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vignettes/dx.Rmd

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@@ -31,7 +31,7 @@ To cite the `stokes` package in publications, please use
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@hankin2022_stokes. Convenience objects `dx`, `dy`, and `dz`,
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corresponding to elementary differential forms, are discussed here
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(basis vectors $e_1$, $e_2$, $e_2$ are discussed in vignette
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`ex.Rmd`).
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[ex](ex.html)).
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@spivak1965, in a memorable passage, states:
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hodge(dx,3)
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```
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This is further discussed in the `dovs` vignette.
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This is further discussed in the [dovs](dovs.html) vignette.
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## Creating elementary one-forms
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vignettes/hodge.Rmd

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```
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showing agreement (above, we use function `volume()` in lieu of
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calculating the permutation's sign explicitly. See the `volume`
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vignette for more details). We may work more formally by defining a
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function that returns `TRUE` if the left and right hand sides match
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calculating the permutation's sign explicitly. See the
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[volume](volume.html) vignette for more details). We may work more
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formally by defining a function that returns `TRUE` if the left and
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right hand sides match
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```{r defdif}
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diff <- function(a,b){a^hodge(b) == kinner(a,b)*volume(dovs(a))}
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hodge(dx,3)
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```
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This is further discussed in the `dovs` vignette.
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This is further discussed in the [dovs](dovs.html) vignette.
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## Vector cross product identities
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vignettes/phi.Rmd

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To cite the `stokes` package in publications, please use
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@hankin2022_stokes. Function `phi()` returns a tensor dual to the
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standard basis of $V=\mathbb{R}^n$. Here I discuss `phi()` but there
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is some overlap between this vignette and the `tensorprod` vignette.
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is some overlap between this vignette and the
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[tensorprod](tensorprod.html) vignette.
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In a memorable passage, @spivak1965 states (theorem 4.1):
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## Function `Alt()`
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Function `Alt()` returns an alternating tensor as documented in the
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`Alt` vignette in the package. It works nicely with `phi()`:
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[Alt](Alt.hmtl) vignette in the package. It works nicely with `phi()`:
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```{r}
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phi(1:3)

vignettes/volume.Rmd

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\noindent In other words, if $A^T$ denotes the transpose of the matrix
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$A$, then we have $A\cdot A^T=I$, so $\operatorname{det}A=\pm 1$. It
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follows from Theorem 4-6 [see vignette `det.Rmd`] that if
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follows from Theorem 4-6 that if
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$\omega\in\Lambda^n(V)$ satisfies
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$\omega{\left(v_1,\ldots,v_n\right)}=\pm 1$, then
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$\omega{\left(w_1,\ldots,w_n\right)}=\pm 1$. If an orientation $\mu$

vignettes/wedge.Rmd

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# Algebraic properties
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First of all we should note that $\Lambda^k(V)$ is a vector space
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(this is considered in the `kform` vignette). If
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(this is considered in the [kform](kform.html) vignette). If
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$\omega,\omega_i\in\Lambda^k(V)$ and $\eta,\eta_i\in\Lambda^l(V)$ then
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\begin{eqnarray}

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