with(DifferentialGeometry):

Preferences("DisplayFrameDefinitions");

$\mathrm{false}$

DGsetup([x, y, z], E3);

$\mathrm{frame\; name:\; E3}$

Preferences("DisplayFrameDefinitions", true);

$\mathrm{false}$

DGsetup([x, y, z], E3);

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\,z\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E3}\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E3}\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E3}\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E3}\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{frame\; name:\; E3}$

with(DifferentialGeometry):

Test := proc(n)
local i;
for i to n do
Tools:-CalculationHistory:-Update("Test", [i, i^2])
od; n^2 end:

Test(25);

$625$

Tools:-CalculationHistory:-Get("Test");

$\left[\left[25\,625\right]\,\left[24\,576\right]\,\left[23\,529\right]\,\left[22\,484\right]\,\left[21\,441\right]\,\left[20\,400\right]\,\left[19\,361\right]\,\left[18\,324\right]\,\left[17\,289\right]\,\left[16\,256\right]\right]$

Preferences("HistoryDepth", 3);

$10$

Rerun the program Test and retrieve the calculated values.  This time only 3 values are stored.

Test(25);

$625$

Tools:-CalculationHistory:-Get("Test");

$\left[\left[25\,625\right]\,\left[24\,576\right]\,\left[23\,529\right]\right]$

The JetNotation preference determines which notation is used by the DifferentialGeometry package (and the JetCalculus subpackage).  The default is JetNotation1.

with(DifferentialGeometry):

Preferences("JetNotation");

$''JetNotation1''$

DGsetup([x,y], [u], Jet, 3, verbose);

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\,u\left[\right]\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\,u_{1,1,1}\,u_{1,1,2}\,u_{1,2,2}\,u_{2,2,2}\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[9\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[10\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[11\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Jet}\,\left[\right]\right]\,\left[\left[\left[12\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[9\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[10\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[11\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Jet}\,1\right]\,\left[\left[\left[12\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; type\; [1,0]\; biforms\; have\; been\; defined\; and\; protected::}$

$\left[\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[1\,0\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; type\; [0,1]\; biforms\; (contact\; 1-forms)\; have\; been\; defined\; and\; protected::}$

$\left[\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[5\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[6\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[7\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[8\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[9\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[10\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[11\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{Jet}\,\left[0\,1\right]\right]\,\left[\left[\left[12\right]\,1\right]\right]\right]\right)\right]$

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

Preferences("JetNotation", "JetNotation2");

$''JetNotation1''$

DGsetup([x, y], [u], Jet, 3, verbose);