Returns a character string representing \(k\)-tensor and
\(k\)-form objects in symbolic form. Used by the print method if
either option kform_symbolic_print or
ktensor_symbolic_print is non-null.
as.symbolic(M,symbols=letters,d="")Returns a “noquote” character string.
Object of class kform or ktensor; a map from
\(V^k\) to \(\mathbb{R}\), where
\(V=\mathbb{R}^n\)
A character vector giving the names of the symbols
String specifying the appearance of the differential operator
Robin K. S. Hankin
Spivak (p89), in archetypically terse writing, states:
A function \(f\) is considered to be a 0-form and \(f\cdot\omega\) is also written \(f\wedge\omega\). If \(f\colon\mathbb{R}^n\longrightarrow\mathbb{R}\) is differentiable, then \(Df(p)\in\Lambda^1\left(\mathbb{R}^n\right)\). By a minor modification we therefore obtain a 1-form \(\mathrm{d}f\), defined by $$\mathrm{d}f(p)\left(v_p\right)=Df(p)(v).$$
Let us consider in particular the 1-forms \(\mathrm{d}\pi^i\). 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 \(z\)). This standard notation has obvious disadvantages but it allows many classical results to be expressed by formulas of equally classical appearance. 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 \((e_1)_p,\ldots,(e_n)_p\). Thus every k-form \(\omega\) can be written
$$ \omega=\sum_{i_1 < \cdots < i_k}\omega_{i_1,\ldots,i_k} \mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}.$$
Function as.symbolic() uses this format. For completeness, we
add (p77) that \(k\)-tensors may be expressed in the form
$$ \sum_{i_1,\ldots, i_k=1}^n a_{i_1,\ldots,i_k}\cdot \phi_{i_1}\otimes\cdots\otimes\phi_{i_k}.$$
and this form is used for \(k\)-tensors.
print.stokes,dx
(o <- kform_general(3,2,1:3))
as.symbolic(o,d="d",symbols=letters[23:26])
(a <- rform(n=50))
as.symbolic(a,symbols=state.abb)
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