\qquad 1\leq i_i,i_2,\ldots,i_k\leq n ]
where $v_1,\ldots,v_k$ is a basis for $V$ and $\phi_i\left(v_j\right)=\delta_{ij}$; we can therefore write
[ S=\sum_{1\leq i_1,\ldots,i_k\leq n} a_{i_1\ldots i_k} \phi_{i_1}\otimes\ldots\otimes\phi_{i_k}. ]
The space spanned by such products has a natural representation in R
as an array of dimensions $n\times\cdots\times n$. If A is such
an array, then the element A[i_1,i_2,...,i_k] is the coefficient
of $\phi_{i_1}\otimes\ldots\otimes\phi_{i_k}$. However, it is more
efficient and conceptually cleaner to consider a sparse array, as
implemented by the spray package. We will consider the case
$n=5,k=4$, so we have multilinear maps from
$\left(\mathbb{R}^5\right)^4$ to $\mathbb{R}$. Below, we will test
algebraic identities in R using the idiom furnished by the wedge
package. For our example we will define
$S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2$
using a matrix with three rows, one per term, and whose rows
correspond to each terms's cross products of the $\phi$'s. We first
have to load the wedge library:
library("wedge")
Then the idiom is straightforward:
k <- 4 n <- 5 M <- matrix(c(5,1,1,1, 1,1,2,3, 1,3,4,2),3,4,byrow=TRUE) M S <- as.ktensor(M,coeffs= 0.5 + 1:3) S
Observe that, if stored as an array of size $n^k$, $S$ would have
$5^4=625$ elements, all but three of which are zero. So $S$ is a
4-tensor, mapping $V^4$ to $\mathbb{R}$, where $V=\mathbb{R}^5$. Here
we have
$S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2$.
Note that in some implementations the row order of object S will
differ from that of M. We will define $E$ to be a random point
in $V^k$ in terms of a matrix:
E <- matrix(rnorm(n*k),n,k) # A random point in V^k
Then
f <- as.function(S) f(E)
Testing multilinearity is straightforward in the package:
E1 <- E E2 <- E E3 <- E x1 <- rnorm(n) x2 <- rnorm(n) r1 <- rnorm(1) r2 <- rnorm(1) E1[,2] <- x1 E2[,2] <- x2 E3[,2] <- r1*x1 + r2*x2
Then we can verify the multilinearity of $S$ by coercing to a function
which is applied to E1, E2, E3:
f <- as.function(S) way1 <- r1*f(E1) + r2*f(E2) way2 <- f(E3) c(way1,way2)
(that is, identical up to numerical precision). Note that this is not equivalent to linearity over $V^{nk}$:
E1 <- matrix(rnorm(n*k),n,k) E2 <- matrix(rnorm(n*k),n,k) way1 <- f(r1*E1+r2*E2) way2 <- r1*f(E1)+r2*f(E2) c(way1,way2)
Given two k-tensor objects $S,T$ we can form the cross product $S\otimes T$, defined as
[ S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)= S\left(v_1,\ldots v_k\right)\cdot T\left(v_{k+1},\ldots v_{k+l}\right) ]
S1 <- ktensor(spray(cbind(1:3,2:4),1:3)) S2 <- as.ktensor(matrix(1:6,2,3)) S1 S2
The R idiom for $S1\otimes S2$ would be cross(), or %X%:
cross(S1,S2)
Then, for example:
as.function(cross(S1,S2))(matrix(rnorm(30),6,5))
Alternating forms
An alternating form is a multilinear map $T$ satisfying
[ \mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)= -\mathrm{T}\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right) ]
(or, equivalently, $\mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_i,\ldots,v_k\right)= 0$). We write $\Lambda^k(V)$ for the space of all alternating multilinear maps from $V^k$ to $\mathbb{R}$. Spivak gives $\operatorname{Alt}\colon\mathcal{J}^k(V)\longrightarrow\Lambda^k(V)$ defined by
[\operatorname{Alt}(T)\left(v_1,\ldots,v_k\right)= \frac{1}{k!}\sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\cdot T\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right) ]
where the sum ranges over all permutations of $\left[n\right]=\left{1,2,\ldots,n\right}$ and $\operatorname{sgn}(\sigma)\in\pm 1$ is the sign of the permutation. If $T\in\mathcal{J}^k(V)$ and $\omega\in\Lambda^k(V)$, it is straightforward to prove that $\operatorname{Alt}(T)\in\Lambda^k(V)$, $\operatorname{Alt}\left(\operatorname{Alt}\left(T\right)\right)=\operatorname{Alt}\left(T\right)$, and $\operatorname{Alt}\left(\omega\right)=\omega$.
In the wedge package, this is effected by the Alt() function:
S1
Alt(S1)
Verifying that S1 is in fact alternating is straightforward:
E <- matrix(rnorm(8),4,2) Erev <- E[,2:1] as.function(Alt(S1))(E) + as.function(Alt(S1))(Erev) # should be zero
However, we can see that this form for alternating tensors (here
called $k$-forms) is inefficient and highly redundant: in this example
there is a 1 2 term and a 2 1 term (the coefficients are
equal and opposite). In this example we have $k=2$ but in general
there would be potentially $k!$ essentially repeated terms which
collectively require only a single coefficient. The package provides
kform objects which are inherently alternating using a more
efficient representation; they are described using wedge products
which are discussed next.
Wedge products and the exterior calculus
This section follows the exposition of Hubbard and Hubbard, who introduce the exterior calculus starting with a discussion of elementary forms, which are alternating forms with a particularly simple structure. An elementary form is an alternating multilinear map with a particularly simple structure. An example of an elementary form would be $dx_1\wedge dx_3$ [treated as an indivisible entity], which is an alternating multilinear map from $\mathbb{R}^n\times\mathbb{R}^n$ to $\mathbb{R}$ with
[ \left( dx_1\wedge dx_3 \right)\left( \begin{pmatrix}a_1\a_2\a_3\ \vdots\ a_n\end{pmatrix}, \begin{pmatrix}b_1\b_3\b_3\ \vdots\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_1 & b_1 \ a_3 & b_3\end{pmatrix} =a_1b_3-a_3b_1 ]
That this is alternating follows from the properties of the determinant. In general of course, $dx_i\wedge dx_j\left( \begin{pmatrix}a_1\ \vdots\ a_n\end{pmatrix}, \begin{pmatrix}b_1\ \vdots\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_i & b_i \ a_j & b_j\end{pmatrix}$. Because such objects are linear, it is possible to consider sums of elementary forms, such as $dx_1\wedge dx_2 + 3 dx_2\wedge dx_3$ with
[ \left( dx_1\wedge dx_2 + 3dx_2\wedge dx_3 \right)\left( \begin{pmatrix}a_1\a_2\ \vdots\ a_n\end{pmatrix}, \begin{pmatrix}b_1\b_2\ \vdots\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_1 & b_1\ a_2 & b_2\end{pmatrix} +3\mathrm{det} \begin{pmatrix} a_2 & b_2\ a_3 & b_3\end{pmatrix} ]
or even $K=dx_1\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4$ which would be a linear map from $\left(\mathbb{R}^n\right)^3$ to $\mathbb{R}$ with
[ \left( dx_4\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4 \right)\left( \begin{pmatrix}a_1\a_2\ \vdots\ a_n\end{pmatrix}, \begin{pmatrix}b_1\b_2\ \vdots\ b_n\end{pmatrix}, \begin{pmatrix}c_1\c_2\ \vdots\ c_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_4 & b_4 & c_4\ a_2 & b_2 & c_2\ a_3 & b_3 & c_3 \end{pmatrix} +5\mathrm{det} \begin{pmatrix} a_1 & b_1 & c_1\ a_2 & b_2 & c_2\ a_4 & b_4 & c_4 \end{pmatrix}. ]
Defining $K$ has ready R idiom in which we define a matrix whose rows correspond to the differentials in each term:
M <- matrix(c(4,2,3,1,2,4),2,3,byrow=TRUE) M K <- as.kform(M,c(1,5)) K
Note that the order of the rows in K is immaterial and indeed in
some implementations will appear in a different order: the wedge
package uses the spray package, which in turn utilises the STL
map class of C++.
Formal definition of wedge product
In the previous section we defined objects such as "$dx_1\wedge dx_6$" as a single entity. Here I define the elementary form $dx_i$ and the wedge product $\wedge$ formally. The elementary form $dx_i$ is simply a map from $\mathbb{R}^n$ to $\mathbb{R}$ with $dx_i\left(x_1,x_2,\ldots,x_n\right)=x_i$. The wedge product maps two alternating forms to another alternating form; given $\omega\in\Lambda^k(V)$ and $\eta\in\Lambda^l(V)$, Spivak defines the wedge product $\omega\wedge\eta\in\Lambda^{k+l}(V)$ as
[ \omega\wedge\eta={k+l\choose k\,l}\operatorname{Alt}(\omega\otimes\eta) ]
and this is implemented in the package by function wedge(), or,
more idiomatically, %^%:
M1 <- matrix(c(3,4,5, 4,6,1),2,3,byrow=TRUE) K1 <- as.kform(M1,c(2,7)) K1 M2 <- cbind(1:5,3:7) K2 <- as.kform(M2,1:5) K2 K1 %^% K2
(we might write the product as $-35dx_1\wedge dx_4\wedge dx_5\wedge dx_6\wedge dx_7 -21dx_1\wedge dx_3\wedge dx_4\wedge dx_5\wedge dx_6$). See how the wedge product eliminates rows with repeated entries, gathers permuted rows together (respecting the sign of the permutation), and expresses the result in terms of elementary forms. The product is a linear combination of two elementary forms but only two coefficients are nonzero out of a possible ${7\choose 5}=21$ terms. Note again that the order of the rows in the product is arbitrary.
The wedge product has formal properties such as distributivity but by far the most interesting one is associativity, which I will demonstrate below:
F1 <- as.kform(matrix(c(3,4,5, 4,6,1,3,2,1),3,3,byrow=TRUE)) F2 <- as.kform(cbind(1:6,3:8),1:6) F3 <- kform_general(1:8,2) (F1 %^% F2) %^% F3 F1 %^% (F2 %^% F3)
Note carefully in the above that the terms in (F1 %^% F2) %^% F3
and F1 %^% (F2 %^% F3) appear in a different order. They are
nevertheless algebraically identical:
(F1 %^% F2) %^% F3 - F1 %^% (F2 %^% F3)
Spivak observes that $\Lambda^k(V)$ is spanned by the $n\choose k$ wedge products of the form
[ dx_{i_1}\wedge dx_{i_2}\wedge\ldots\wedge dx_{i_k}\qquad 1\leq i_i<i_2<\cdots <i_k\leq n ]
where these products are the elementary forms: every element of the
space $\Lambda^k(V)$ is a linear combination of elementary forms, as
illustrated in the package by function kform_general().
Consider the following idiom:
Krel <- kform_general(4,2,1:6) Krel
Object Krel is a two-form, specifically a map from
$\left(\mathbb{R}^4\right)^2$ to $\mathbb{R}$. Observe that
Krel has ${4\choose 2}=6$ components, which do not appear in any
particular order. Addition of such $k$-forms is straightforward in R
idiom but algebraically nontrivial:
K1 <- as.kform(matrix(1:4,2,2),1:2) K2 <- as.kform(matrix(c(1,3,7,8,2,4),ncol=2,byrow=TRUE),c(-1,5,4)) K1 K2 K1+K2
In the above, note how the $dx_2\wedge dx_4$ terms combine [to give
2 4 = 6] and the $dx_1\wedge dx_3$ term vanishes.
Print methods
Although the spray form used above is probably the most direct and natural representation of differential forms in numerical work, sometimes we need a more algebraic print method.
U <- ktensor(spray(cbind(1:4,2:5),1:4)) U
we can represent this more algebraically using the as.symbolic() function:
as.symbolic(U)
In the above, U is a multilinear map from
$\left(\mathbb{R}^5\right)^2$ to $\mathbb{R}$. Symbolically, a
represents the map that takes $(a,b,c,d,e)$ to $a$, b the map that
takes $(a,b,c,d,e)$ to b, and so on. The asterisk * represents
the tensor product $\otimes$. Alternating forms work similarly but
$k$-forms have different defaults:
K <- kform_general(3,2,1:3) K as.symbolic(K,d="d",symbols=letters[23:26])
Note that the wedge product $\wedge$, although implemented in package
idiom as %^%, appears in the symbolic representation as an ascii
caret, ^.
Contractions
Given a $k$-form $\phi\colon V^k\longrightarrow\mathbb{R}$ and a vector $\mathbf{v}\in V$, the contraction $\phi_\mathbf{v}$ of $\phi$ and $\mathbf{v}$ is a $k-1$-form with
[ \phi_\mathbf{v}\left(\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) = \phi\left(\mathbf{v},\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) ]
if $k>1$; we specify $\phi_\mathbf{v}=\phi(\mathbf{v})$ if $k=1$. Verification is straightforward:
(o <- rform()) # a random 3-form V <- matrix(runif(21),ncol=3) LHS <- as.function(o)(V) RHS <- as.function(contract(o,V[,1]))(V[,-1]) c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
It is possible to iterate the contraction process; if we pass a matrix
$V$ to contract() then this is interpreted as repeated contraction
with the columns of $V$:
as.function(contract(o,V[,1:2]))(V[,-(1:2),drop=FALSE])
If we pass three columns to contract() the result is a $0$-form:
contract(o,V)
In the above, the result is coerced to a scalar; in order to work with
a formal $0$-form (which is represented in the package as a spray
with a zero-column index matrix) we can use the lose=FALSE argument:
contract(o,V,lose=FALSE)
Exterior derivatives
Given a $k$-form $\omega$, Spivak defines the differential of $\omega$ to be a $(k+1)$-form $d\omega$ as follows. If
[ \omega = \sum_{ i_1 < i_2 <\cdots<i_k} \omega_{i_1i_2\ldots i_k} dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} ]
then
[ d\omega = \sum_{ i_1 < i_2 <\cdots<i_k} \sum_{\alpha=1}^n D_\alpha\left(\omega_{i_1i_2\ldots i_k}\right) \cdot dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} ]
Hubbard and Hubbard define the exterior derivative $d\phi$ (they use a bold font, $\mathbf{d}\phi$) of the $k$-form $\phi$ as the $(k+1)$-form given by
[ {d}\phi \left({v}i,\ldots,{v}{k+1}\right) = \lim_{h\longrightarrow 0}\frac{1}{h^{k+1}}\int_{\partial P_{x}\left(h{v}1,\ldots,h{v}{k+1}\right)}\phi ]
which, by their own account, is a rather opaque mathematical idiom. However, the definition makes sense and it is consistent with Spivak's definition above. The definition allows one to express the fundamental theorem of calculus in an arbitrary number of dimensions without modification.
It can be shown that
[ {d}\left(f\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\right)= {d}f\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k} ]
where $f\colon\mathbb{R}^n\longrightarrow\mathbb{R}$ is a scalar
function of position. The package provides grad(), which when
given a vector $x_1,\ldots,x_n$ returns the one-form
[ \sum_{i=1}^n x_idx_i ]
This is useful because $df=\sum_{j=1}^n\left(D_j f\right)\,dx_j$. Thus
grad(c(0.4,0.1,-3.2,1.5))
We will use the grad() function to verify that, in
$\mathbb{R}^n$, a certain $(k-1)$-form has zero work function.
Motivated by the fact that
[ F_3=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}} \begin{pmatrix}x\y\z\end{pmatrix} ]
is a divergenceless velocity field in $\mathbb{R}^3$, H\&H go on to define
[ \omega_n= d\frac{1}{\left(x_1^2+\ldots +x_n^2\right){^n/2}}\sum_{i=1}^{n}(-1)^{i-1} x_idx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n ]
(where a hat indicates the absence of a term), and show analyically that $d\omega=0$. Here I show this using R idiom. The first thing is to define a function that implements the hat:
f <- function(x){ n <- length(x) as.kform(t(apply(diag(n)<1,2,which))) }
So, for example:
f(1:5)
Then we can use the grad() function to calculate $d\omega$,
using the quotient law to express the derivatives analytically:
df <- function(x){ n <- length(x) S <- sum(x^2) grad(rep(c(1,-1),length=n)*(S^(n/2) - n*x^2*S^(n/2-1))/S^n ) }
Thus
df(1:5)
Now we can use the wedge product of the two parts to show that the exterior derivative is zero:
x <- rnorm(9) print(df(x) %^% f(x)) # should be zero
Differential of the differential, $d^2=0$
We can use the package to verify the celebrated fact that, for any
$k$-form $\phi$, $d\left(d\phi\right)=0$. The first step is to define
scalar functions f1(), f2(), f3(), all $0$-forms:
f1 <- function(w,x,y,z){x + y^3 + x*y*w*z} f2 <- function(w,x,y,z){w^2*x*y*z + sin(w) + w+z} f3 <- function(w,x,y,z){w*x*y*z + sin(x) + cos(w)}
Now we need to define elementary $1$-forms:
dw <- as.kform(1) dx <- as.kform(2) dy <- as.kform(3) dz <- as.kform(4)
I will demonstrate the theorem by defining a $2$-form which is the sum of three elementary two-forms, evaluated at a particular point in $\mathbb{R}^4$:
phi <- ( +f1(1,2,3,4) %^% dw %^% dx +f2(1,2,3,4) %^% dw %^% dy +f3(1,2,3,4) %^% dy %^% dz )
We can use slightly slicker R idiom by defining elementary forms
e1,e2,e3 and then defining phi to be a linear sum, weighted with
$0$-forms given by the (scalar) functions:
e1 <- dw %^% dx e2 <- dw %^% dy e3 <- dy %^% dz phi <- ( +f1(1,2,3,4) %^% e1 +f2(1,2,3,4) %^% e2 +f3(1,2,3,4) %^% e3 ) phi
Now to evaluate first derivatives of f1() etc at point
$(1,2,3,4)$, using Deriv() from the Deriv package:
library("Deriv") Df1 <- Deriv(f1)(1,2,3,4) Df2 <- Deriv(f2)(1,2,3,4) Df3 <- Deriv(f3)(1,2,3,4)
So Df1 etc are numeric vectors of length 4, for example:
Df1
Calculating dphi, or $d\phi$, is easy:
dphi <- ( +grad(Df1) %^% e1 +grad(Df2) %^% e2 +grad(Df3) %^% e3 )
Now work on the differential of the differential. First evaluate the Hessians (4x4 numeric matrices) at the same point:
Hf1 <- matrix(Deriv(f1,nderiv=2)(1,2,3,4),4,4) Hf2 <- matrix(Deriv(f2,nderiv=2)(1,2,3,4),4,4) Hf3 <- matrix(Deriv(f3,nderiv=2)(1,2,3,4),4,4)
rownames(Hf1) <- c("w","x","y","z") colnames(Hf1) <- c("w","x","y","z")
For example
Hf1
(note the nonzero diagonal term). But $dd\phi$ is clearly zero as the Hessians are symmetrical:
ddphi <- # should be zero ( +as.kform(which(!is.na(Hf1),arr.ind=TRUE),c(Hf1)) +as.kform(which(!is.na(Hf2),arr.ind=TRUE),c(Hf2)) +as.kform(which(!is.na(Hf3),arr.ind=TRUE),c(Hf3)) ) ddphi
as expected.
Stokes's theorem
In its most general form, Stokes's theorem states
[ \int_{\partial X}\phi=\int_Xd\phi ]
where $X\subset\mathbb{R}^n$ is a compact oriented $(k+1)$-dimensional manifold with boundary $\partial X$ and $\phi$ is a $k$-form defined on a neighborhood of $X$.
We will verify Stokes, following 6.9.5 of Hubbard in which
[ \phi= \left(x_1-x_2^2+x_3^3-\cdots\pm x_n^n\right) \left( \sum_{i=1}^n dx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n \right) ]
(a hat indicates that a term is absent), and we wish to evaluate $\int_{\partial C_a}\phi$ where $C_a$ is the cube $0\leq x_j\leq a, 1\leq j\leq n$. Stokes tells us that this is equal to $\int_{C_a} d\phi$, which is given by
[ d\phi = \left( 1+2x_2+\cdots + nx_n^{n-1}\right) dx_1\wedge\cdots\wedge dx_n ]
and so the volume integral is just
[ \sum_{j=1}^n \int_{x_1=0}^a \int_{x_2=0}^a \cdots \int_{x_i=0}^a jx_j^{j-1} dx_1 dx_2\ldots dx_n= a^{n-1}\left(a+a^2+\cdots+a^n\right). ]
Stokes's theorem, being trivial, is not amenable to direct numerical verification but the package does allow slick creation of $\phi$:
phi <- function(x){ n <- length(x) sum(x^seq_len(n)*rep_len(c(1,-1),n)) * as.kform(t(apply(diag(n)<1,2,which))) } phi(1:9)
Further, $d\phi$ is given by
dphi <- function(x){ nn <- seq_along(x) sum(nn*x^(nn-1)) * as.kform(seq_along(x)) } dphi(1:9)
References
- J. H. Hubbard and B. B. Hubbard 2015. Vector calculus, linear algebra and differential forms: a unified aproach, Ithaca, NY.
- M. Spivak 1971. Calculus on manifolds, Addison-Wesley.
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