`dx`

, `dy`

, and `dz`

in the `stokes`

package

```
d(1)
dx <- d(2)
dy <- d(3) dz <-
```

Convenience objects `dx`

, `dy`

, and `dz`

are discussed here.

Spivak, in a memorable passage, states (p89):

**Fields and forms**

If \(f\colon\mathbb{R}^n\longrightarrow\mathbb{R}\) is differentiable, then \(Df(p)\in\Lambda^1(\mathbb{R}^n)\). By a minor modification we therefore obtain a \(1\)-form \(df\), defined by

\[df(p)(v_p)=Df(p)(v).\]

Let us consider in particular the \(1\)-forms \(d\pi^i\) ^{1} It is customary to let \(x^i\) denote the *function* \(\pi^i\) (on \(\mathbb{R}^3\) we often denote \(x^1\), \(x^2\), and \(x^3\) by \(x\), \(y\), and \(z\)) \(\ldots\) Since \(dx^i(p)(v_p)=d\pi^i(p)(v_p)=D\pi^i(p)(v)=v^i\), we see that \(dx^1(p),\ldots,dx^n(p)\) is just the dual basis to \((e_1)_p,\ldots, (e_n)_p\).

**- Michael Spivak, 1969** *(Calculus on Manifolds, Perseus books). Page 89*

Spivak goes on to observe that every \(k\)-form \(\omega\) can be written \(\omega=\sum_{i_1 < \cdots < i_k}\omega_{i_1,\ldots i_k}dx^{i_1}\wedge\cdots\wedge dx^{i_k}\). If working in \(\mathbb{R}^3\), we have three elementary forms \(dx\), \(dy\), and \(dz\); in the package we have the pre-defined objects `dx`

, `dy`

, and `dz`

. These are convenient for reproducing textbook results. We start with some illustrations of the package print method.

` dx`

```
## An alternating linear map from V^1 to R with V=R^1:
## val
## 1 = 1
```

This is somewhat opaque and difficult to understand. It is easier to start with a more complicated example: take \(dx\wedge dy -7dx\wedge dz + 3dy\wedge dz\):

`^dy -7*dx^dz + 3*dy^dz dx`

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

We see three rows for the three elementary components. Taking the row with coefficient \(-7\) [which would be \(-7dx\wedge dz\)], this maps \(\left(\mathbb{R}^3\right)^2\) to \(\mathbb{R}\) and we have

\[(-7dx\wedge dz)\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix}, \begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\right)= -7\det\begin{pmatrix}u_1&v_1\\u_3&v_3\end{pmatrix}\]

Armed with this familiar fact, we can interpret \(dx\) as a map from \(\left(\mathbb{R}^3\right)^1\) to \(\mathbb{R}\) with

\[dx\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix} \right)= \det\begin{pmatrix}u_1\end{pmatrix}=u_1\]

or, in other words, \(dx\) picks out the first component of its vector (as the print method gives, albeit obscurely). This is easily shown in the package:

`as.function(dx)(c(113,3,6))`

`## [1] 113`

We might want to verify that \(dx\wedge dy=-dy\wedge dx\):

`^ dy == -dy ^ dx dx `

`## [1] TRUE`

The print method is configurable and can display kforms in symbolic form. For working with `dx dy dz`

we may set option `kform_symbolic_print`

to `dx`

:

`options(kform_symbolic_print = 'dx')`

Then the results of calculations are more natural:

` dx`

```
## An alternating linear map from V^1 to R with V=R^1:
## + dx
```

`^dy + 56*dy^dz dx`

```
## An alternating linear map from V^2 to R with V=R^3:
## + dx^dy +56 dy^dz
```

However, this setting can be confusing if we work with \(dx^i,i>3\), for the print method runs out of alphabet:

`rform()`

```
## An alternating linear map from V^3 to R with V=R^7:
## +6 dy^dNA^dNA +5 dy^dNA^dNA -9 dNA^dNA^dNA +4 dx^dz^dNA +7 dx^dNA^dNA -3 dy^dz^dNA -8 dx^dNA^dNA +2 dx^dy^dNA + dx^dNA^dNA
```

Above, we see the use of `NA`

because there is no defined symbol.

Function `hodge()`

returns the Hodge dual:

`hodge(dx^dy + 13*dy^dz)`

```
## An alternating linear map from V^1 to R with V=R^3:
## +13 dx + dz
```

Note that calling `hodge(dx)`

can be confusing:

`hodge(dx)`

`## [1] 1`

This returns a scalar because `dx`

is interpreted as a one-form on one-dimensional space, which is a scalar form. One usually wants the result in three dimensions:

`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.

It is possible to create these objects using package idiom:

`d(1) == dx`

`## [1] TRUE`

Following lines create `dx.rda`

, residing in the `data/`

directory of the package.

`save(dx,dy,dz,file="dx.rda")`

Spivak introduces the \(\pi^i\) notation on page 11: “if \(\pi\colon\mathbb{R}^n\longrightarrow\mathbb{R}^n\) is the identity function, \(\pi(x)=x\), then [its components are] \(\pi^i(x)=x^i\); the function \(\pi^i\) is called the \(i^\mathrm{th}\)

*projection function*”↩︎