Interpret the internal representation of a differential geometry object.

Find the frame in which a vector field, differential form, or tensor is defined.

Find the domain and range frames of a transformation.

Determine the DifferentialGeometry type of a object (ie, vector, differential form, tensor ...).

Find the degree of a differential form and the index type of a tensor.

Find the differential order of a function, vector field, or differential form defined on a jet space.

Obtain the list or set of coefficients of a given vector field, differential form, or tensor.

List the coordinates defined for a given frame.

Obtain the bases for the tangent space and cotangent space for a given frame.

Obtain the structure equations for an anholonomic frame.

Here A is the attribute list and C is the component list.

The first element of the attribute list A always specifies the object type.

The second element of the attribute list A always specifies the frame information for the object.

Other attributes include form degree, tensor index type, Jacobian, etc...

The component list is a list of two-element sublists C = [C1, C2, C3 , ...].  The first element of the sublist Ck specifies the component and the second element of Ck gives the coefficient for that component.

The use of the unevaluated function _DG enables the DifferentialGeometry package to control the printing (and LaTeX formatting) of all the various DifferentialGeometry objects (see print).

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([t], R):

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

$X≔a\mathrm{D\_x}\+b\mathrm{D\_y}\+c\mathrm{D\_z}$

op(0, X);

$\mathrm{\_DG}$

internalX := op(X);

$\mathrm{internalX}≔\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,b\right]\,\left[\left[3\right]\,c\right]\right]\right]$

nops(internalX);

$2$

AttributeList := internalX[1];

$\mathrm{AttributeList}≔\left[''vector''\,M\,\left[\right]\right]$

ComponentList := internalX[2];

$\mathrm{ComponentList}≔\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,b\right]\,\left[\left[3\right]\,c\right]\right]$

DGinfo(X, "ObjectAttributes");

$\left[''vector''\,M\,\left[\right]\right]$

DGinfo(X, "ObjectComponents");

$\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,b\right]\,\left[\left[3\right]\,c\right]\right]$

alpha := evalDG(x^2*dx &w dy - y^2*dy &w dz);

$\mathrm{\α}≔x^2\mathrm{dx}\mathrm{`^`}\mathrm{dy}-y^2\mathrm{dy}\mathrm{`^`}\mathrm{dz}$

DGinfo(alpha, "ObjectAttributes");

$\left[''form''\,M\,2\right]$

DGinfo(alpha, "ObjectComponents");

$\left[\left[\left[1\,2\right]\,x^2\right]\,\left[\left[2\,3\right]\,-y^2\right]\right]$

T := evalDG(dx &t dy &t D_z - dy &t dz &t D_x);

$T≔\mathrm{dx}\mathrm{dy}\mathrm{D\_z}-\mathrm{dy}\mathrm{dz}\mathrm{D\_x}$

DGinfo(T, "ObjectAttributes");

$\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]$

DGinfo(T, "ObjectComponents");

$\left[\left[\left[1\,2\,3\right]\,1\right]\,\left[\left[2\,3\,1\right]\,-1\right]\right]$

Phi := Transformation(R, M, [x = t*cos(t), y = t*sin(t), z = t^2]);

$\mathrm{\Φ}≔\left[x\=t\cos \left(t\right)\,y\=t\sin \left(t\right)\,z\=t^2\right]$

DGinfo(Phi, "ObjectAttributes");

DGinfo(Phi, "ObjectComponents");

$\left[\left[t\cos \left(t\right)\,x\right]\,\left[t\sin \left(t\right)\,y\right]\,\left[t^2\,z\right]\right]$

with(DifferentialGeometry): with(LieAlgebras): with(Tools):

DGsetup([x, y], M):

lprint(dx);

_DG([["form", M, 1], [[[1], 1]]])

DGinfo(dx, "ObjectType");

$''form''$

lprint(D_x);

_DG([["vector", M, []], [[[1], 1]]])

DGinfo(D_x, "ObjectType");

$''vector''$

lprint(D_x &tensor dy);

_DG([["tensor", M, [["con_bas", "cov_bas"], []]], [[[1, 2], 1]]])

DGinfo(D_x &tensor dy, "ObjectType");

$''tensor''$

f := Transformation(M, M, [x= 1, y = y]);

$f≔\left[x\=1\,y\=y\right]$

lprint(f);

_DG([["transformation", [[M, 0], [M, 0]], [], [Matrix(2,2,{(2, 2) = 1},datatype = anything,storage = rectangular,order = Fortran_order,shape = [])]], [[1, x], [y, y]]])

DGinfo(f, "ObjectType");

$''transformation''$

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v], N):

DGinfo(D_x, "ObjectFrame");

$M$

DGinfo(du, "ObjectFrame");

$N$

f := Transformation(M, N, [u = x^2 + y^2, v = x/y]);

$f≔\left[u\=x^2\+y^2\,v\=\frac{x}{y}\right]$

DGinfo(f, "DomainFrame");

$M$

DGinfo(f, "RangeFrame");

$N$

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

DGsetup([x, y], [u], M):

alpha := evalDG(dx &w dy);

$\mathrm{\α}≔\mathrm{dx}\mathrm{`^`}\mathrm{dy}$

DGinfo(alpha, "FormDegree");

$2$

T := evalDG(D_x &t dy &t PermutationSymbol("con_vrt"));

$T≔\mathrm{D\_x}\mathrm{dy}\mathrm{D\_u}$

DGinfo(T, "TensorType");

$\left[\left[''con\_bas''\,''cov\_bas''\,''con\_vrt''\right]\,\left[\left[''vrt''\,1\right]\right]\right]$

DGinfo(T, "TensorIndexType");

$\left[''con\_bas''\,''cov\_bas''\,''con\_vrt''\right]$

DGinfo(T, "TensorDensityType");

$\left[\left[''vrt''\,1\right]\right]$

DGinfo("con_bas", "TensorIndexPart1");

$''con''$

DGinfo("con_bas", "TensorIndexPart2");

$''bas''$

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([u, v], N):

Phi := Transformation(N, M, [x = u, y = v, z = 1 - u^2 - v^2]);

$\mathrm{\Φ}≔\left[x\=u\,y\=v\,z\=1-u^2-v^2\right]$

DGinfo(Phi, "JacobianMatrix");

with(DifferentialGeometry): with(Tools):

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

omega := evalDG(a*dx &w dy + b*dx &w dz + b*dy &w dz + d*dx &w dw + e*dz &w dw);

$\mathrm{\ω}≔a\mathrm{dx}\mathrm{`^`}\mathrm{dy}\+b\mathrm{dx}\mathrm{`^`}\mathrm{dz}\+d\mathrm{dx}\mathrm{`^`}\mathrm{dw}\+b\mathrm{dy}\mathrm{`^`}\mathrm{dz}\+e\mathrm{dz}\mathrm{`^`}\mathrm{dw}$

DGinfo(omega, "CoefficientSet");

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

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

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

DGinfo(omega, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);

$\left[a\,b\,0\right]$

DGinfo(omega, "CoefficientList", [[1, 2], [1, 3], [2, 4]]);

$\left[a\,b\,0\right]$

restart: with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M):

DGsetup([t], R):

DGsetup([x, y], [z], E):

DGinfo("CurrentFrame");

$E$

DGinfo("FrameNames");

$\left[E\,M\,R\right]$

DGinfo(E,"FrameGlobals");

$\left\{x\,E\,y\,z\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\,\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right\}$

DGinfo(M, "FrameInformation");

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

$\mathrm{library\; name:\; CoordinateProtocol}$

$\mathrm{Frame\; Variables:}$

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

$\mathrm{Frame\; Labels}$

$\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]$

$\mathrm{Coframe\; Labels}$

$\left[\mathrm{dx}\,\mathrm{dy}\right]$

$\mathrm{--------------}$

ChangeFrame(M):

Omega := FrameData([1/y*dx, 1/x*dy], P);

$\mathrm{\Ω}≔\left[d\mathrm{\Θ1}\=\frac{x\mathrm{\Θ1}\mathrm{`^`}\mathrm{\Θ2}}{y}\,d\mathrm{\Θ2}\=-\frac{y\mathrm{\Θ1}\mathrm{`^`}\mathrm{\Θ2}}{x}\right]$

DGsetup(Omega, [E], [beta]);

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

DGinfo(P, "FrameInformation");

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

$\mathrm{library\; name:\; AnholonomicFrameProtocol}$

$\mathrm{Frame\; Variables:}$

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

$\mathrm{Frame\; Labels}$

$\left[\mathrm{E1}\,\mathrm{E2}\right]$

$\mathrm{Coframe\; Labels}$

$\left[\mathrm{\β1}\,\mathrm{\β2}\right]$

$\mathrm{ext\_d:}\,x\,y\mathrm{\β1}$

$\mathrm{ext\_d:}\,y\,x\mathrm{\β2}$

$\mathrm{ext\_d:}\,\mathrm{\β1}\,\frac{x\mathrm{\β1}\mathrm{`^`}\mathrm{\β2}}{y}$

$\mathrm{ext\_d:}\,\mathrm{\β2}\,-\frac{y\mathrm{\β1}\mathrm{`^`}\mathrm{\β2}}{x}$

$\mathrm{--------------}$

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([x, y], [u, v, w, z], E):

DGinfo(M, "FrameBaseDimension");

$3$

DGinfo(E, "FrameBaseDimension");

$2$

DGinfo(E, "FrameFiberDimension");

$4$

with(DifferentialGeometry): with(Tools):

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

DGsetup([x, y], [u, v, w, z], E):

DGinfo(M, "FrameIndependentVariables");

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

DGinfo(E, "FrameIndependentVariables");

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

DGinfo(E, "FrameDependentVariables");

$\left[u\,v\,w\,z\right]$

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([x, y], [u, v, w, z], E):

DGinfo(M, "FrameBaseVectors");

$\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

DGinfo(M, "FrameBaseForms");

$\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

DGinfo(E, "FrameFiberVectors");

$\left[\mathrm{D\_u}\,\mathrm{D\_v}\,\mathrm{D\_w}\,\mathrm{D\_z}\right]$

DGinfo(E, "FrameFiberForms");

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

F := DGinfo(M, "FrameBaseVectors");

$F≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

X := DGzip([a, b, c], F, "plus");

$X≔a\mathrm{D\_x}\+b\mathrm{D\_y}\+c\mathrm{D\_z}$

C := GetComponents(X, F);

$C≔\left[a\,b\,c\right]$