The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.
dx
, dy
,
and dz
in the stokes
packageTo cite the stokes
package in publications, please use
Hankin (2022). Convenience objects
dx
, dy
, and dz
, corresponding to
elementary differential forms, are discussed here (basis vectors \(e_1\), \(e_2\), \(e_2\) are discussed in
ex.Rmd
). Spivak (1965), in a
memorable passage, states:
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 \(\mathrm{d}f\), defined by
\[\mathrm{d}f(p)(v_p)=Df(p)(v).\]
Let us consider in particular the \(1\)-forms \(\mathrm{d}\pi^i\) 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 \(\mathrm{d}x^i(p)(v_p)=\mathrm{d}\pi^i(p)(v_p)=D\pi^i(p)(v)=v^i\), we see that \(\mathrm{d}x^1(p),\ldots,\mathrm{d}x^n(p)\) 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 \(\omega=\sum_{i_1 < \cdots <
i_k}\omega_{i_1,\ldots
i_k}\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}\). If
working in \(\mathbb{R}^3\), we have
three elementary forms \(\mathrm{d}x\),
\(\mathrm{d}y\), and \(\mathrm{d}z\); 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.
## 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 \(\mathrm{d}x\wedge\mathrm{d}y -7\mathrm{d}x\wedge\mathrm{d}z + 3\mathrm{d}y\wedge\mathrm{d}z\):
## 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 \(-7\mathrm{d}x\wedge\mathrm{d}z\)], this maps \(\left(\mathbb{R}^3\right)^2\) to \(\mathbb{R}\) and we have
\[(-7\mathrm{d}x\wedge\mathrm{d}z)\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix}, \begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\right)= -7\det\begin{pmatrix}u_1&v_1\\u_3&v_3\end{pmatrix}\]
Armed with this familiar fact, we can interpret \(dx\) as a map from \(\left(\mathbb{R}^3\right)^1\) to \(\mathbb{R}\) with
\[\mathrm{d}x\left(\begin{pmatrix} u_1\\u_2\\u_3\end{pmatrix} \right)= \det\begin{pmatrix}u_1\end{pmatrix}=u_1\]
or, in other words, \(\mathrm{d}x\) picks out the first component of its vector (as the print method gives, albeit obscurely). This is easily shown in the package:
## [1] 113
We might want to verify that \(\mathrm{d}x\wedge\mathrm{d}y=-\mathrm{d}y\wedge\mathrm{d}x\):
## [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
:
Then the results of calculations are more natural:
## An alternating linear map from V^1 to R with V=R^1:
## + 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 \(\mathrm{d}x^i,i>3\), for the print method runs out of alphabet:
## 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:
## An alternating linear map from V^1 to R with V=R^3:
## +13 dx + dz
Note that calling hodge(dx)
can be confusing:
## [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:
## 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:
## [1] TRUE
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”↩︎
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.