symbolic: Symbolic form

In stokes: The Exterior Calculus

Symbolic form

Description

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.

Usage

as.symbolic(M,symbols=letters,d="")

Arguments

M	Object of class kform or ktensor; a map from V^k to \mathbb{R}, where V=\mathbb{R}^n
symbols	A character vector giving the names of the symbols
d	String specifying the appearance of the differential operator

Details

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. The print method for tensors, print.ktensor(), writes d1 for \phi_1, d2 for \phi_2 [where \phi_i(x^j)=\delta_i^j].

Value

Returns a “noquote” character string.

Author(s)

Robin K. S. Hankin

See Also

print.stokes,dx

Examples

(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)

stokes documentation built on April 4, 2025, 1:48 a.m.