symbolic: Symbolic form

In stokes: The Exterior Calculus

Symbolic form

Description

\loadmathjax

Returns a character string representing \mjseqnk-tensor and \mjseqnk-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 \mjeqnV^kV^k to \mjseqnR, where \mjeqnV=R^nV=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 \mjseqnf is considered to be a 0-form and \mjeqnf\cdot\omegaomitted is also written \mjeqnf\wedge\omegaomitted. If \mjeqnf\colon\mathcalR^n\longrightarrow\mathcalRf: R^n -> R is differentiable, then \mjeqnDf(p)\in\Lambda^1\left(\mathcalR^n\right)omitted; see PDF. By a minor modification we therefore obtain a 1-form \mjseqndf, defined by

\mjdeqn

df(p)\left(v_p\right)=Df(p)(v)df(p)(v_p)=Df(p)(v).

Let us consider in particular the 1-forms \mjeqnd\pi^iomitted; see PDF. It is customary to let \mjseqnx^i denote the function \mjeqn\pi^iomitted; see PDF (On \mjeqn\mathcalR^3R^3 we often denote \mjseqnx^1, \mjseqnx^2, and \mjseqnx^3 by \mjseqnx, \mjseqny, and \mjseqnz). This standard notation has obvious disadvantages but it allows many classical results to be expressed by formulas of equally classical appearance. Since \mjeqndx^i(p)(v_p)=d\pi^i(p)(v_p)=D\pi^i(p)(v)=v^i(omitted; see PDF), we see that \mjeqndx^1(p),...,dx^n(p)dx^1(p),...,dx^n(p) is just the dual basis to \mjeqn(e_1)_p,...,(e_n)_p(e_1)_n,...,(e_n)_p. Thus every k-form \mjeqn\omegaomitted can be written

\mjdeqn \omega

=\sum_i_1 < \cdots < i_k\omega_i_1,...,i_k dx^i_1\wedge\cdots\wedge dx^i_k.omitted.

Function as.symbolic() uses this format. For completeness, we add (p77) that k-tensors may be expressed in the form

\mjdeqn \sum

_i_1,..., i_k=1^n a_i_1,...,i_k\cdot \phi_i_1\otimes\cdots\otimes\phi_i_k.omitted.

and this form is used for k-tensors.

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 Aug. 19, 2023, 1:07 a.m.