The module DGmain contains DifferentialGeometry procedures identical to procedures in the DifferentialGeometry package -- the sole difference being that no argument checking or validation is performed by DGmain procedures.  The DGmain procedures can therefore be used in programming situations where the arguments to the procedures &mult, &plus, &minus, &tensor, &wedge, Hook, DGzip are known to be valid.

This command is part of the DifferentialGeometry:-Tools package, and so can be used in the form DGmain:-&mult only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGmain:-&mult.  The other valid exports work the same way.

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

$\mathrm{exports}\left(\mathrm{DGmain}\right)$

$\mathrm{\&mult},\mathrm{\&algmult},\mathrm{\&plus},\mathrm{\&minus},\mathrm{\&tensor},\mathrm{\&wedge},\mathrm{Hook},\mathrm{DGzip}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

Test1 := proc(n)
local T, i;
T := time();
for i to n do
D_x &plus D_y &plus D_z &plus D_w;
od;
time() - T;
end:

Test2 := proc(n)
local T, i;
T := time();
for i to n do
DGmain:-`&plus`(DGmain:-`&plus`(DGmain:-`&plus`(D_x, D_y), D_z), D_w);
od;
time() - T;
end:

$\mathrm{Test1}\left(1000\right)$

$0.416$

$\mathrm{Test2}\left(1000\right)$

$0.392$