stokes (version 1.2-0)

symbolic: 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="")

Value

Returns a “noquote” character string.

Arguments

M

Object of class kform or ktensor; a map from V^kV^k to R, where V=R^nV=R^n

symbols

A character vector giving the names of the symbols

d

String specifying the appearance of the differential operator

Author

Robin K. S. Hankin

Details

Spivak (p89), in archetypically terse writing, states:

A function f is considered to be a 0-form and fomitted is also written fomitted. If fR^nRf: R^n -> R is differentiable, then Df(p)^1(R^n)omitted; see PDF. By a minor modification we therefore obtain a 1-form df, defined by

df(p)(v_p)=Df(p)(v)df(p)(v_p)=Df(p)(v).

Let us consider in particular the 1-forms d^iomitted; see PDF. It is customary to let x^i denote the function ^iomitted; see PDF (On R^3R^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 dx^i(p)(v_p)=d^i(p)(v_p)=D^i(p)(v)=v^i(omitted; see PDF), we see that dx^1(p),...,dx^n(p)dx^1(p),...,dx^n(p) is just the dual basis to (e_1)_p,...,(e_n)_p(e_1)_n,...,(e_n)_p. Thus every k-form omitted can be written

=_i_1 < < i_k_i_1,...,i_k dx^i_1 dx^i_k.omitted.

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

_i_1,..., i_k=1^n a_i_1,...,i_k _i_1_i_k.omitted.

and this form is used for k-tensors.

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)