---
title: "Exterior calculus with R"
author: "Robin K. S. Hankin"
output: html_vignette
bibliography: stokes.bib
vignette: >
  %\VignetteIndexEntry{The exterior calculus}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# The `stokes` package

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
options(rmarkdown.html_vignette.check_title = FALSE)
library("stokes")
set.seed(1)
```

```{r out.width='20%', out.extra='style="float:right; padding:10px"',echo=FALSE}
knitr::include_graphics(system.file("help/figures/stokes.png", package = "stokes"))
```

To cite the `stokes` package in publications please use
[@hankin2022_stokes].  Ordinary differential calculus may be
formalized and generalized to arbitrary-dimensional oriented manifolds
using the exterior calculus.  Here I show how the `stokes` package
furnishes functionality for working with the exterior calculus, and
provide numerical verification of a number of theorems.  Notation
follows that of @spivak1965, and @hubbard2015.

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

\[
\phi_{i_1}\otimes\phi_{i_2}\otimes\cdots\otimes\phi_{i_k}\qquad
1\leq i_i,i_2,\ldots,i_k\leq n
\]

where $v_1,\ldots,v_k$ is a basis for $V$ and
$\phi_i{\left(v_j\right)}=\delta_{ij}$; we can therefore write

\[
S=\sum_{1\leq i_1,\ldots,i_k\leq n} a_{i_1\ldots i_k}
\phi_{i_1}\otimes\cdots\otimes\phi_{i_k}.
\]

The space spanned by such products has a natural representation in R
as an array of dimensions $n\times\cdots\times n=n^k$.  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 stokes
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 term's tensor products of the $\phi$'s.  We first
have to load the `stokes` package:

```{r,label=loadstokeslibrary,message=FALSE}
library("stokes")
```

Then the idiom is straightforward:


```{r label=straightforwardidiom}
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`; this phenomenon is due to the underlying `C`
implementation using the `STL map` class; see the `disordR` package
[@hankin2022_disordR] and is discussed in more detail in the `mvp`
package [@hankin2022_mvp].

### Package idiom for evaluation of a tensor

First, we will define $E$ to be a random point in $V^k$ in terms of a
matrix:

```{r setupsetseed}
set.seed(0)
(E <- matrix(rnorm(n*k),n,k))   # A random point in V^k
```

Recall that $n=5$, $k=4$, so $E\in\left(\mathbb{R}^5\right)^4$.  We
can evaluate $S$ at $E$ as follows:

```{r showfandfE}
f <- as.function(S)
f(E)
```

### Vector space structure of tensors

Tensors have a natural vector space structure; they may be added and
subtracted, and multiplied by a scalar, the same as any other vector
space.  Below, we define a new tensor $S_1$ and work with $2S-3S_1$:

```{r simpletensorarith}
S1 <- as.ktensor(1+diag(4),1:4)
2*S-3*S1
```

We may verify that tensors are linear using package idiom:

```{r verifylinearity}
LHS <- as.function(2*S-3*S1)(E)
RHS <- 2*as.function(S)(E) -3*as.function(S1)(E)
c(lhs=LHS,rhs=RHS,diff=LHS-RHS)
```

(that is, identical up to numerical precision).

### Numerical verification of multilinearity in the package

Testing multilinearity is straightforward in the package.  To do this,
we need to define three matrices `E1,E2,E3` corresponding to
points in $\left(\mathbb{R}^5\right)^4$ which are identical except for
one column.  In `E3`, this column is a linear combination of the
corresponding column in `E2` and `E3`:

```{r E1E2E3columns}

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`:

```{r asfuncS}
f <- as.function(S)
LHS <- r1*f(E1) + r2*f(E2)
RHS <- f(E3)
c(lhs=LHS,rhs=RHS,diff=LHS-RHS)
```

(that is, identical up to numerical precision).  Note that this is
*not* equivalent to linearity over $V^{nk}$:


```{r E1E2matrix}
E1 <- matrix(rnorm(n*k),n,k)
E2 <- matrix(rnorm(n*k),n,k)
LHS <- f(r1*E1+r2*E2)
RHS <- r1*f(E1)+r2*f(E2)
c(lhs=LHS,rhs=RHS,diff=LHS-RHS)
```

### Tensor product of general tensors

Given two k-tensor objects $S,T$ we can form the tensor product
$S\otimes T$, defined as

\[
S\otimes T{\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)}=
  S{\left(v_1,\ldots v_k\right)} \cdot T{\left(v_{k+1},\ldots
  v_{k+l}\right)}
\]

We will calculate the tensor product of two tensors `S1,S2` defined as follows:

```{r S1S2tensors}
(S1 <- ktensor(spray(cbind(1:3,2:4),1:3)))
(S2 <- as.ktensor(matrix(1:6,2,3)))
```

The R idiom for $S1\otimes S2$ would be `tensorprod()`, or `%X%`:


```{r S1S2tensorprod}
tensorprod(S1,S2)
```

Then, for example:

```{r tensorS1S2verify}
E <- matrix(rnorm(30),6,5)
LHS <- as.function(tensorprod(S1,S2))(E)
RHS <- as.function(S1)(E[,1:2]) * as.function(S2)(E[,3:5])
c(lhs=LHS,rhs=RHS,diff=LHS-RHS)
```

(that is, identical up to numerical precision).

# Alternating forms

An alternating form is a multilinear map $T$ satisfying

\[
T{\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)}=
    -T{\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right)}
\]

(or, equivalently,
$T{\left(v_1,\ldots,v_i,\ldots,v_i,\ldots,v_k\right)}= 0$).  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 stokes package, this is effected by the `Alt()` function:


```{r showAltS1}
S1
Alt(S1)
```

Verifying that `S1` is in fact alternating is straightforward:

```{r verifyAltisAlt}
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 example of an elementary form would be
$\mathrm{d}x_1\wedge\mathrm{d}x_3$ [treated as an indivisible entity],
which is an alternating multilinear map from
$\mathbb{R}^n\times\mathbb{R}^n$ to $\mathbb{R}$ with

\[
\left(
\mathrm{d}x_1\wedge\mathrm{d}x_3
\right)\left(
\begin{pmatrix}a_1\\a_2\\a_3\\ \vdots\\ a_n\end{pmatrix},
\begin{pmatrix}b_1\\b_3\\b_3\\ \vdots\\ b_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix} a_1 & b_1 \\ a_3 & b_3\end{pmatrix}
=a_1b_3-a_3b_1
\]

That this is alternating follows from the properties of the
determinant.  In general of course,
$\mathrm{d}x_i\wedge\mathrm{d}x_j\left( \begin{pmatrix}a_1\\ \vdots\\
a_n\end{pmatrix}, \begin{pmatrix}b_1\\ \vdots\\ b_n\end{pmatrix}
\right)=\mathrm{det} \begin{pmatrix} a_i & b_i \\ a_j &
b_j\end{pmatrix}$.  Because such objects are linear, it is possible to
consider sums of elementary forms, such as $d\mathrm{x}_1\wedge\mathrm{d}x_2 + 3
\mathrm{d}x_2\wedge\mathrm{d}x_3$ with

\[
\left(
\mathrm{d}x_1\wedge\mathrm{d}x_2 + 3\mathrm{d}x_2\wedge\mathrm{d}x_3
\right)\left(
\begin{pmatrix}a_1\\a_2\\ \vdots\\ a_n\end{pmatrix},
\begin{pmatrix}b_1\\b_2\\ \vdots\\ b_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix} a_1 & b_1\\ a_2 & b_2\end{pmatrix}
+3\mathrm{det}
\begin{pmatrix} a_2 & b_2\\ a_3 & b_3\end{pmatrix}
\]

or even $K=\mathrm{d}x_1\wedge\mathrm{d}x_2\wedge\mathrm{d}x_3 +5\mathrm{d}x_1\wedge\mathrm{d}x_2\wedge\mathrm{d}x_4$
which would be a linear map from $\left(\mathbb{R}^n\right)^3$ to
$\mathbb{R}$ with


\[
\left(
\mathrm{d}x_4\wedge\mathrm{d}x_2\wedge\mathrm{d}x_3 +5\mathrm{d}x_1\wedge\mathrm{d}x_2\wedge\mathrm{d}x_4
\right)\left(
\begin{pmatrix}a_1\\a_2\\ \vdots\\ a_n\end{pmatrix},
\begin{pmatrix}b_1\\b_2\\ \vdots\\ b_n\end{pmatrix},
\begin{pmatrix}c_1\\c_2\\ \vdots\\ c_n\end{pmatrix}
\right)=\mathrm{det}
\begin{pmatrix}
	a_4 & b_4 & c_4\\
	a_2 & b_2 & c_2\\
	a_3 & b_3 & c_3
\end{pmatrix}
+5\mathrm{det}
\begin{pmatrix}
	a_1 & b_1 & c_1\\
	a_2 & b_2 & c_2\\
	a_4 & b_4 & c_4
\end{pmatrix}.
\]


Defining $K$ has ready R idiom in which we define a matrix whose rows
correspond to the differentials in each term:

```{r usedifferentials}
M <- matrix(c(4,2,3,1,4,2),2,3,byrow=TRUE)
M
K <- as.kform(M,c(1,5))
K
```

Function `as.kform()` takes each row of `M` and places the elements in
increasing order; the coefficient will change sign if the permutation
is odd.  Note that the order of the rows in `K` is immaterial and
indeed in some implementations will appear in a different order: the
stokes package uses the `spray` package, which in turn utilises the
STL map class of C++.

## Formal definition of dx

In the previous section we defined objects such as "$\mathrm{d}x_1\wedge\mathrm{d}x_6$"
as a single entity.  Here I define the elementary form $\mathrm{d}x_i$ formally
and in the next section discuss the wedge product $\wedge$.  The
elementary form $\mathrm{d}x_i$ is simply a map from $\mathbb{R}^n$ to
$\mathbb{R}$ with $\mathrm{d}x_i{\left(x_1,x_2,\ldots,x_n\right)}=x_i$.  Observe
that $\mathrm{d}x_i$ is an alternating form, even though we cannot swap
arguments (because there is only one).  Package idiom for creating an
elementary form appears somewhat cryptic at first sight, but is
consistent (it is easier to understand package idiom for creating more
complicated alternating forms, as in the next section).  Suppose we
wish to work with $\mathrm{d}x_3$:

```{r dx3inmatrixform}
dx3 <- as.kform(matrix(3,1,1),1)
options(kform_symbolic_print = NULL) # revert to default print method
dx3
```

Interpretation of the output above is not obvious (it is easier to
understand the output from more complicated alternating forms, as in
the next section), but for the moment observe that $\mathrm{d}x_3$ is indeed an
alternating form, mapping $\mathbb{R}^n$ to $\mathbb{R}$ with
$\mathrm{d}x_3{\left(x_1,x_2,\ldots,x_n\right)}=x_3$.  Thus, for example:


```{r showdx3beingused}
as.function(dx3)(c(14,15,16))
as.function(dx3)(c(14,15,16,17,18))  # idiom can deal with arbitrary vectors
```

and we see that $\mathrm{d}x_3$ picks out the third element of a vector.  These
are linear in the sense that we may add and subtract these elementary
forms:

```{r dx5askform}
dx5 <- as.kform(matrix(5,1,1),1)
as.function(dx3 + 2*dx5)(1:10)  # picks out element 3 + 2*element 5
```

## Formal definition of wedge product


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

\[
\omega\wedge\eta={k+l\choose k\quad l}\operatorname{Alt}(\omega\otimes\eta)
\]

and this is implemented in the package by function `wedge()`, or,
more idiomatically, `^`:

```{r M1K1M2K2}
M1 <- matrix(c(3,5,4, 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
```

In symbolic notation, `K1` is equal to $7\mathrm{d}x_1\wedge\mathrm{d}x_4\wedge\mathrm{d}x_6
-2\mathrm{d}x_3\wedge\mathrm{d}x_4\wedge\mathrm{d}x_5$. and `K2` is $\mathrm{d}x_1\wedge\mathrm{d}x_3+
2\mathrm{d}x_2\wedge\mathrm{d}x_4+ 3\mathrm{d}x_3\wedge\mathrm{d}x_5+ 4\mathrm{d}x_4\wedge\mathrm{d}x_6+ 5\mathrm{d}x_5\wedge
\mathrm{d}x_7$.  Package idiom for wedge products is straightforward:

```{r K1wedgeK2}
K1 ^ K2
```

(we might write the product as $-35\mathrm{d}x_1\wedge\mathrm{d}x_4\wedge\mathrm{d}x_5\wedge
\mathrm{d}x_6\wedge\mathrm{d}x_7 -21\mathrm{d}x_1\wedge\mathrm{d}x_3\wedge\mathrm{d}x_4\wedge\mathrm{d}x_5\wedge\mathrm{d}x_6$).
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; note that
only two coefficients out of a possible ${7\choose 5}=21$ are nonzero.
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:

```{r F1F2F3kforms}
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, as we may demonstrate by
calculating their difference:


```{r wedgearithmeticF1F2F3}
(F1 ^ F2) ^ F3 - F1 ^ (F2 ^ F3)
```

Spivak observes that $\Lambda^k(V)$ is spanned by
the $n\choose k$ wedge products of the form

\[
\mathrm{d}x_{i_1}\wedge\mathrm{d}x_{i_2}\wedge\ldots\wedge\mathrm{d}x_{i_k}\qquad
1\leq i_i<i_2<\cdots <i_k\leq n
\]

where these products are the elementary forms (compare
$\mathcal{J}^k(V)$, which is spanned by $n^k$ elementary forms).
Formally, multilinearity means 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:

```{r kformgeneralandrel}
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:

```{r addkformsnontrivial}
K1 <- as.kform(matrix(1:4,2,2),c(1,109))
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 $\mathrm{d}x_2\wedge\mathrm{d}x_4$ terms combine [to give `2
4 = 113`] and the $\mathrm{d}x_1\wedge\mathrm{d}x_3$ term vanishes by cancellation.

## 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.

```{r definetensorU}
U <- ktensor(spray(cbind(1:4,2:5),1:4))
U
```

we can represent this more algebraically using the `as.symbolic()` function:

```{r coerceUsymbolic}
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:

```{r showassymbolicK}
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 `^` or `%^%`, appears in the symbolic representation as an
ascii caret, `^`.

We can alter the default print method with the `kform_symbolic_print`
option, which uses `as.symbolic()`:

```{r useprintmethodsymbolic}
options(kform_symbolic_print = "d")
K
```

This print option works nicely with the `d()` function for elementary
forms:

```{r printshowofelementaryforms}
(d(1) + d(5)) ^ (d(3)-5*d(2)) ^ d(7)
options(kform_symbolic_print = NULL) # restore default
```



## 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:

```{r lookatfunc}
(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 $V$:

```{r coerceotoafunction}
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:

```{r contractovlosetrue}
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:

```{r contractovlosefalse}
contract(o,V,lose=FALSE)
```

## Transformations and pullback

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

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

Function `pullback()` function does all this:

```{r usetransform}
options(kform_symbolic_print = "dx")   # uses dx etc in print method
pullback(dx^dy+5*dx^dz, matrix(1:9,3,3))
options(kform_symbolic_print = NULL) # revert to default
```

However, it is slow and 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`.  Here I show that transformations may be inverted
using matrix inverses:

```{r defineoandM}
(o <- 2 * as.kform(2) ^ as.kform(4) ^ as.kform(5))
M <- matrix(rnorm(25),5,5)
```

Then we will transform according to matrix `M` and then transform
according to the matrix inverse; the functionality works nicely with
magrittr pipes:

```{r transformbackandforward}
o |> pullback(M) |> pullback(solve(M))
```

Above we see many rows with values small enough for the print method
to print an exact zero, but not sufficiently small to be eliminated by
the `spray` internals.  We can remove the small entries with `zap()`:

```{r zapsmallroundofferrors}
o |> pullback(M) |> pullback(solve(M)) |> zap()
```

See how the result is equal to the original $k$-form $2dy_2\wedge
dy_4\wedge dy_5$.

## Exterior derivatives

Given a $k$-form $\omega$, Spivak defines the differential of $\omega$
to be a $(k+1)$-form $\mathrm{d}\omega$ as follows.  If

\[
\omega =
\sum_{
i_1 < i_2 <\cdots<i_k}
\omega_{i_1i_2\ldots i_k}
\mathrm{d}x^{i_1}\wedge
\mathrm{d}x^{i_2}\wedge\cdots\wedge\mathrm{d}x^{i_k}
\]

then

\[
\mathrm{d}\omega =
\sum_{
i_1 < i_2 <\cdots<i_k}
\sum_{\alpha=1}^n D_\alpha\left(\omega_{i_1i_2\ldots i_k}\right)
\cdot
\mathrm{d}x^{i_1}\wedge
\mathrm{d}x^{i_2}\wedge\cdots\wedge\mathrm{d}x^{i_k}
\]


where $D_if(a)=\lim_{h\longrightarrow
0}\frac{f(a^1,\ldots,a^i+h,\ldots,a^n)-f(a^1,\ldots,a^i,\ldots,a^n)}{h}$
is the ordinary $i^\mathrm{th}$ partial derivative (Spivak, p25).
Hubbard and Hubbard take a conceptually distinct approach and 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

\[
    \mathrm{d}{\left(f\,dx_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}\right)}=
    \mathrm{d}f\wedge\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}
\]

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 $\mathrm{d}f=\sum_{j=1}^n\left(D_j
f\right)\,\mathrm{d}x_j$. Thus

```{r showgradinuse}
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 [page 548, equation 6.7.16]

\[
\omega_{n}=\mathrm{d}\frac{1}{\left(x_1^2+\ldots +x_n^2\right)^{n/2}}\sum_{i=1}^{n}(-1)^{i-1}
x_i\mathrm{d}x_1\wedge\cdots\wedge\widehat{\mathrm{d}x_i}\wedge\cdots\wedge\mathrm{d}x_n
\]

(where a hat indicates the absence of a term), and show analytically
that $\mathrm{d}\omega=0$.  Here I show this using R idiom.  The first
thing is to define a function that implements the hat:


```{r definefofakform}
f <- function(x){
    n <- length(x)
    as.kform(t(apply(diag(n)<1,2,which)))
}
```

So, for example:

```{r exampleuseoff}
f(1:5)
```

Then we can use the `grad()` function to calculate $\mathrm{d}\omega$,
using the quotient law to express the derivatives analytically:

```{r definedfanduseit}
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
```{r dfevaluatedatonetofive}
df(1:5)
```

Now we can use the wedge product of the two parts to show that the
exterior derivative is zero:

```{r dfevaluatedatrandompoint}
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 $\phi$, $\mathrm{d}\left(\mathrm{d}\phi\right)=0$.  The first
step is to define scalar functions `f1(), f2(), f3()`, all $0$-forms:

```{r definef1f2f3usingarith}
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:

```{r definedwdxdydzofkforms}
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
$\mathbb{R}^4$:

```{r definephi}
phi <-
  (
    +f1(1,2,3,4) ^ dw ^ dx
    +f2(1,2,3,4) ^ dw ^ dy
    +f3(1,2,3,4) ^ dy ^ dz
  )
```

We could 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 `f1,f2,f3`:

```{r e1e2e3usingwedge}
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:

```{r demonstrateDerivpackage}
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:

```{r showDf1fromderiv}
Df1
```

To calculate `dphi`, or $\mathrm{d}\phi$, we can use function `grad()`:

```{r calculatedphi}
dphi <-
  (
    +grad(Df1) ^ e1
    +grad(Df2) ^ e2
    +grad(Df3) ^ e3
  )
dphi
```

Now work on the differential of the differential.  First evaluate
the Hessians (4x4 numeric matrices) at the same point:

```{r diffofdiff}
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)
```

```{r setrownames, echo=FALSE}
rownames(Hf1) <- c("w","x","y","z")
colnames(Hf1) <- c("w","x","y","z")
```

For example

```{r Hf1showexample}
Hf1
```

(note the matrix is symmetric; also note carefully the nonzero
diagonal term).  But $dd\phi$ is clearly zero as the Hessians are
symmetrical:

```{r ddphishouldbezero}
ij <- expand.grid(seq_len(nrow(Hf1)),seq_len(ncol(Hf1)))

ddphi <- # should be zero
  (
    +as.kform(ij,c(Hf1))
    +as.kform(ij,c(Hf2))
    +as.kform(ij,c(Hf3))
  )

ddphi
```

as expected.

## Stokes's theorem

In its most general form, Stokes's theorem states

\[
\int_{\partial X}\phi=\int_X\mathrm{d}\phi
\]

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

\[
\phi=
\left(x_1-x_2^2+x_3^3-\cdots\pm x_n^n\right)
\left(
\sum_{i=1}^n
\mathrm{d}x_1\wedge\cdots\wedge\widehat{\mathrm{d}x_i}\wedge\cdots\wedge\mathrm{d}x_n
\right)
\]

(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
$\int_{C_a}\mathrm{d}\phi$, which is given by

\[
d\phi = \left(
1+2x_2+\cdots + nx_n^{n-1}\right)
\mathrm{d}x_1\wedge\cdots\wedge\mathrm{d}x_n
\]

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 $\phi$:

```{r slickcreationofphi}
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)
```

(recall that `phi` is a function that maps $\mathbb{R}^9$ to 8-forms.
Here we choose $\left(1,2,\ldots,9\right)\in\mathbb{R}^9$ and
`phi(1:9)` as shown above is the resulting 8-form.  Thus, if we write
$\phi_{1:9}$ for `phi(1:9)` we would have
$\phi_{1:9}\colon\left(\mathbb{R}^9\right)^8\longrightarrow\mathbb{R}$,
with package idiom as follows:

```{r useslickdefforf}
f <- as.function(phi(1:9))
E <- matrix(runif(72),9,8)   # (R^9)^8
f(E)
```

Further, $\mathrm{d}\phi$ is given by

```{r dphiusingpowerdef}
dphi <- function(x){
    nn <- seq_along(x)
    sum(nn*x^(nn-1)) * as.kform(seq_along(x))
}
dphi(1:9)
```

(observe that `dphi(1:9)` is a 9-form, with
$\mathrm{d}\phi_{1:9}\colon\left(\mathbb{R}^9\right)^9\longrightarrow\mathbb{R}$).
Now consider Spivak's theorem 4.6 (page 82), which in this context
states that a 9-form is proportional to the determinant of the
$9\times 9$ matrix formed from its arguments, with constant of
proportionality equal to the form evaluated on the identity matrix
$I_9$ [formally and more generally, if $v_1,\ldots,v_n$ is a basis for
$V$, $\omega\in\Lambda^n(V)$ and $w_i=\sum a_{ij}v_j$ then
$\omega\left(w_1,\ldots,w_n\right) =
\det\left(a_{ij}\right)\cdot\omega\left(v_1,\ldots v_n\right)$].
Numerically:


```{r verifyfEbothways}
f <- as.function(dphi(1:9))
E <- matrix(runif(81),9,9)
f(E)
det(E)*f(diag(9))  # should match f(E) by Spivak's 4.6
```

```{r,tidyup,include=FALSE}
i <- 0
j <- 0
k <- 0
rm(i)
rm(j)
rm(k)
```

## References
