DGmap - Maple Help

DifferentialGeometry[Tools]

 DGmap

 Calling Sequence DGmap(n, f, X, arg1, arg2, ..., argN)

Parameters

 n - a positive integer f - a Maple procedure X - any DifferentialGeometry object argN - (optional) arguments for the procedure f

Description

 • The command DGmap is similar to the command map.  DGmap will apply the procedure f to the coefficients of the object X.  The integer n indicates the position of the coefficients of X in the argument list of  f.  Thus DGmap(1, f, X, arg1, arg2, ..., argN) will replace the coefficient C of X with  f(C, arg1, arg2, ..., argN); DGmap(2, f, X, arg1, arg2, ..., argN) will replace the coefficient C of X with f(arg1, C, arg2, ..., argN); and so on.
 • This command is part of the DifferentialGeometry:-Tools package, and so can be used in the form DGmap(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGmap.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$

Define a manifold M with local coordinates [x, y].

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (1)

Example 1.

Define a vector field X depending on parameters C1 and C2. Differentiate the coefficients of X with respect to C1 and C2.

 > $X≔\mathrm{evalDG}\left(\left(\mathrm{C1}x+\mathrm{C2}\right)\mathrm{D_x}+\mathrm{C1}y\mathrm{D_y}\right)$
 ${X}{≔}\left({\mathrm{C1}}{}{x}{+}{\mathrm{C2}}\right){}{\mathrm{D_x}}{+}{\mathrm{C1}}{}{y}{}{\mathrm{D_y}}$ (2)
 > $\mathrm{DGmap}\left(1,\mathrm{diff},X,\mathrm{C1}\right)$
 ${x}{}{\mathrm{D_x}}{+}{y}{}{\mathrm{D_y}}$ (3)
 > $\mathrm{DGmap}\left(1,\mathrm{diff},X,\mathrm{C2}\right)$
 ${\mathrm{D_x}}$ (4)

Example 2.

Define a differential 1-form omega depending on a parameter t.  Integrate the coefficients of omega with respect to t from t = 0 .. 1

 > $\mathrm{\omega }≔\mathrm{evalDG}\left({t}^{2}{x}^{2}\mathrm{dx}-{t}^{3}x{y}^{2}\mathrm{dy}\right)$
 ${\mathrm{\omega }}{≔}{{x}}^{{2}}{}{{t}}^{{2}}{}{\mathrm{dx}}{-}{{y}}^{{2}}{}{x}{}{{t}}^{{3}}{}{\mathrm{dy}}$ (5)
 > $\mathrm{DGmap}\left(1,\mathrm{int},\mathrm{\omega },t=0..1\right)$
 $\frac{{{x}}^{{2}}{}{\mathrm{dx}}}{{3}}{-}\frac{{{y}}^{{2}}{}{x}{}{\mathrm{dy}}}{{4}}$ (6)

Example 3.

Evaluate the tensor T at x = 0 by taking the limit of the coefficients as x -> 0.

 > $T≔\mathrm{evalDG}\left(\frac{\mathrm{exp}\left(x\right)-1}{x}\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+\frac{\mathrm{sin}\left(2x\right)}{x}\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${T}{≔}\frac{\left({{ⅇ}}^{{x}}{-}{1}\right){}{\mathrm{dx}}}{{x}}{}{\mathrm{dx}}{+}\frac{{\mathrm{sin}}{}\left({2}{}{x}\right){}{\mathrm{dy}}}{{x}}{}{\mathrm{dy}}$ (7)
 > $\mathrm{DGmap}\left(1,\mathrm{limit},T,x=0\right)$
 ${\mathrm{dx}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (8)

Example 4.

Substitute s = 1 into the transformation Phi:

 > $\mathrm{\Phi }≔\mathrm{Transformation}\left(M,M,\left[x=sx+\left(s-1\right)y,y=sy\right]\right)$
 ${\mathrm{\Phi }}{≔}\left[{x}{=}{s}{}{x}{+}\left({s}{-}{1}\right){}{y}{,}{y}{=}{s}{}{y}\right]$ (9)
 > $\mathrm{DGmap}\left(2,\mathrm{subs},\mathrm{\Phi },\left\{s=1\right\}\right)$
 $\left[{x}{=}{x}{,}{y}{=}{y}\right]$ (10)