Extract the set of coefficients of a DifferentialGeometry VFT.

Extract specific coefficients of a DifferentialGeometry VFT.

Determine if a given VFT is a linear combination of a set of VFT.

Create a spanning list of independent VFT (a basis) from a list of VFT.

Generate a basis for the space of p-forms.

Generate a collection of tensors.

Extend a given set of independent VFT to a basis for a subspace.

Find the dual basis for the cotangent space from a basis for the tangent space.

Find the annihilator subspace for a given set of vectors or forms.

with(DifferentialGeometry): with(Tools):

 DGsetup([x, y ,z], "M");

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

X := evalDG(a*D_x + b*D_y + b*D_z);

$X:=a\mathrm{D\_x}\+b\mathrm{D\_y}\+b\mathrm{D\_z}$

DGinfo(X, "CoefficientSet");

$\left\{a\,b\right\}$

DGinfo(X, "CoefficientList", "all");

$\left[a\,b\,b\right]$

DGinfo(X, "CoefficientList", [[1], [2]]);

$\left[a\,b\right]$

DGinfo(X, "CoefficientList", [D_x, D_y]);

$\left[a\,b\right]$

T := evalDG(a*dx &t D_x + b*dy &t D_z + c*dz &t D_x);

$T:=a\mathrm{dx}\mathrm{D\_x}\+b\mathrm{dy}\mathrm{D\_z}\+c\mathrm{dz}\mathrm{D\_x}$

DGinfo(T, "CoefficientList", [[1, 1]]);

$\left[a\right]$

DGinfo(alpha, "CoefficientList", [dy &tensor D_z, dz &tensor D_z]);

$\left[\mathrm{\α}\right]$

with(DifferentialGeometry): with(Tools):

DGsetup([x, y ,z], "M"):

X := evalDG(2*D_x + D_y - D_z);

$X:=2\mathrm{D\_x}\+\mathrm{D\_y}-\mathrm{D\_z}$

B := evalDG([D_x - D_y, D_y - D_z, D_z + D_x]);

$B:=\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_y}-\mathrm{D\_z}\,\mathrm{D\_x}\+\mathrm{D\_z}\right]$

C := GetComponents(X, B);

$C:=\left[1\,2\,1\right]$

DGzip(C, B, "plus");

$2\mathrm{D\_x}\+\mathrm{D\_y}-\mathrm{D\_z}$

alpha := dy &wedge dz;

$\mathrm{\α}:=\mathrm{dy}\bigwedge \mathrm{dz}$

S := [dx &wedge dy, dx &wedge dz];

$S:=\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dz}\right]$

GetComponents(alpha, S);

A := evalDG([D_x + D_y + 2*D_z, D_y + D_z, 2*D_z]);

$A:=\left[\mathrm{D\_x}\+\mathrm{D\_y}\+2\mathrm{D\_z}\,\mathrm{D\_y}\+\mathrm{D\_z}\,2\mathrm{D\_z}\right]$

B := evalDG([D_x - D_y, D_y - D_z, D_z + D_x]);

$B:=\left[\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_y}-\mathrm{D\_z}\,\mathrm{D\_x}\+\mathrm{D\_z}\right]$

C := GetComponents(A, B);

$C:=\left[\left[-1\,0\,2\right]\,\left[-1\,0\,1\right]\,\left[-1\,-1\,1\right]\right]$

P := LinearAlgebra[Transpose](Matrix(C));

$P:=\left[\begin{array}{rrr}-1& -1& -1\\ 0& 0& -1\\ 2& 1& 1\end{array}\right]$

GetComponents(x &mult D_x, [D_x]);

$\left[x\right]$

GetComponents(x &mult D_x, [D_x], method = "real");

GetComponents(x &mult D_x, [D_x, x &mult D_x, (x^2) &mult D_x], method = "real");

$\left[0\,1\,0\right]$

with(DifferentialGeometry): with(Tools): with(Tensor):

DGsetup([u, v, w, x, y], E5);

$\mathrm{frame\; name:\; E5}$

Omega := [du, dv, dw, dx];

$\mathrm{\Ω}:=\left[\mathrm{du}\,\mathrm{dv}\,\mathrm{dw}\,\mathrm{dx}\right]$

GenerateForms(Omega, 2);

$\left[\mathrm{du}\bigwedge \mathrm{dv}\,\mathrm{du}\bigwedge \mathrm{dw}\,\mathrm{du}\bigwedge \mathrm{dx}\,\mathrm{dv}\bigwedge \mathrm{dw}\,\mathrm{dv}\bigwedge \mathrm{dx}\,\mathrm{dw}\bigwedge \mathrm{dx}\right]$

GenerateForms(Omega, 4);

$\left[\mathrm{du}\bigwedge \mathrm{dv}\bigwedge \mathrm{dw}\bigwedge \mathrm{dx}\right]$