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These are replies submitted by prcls

 Hello everyone and thanks for the answers!
  I agree, and in addition to the DG package in Maple, you can find (if desired) commands on the basic representations of differential geometry. Of course, it would be very useful for many to shift a good textbook on this topic in Maple - in the form of work sheets. This would be both a textbook on classical differential geometry and a guide to Maple.

@John Fredsted  The question remained open ... Do you think it is possible to solve it with the authors of these lessons?

@John Fredsted  Yes, indeed, your code works!

But in other cases, say in exercise 5, again the same situation occurs. And how to react to it ?!
On the other hand, the description of geometric objects, as is well known, requires certain coordinate systems (frames), and various properties of objects need different frames. Therefore, in my opinion, the authors use different frames in the Help.

Thank you! Sorry for bothering.

Thanks for the quick response.

But we will analyze a concrete example, because of which the question arose. This is exercise 4, lesson 2, DifferentialGeometry Lessons in Help (Maple16).

TwoFormBasis2 := proc(M) 
local n ,i, j, alpha, beta, Basis;
n := DifferentialGeometry:-Tools:-DGinfo(M, "FrameBaseDimension");

Basis := Array(1 .. n*(n - 1)/2):
k := 0;
for i to n do
alpha := DifferentialGeometry:-Tools:-DGform(i);
for j from i + 1 to n do
k := k + 1;
beta := DifferentialGeometry:-Tools:-DGform(j);
Basis[k] := alpha &wedge beta;
convert(Basis, list);

[dx1 ^ dx2, dx1 ^ dx3, dx1 ^ dx4, dx2 ^ dx3, dx2 ^ dx4, dx3 ^ dx4]

In the first line, the procedure is defined in frame M, and the second line is already in frame E4. If, after the first line, you issue a command to change the frame, then, naturally, an error is produced: it is impossible to interpret.
Further, in other lines, there are switching between M and E4; However, no real a command to change the frame are being introduced by the authors. How they do it ?... it's unclear.


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