Rearrange the indices in a tensor.

Symmetrize and skew-symmetrize the indices of a tensor.

Contract the indices of a tensor.

Find the inverse of a metric.

Raise and lower indices of a tensor using a metric.

Generate spaces of symmetric tensors.

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y, z], E2):

T := evalDG(dx &t D_y &t dz);

$T≔\mathrm{dx}\mathrm{D\_y}\mathrm{dz}$

RearrangeIndices(T, [1, 3, 2]);

$\mathrm{dx}\mathrm{dz}\mathrm{D\_y}$

RearrangeIndices(T, [[3,2]]);

$\mathrm{dx}\mathrm{dz}\mathrm{D\_y}$

RearrangeIndices(T, [[1, 2, 3]]);

$\mathrm{dz}\mathrm{dx}\mathrm{D\_y}$

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y, z], E2):

T := evalDG(D_x &t D_y &t D_z);

$T≔\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}$

W := SymmetrizeIndices(T, [1, 2], "Symmetric");

$W≔\frac{\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}}{2}\+\frac{\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_z}}{2}$

newW := SymmetrizeIndices(W, [1, 2], "Symmetric");

$\mathrm{newW}≔\frac{\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}}{2}\+\frac{\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_z}}{2}$

newW &minus W;

$0\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}$

U := SymmetrizeIndices(T, [1, 2, 3], "Symmetric");

$U≔\frac{\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_z}}{6}\+\frac{\mathrm{D\_x}\mathrm{D\_z}\mathrm{D\_y}}{6}\+\frac{\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_z}}{6}\+\frac{\mathrm{D\_y}\mathrm{D\_z}\mathrm{D\_x}}{6}\+\frac{\mathrm{D\_z}\mathrm{D\_x}\mathrm{D\_y}}{6}\+\frac{\mathrm{D\_z}\mathrm{D\_y}\mathrm{D\_x}}{6}$

SymmetrizeIndices(U, [2, 3], "SkewSymmetric");

$0\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}$

with(DifferentialGeometry): with(Tensor):

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

T:= evalDG(D_x &t dy &t D_z &t dz - 2*D_x &t dx &t D_x &t dx);

$T≔-2\mathrm{D\_x}\mathrm{dx}\mathrm{D\_x}\mathrm{dx}\+\mathrm{D\_x}\mathrm{dy}\mathrm{D\_z}\mathrm{dz}$

ContractIndices(T, [[2, 3]]);

$-2\mathrm{D\_x}\mathrm{dx}$

ContractIndices(T, [[2,3], [1,4]]);

$-2$

S := convert(D_y, DGtensor);

$S≔\mathrm{D\_y}$

ContractIndices(T, S, [[2, 1]]);

$\mathrm{D\_x}\mathrm{D\_z}\mathrm{dz}$

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y], E2):

g := evalDG(2*dx &t dx + dx &t dy + dy &t dx);

$g≔2\mathrm{dx}\mathrm{dx}\+\mathrm{dx}\mathrm{dy}\+\mathrm{dy}\mathrm{dx}$

h := InverseMetric(g);

$h≔\mathrm{D\_x}\mathrm{D\_y}\+\mathrm{D\_y}\mathrm{D\_x}-2\mathrm{D\_y}\mathrm{D\_y}$

A := convert(g, DGArray);

B := convert(h, DGArray);

convert(A,Matrix).convert(B,Matrix);

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y], E2):

g := evalDG(dx &t dy + dy &t dx);

$g≔\mathrm{dx}\mathrm{dy}\+\mathrm{dy}\mathrm{dx}$

h := InverseMetric(g);

$h≔\mathrm{D\_x}\mathrm{D\_y}\+\mathrm{D\_y}\mathrm{D\_x}$

T := evalDG(dx &t D_y &t D_x);

$T≔\mathrm{dx}\mathrm{D\_y}\mathrm{D\_x}$

S := RaiseLowerIndices(g, T, [2]);

$S≔\mathrm{dx}\mathrm{dx}\mathrm{D\_x}$

RaiseLowerIndices(g, T, [2, 3]);

$\mathrm{dx}\mathrm{dx}\mathrm{dy}$

RaiseLowerIndices(h, T, [1]);

$\mathrm{D\_y}\mathrm{D\_y}\mathrm{D\_x}$

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y], E2):

Fr := Tools:-DGinfo(E2, "FrameBaseVectors");

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

Omega := Tools:-DGinfo(E2, "FrameBaseForms");

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

GenerateTensors([Fr, Fr]);

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

GenerateTensors([Fr, Omega]);

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

GenerateSymmetricTensors(Fr, 3);

$\left[\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}\,\frac{\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_y}}{3}\+\frac{\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_x}}{3}\+\frac{\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_x}}{3}\,\frac{\mathrm{D\_x}\mathrm{D\_y}\mathrm{D\_y}}{3}\+\frac{\mathrm{D\_y}\mathrm{D\_x}\mathrm{D\_y}}{3}\+\frac{\mathrm{D\_y}\mathrm{D\_y}\mathrm{D\_x}}{3}\,\mathrm{D\_y}\mathrm{D\_y}\mathrm{D\_y}\right]$

with(DifferentialGeometry): with(Tensor):

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

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

T := map(convert, [D_x, dx, D_u, du] ,DGtensor);

$T≔\left[\mathrm{D\_x}\,\mathrm{dx}\,\mathrm{D\_u}\,\mathrm{du}\right]$

map(Tools:-DGinfo, T, "TensorIndexType");

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

T := evalDG(du &t D_u &t dv);

$T≔\mathrm{du}\mathrm{D\_u}\mathrm{dv}$

RearrangeIndices(T, [[1, 3]]);

$\mathrm{dv}\mathrm{D\_u}\mathrm{du}$

ContractIndices(T, [[1, 2]]);

$\mathrm{dv}$

with(DifferentialGeometry): with(Tensor):

DGsetup([x, y], E2):

DGsetup([r, theta], P):

Phi := Transformation(P, E2,[x = r*cos(theta), y = r*sin(theta)]);

$\mathrm{\Φ}≔\left[x\=r\cos \left(\mathrm{\θ}\right)\,y\=r\sin \left(\mathrm{\θ}\right)\right]$

_EnvExplicit := true:

InvPhi := InverseTransformation(Phi);

$\mathrm{InvPhi}≔\left[r\=\sqrt{y^2\+x^2}\,\mathrm{\θ}\=\arctan \left(\frac{y}{\sqrt{y^2\+x^2}}\,\frac{x}{\sqrt{y^2\+x^2}}\right)\right]$

T1 := evalDG(D_x &t D_x + D_y &t D_y);

$\mathrm{T1}≔\mathrm{D\_x}\mathrm{D\_x}\+\mathrm{D\_y}\mathrm{D\_y}$

T2 := evalDG(D_x &t dy - D_y &t dx);

$\mathrm{T2}≔\mathrm{D\_x}\mathrm{dy}-\mathrm{D\_y}\mathrm{dx}$

T3 := evalDG(dx &t dx + dy &t dy);

$\mathrm{T3}≔\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}$

PushPullTensor(InvPhi, Phi, T1);

$\mathrm{D\_r}\mathrm{D\_r}\+\frac{\mathrm{D\_theta}\mathrm{D\_theta}}{r^2}$

simplify(PushPullTensor(InvPhi, Phi, T2)) assuming r>0;

$r\mathrm{D\_r}\mathrm{dtheta}-\frac{\mathrm{D\_theta}\mathrm{dr}}{r}$

PushPullTensor(Phi,T3);

$\mathrm{dr}\mathrm{dr}\+r^2\mathrm{dtheta}\mathrm{dtheta}$

DGsetup([u, v],E2): DGsetup([x, y, z], E3):

Phi := Transformation(E2, E3, [x = (a + b*sin(v))*cos(u), y = (a + b*sin(v))*sin(u), z = b*cos(v)]);

$\mathrm{\Φ}≔\left[x\=\left(a\+b\sin \left(v\right)\right)\cos \left(u\right)\,y\=\left(a\+b\sin \left(v\right)\right)\sin \left(u\right)\,z\=b\cos \left(v\right)\right]$

g := evalDG(dx &t dx + dy &t dy + dz &t dz);

$g≔\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

gTorus := PushPullTensor(Phi, g);

$\mathrm{gTorus}≔\left(2ab\sin \left(v\right)\+a^2\+b^2-b^2{\cos \left(v\right)}^2\right)\mathrm{du}\mathrm{du}\+b^2\mathrm{dv}\mathrm{dv}$

DGsetup([x, y], E2): DGsetup([x, y, z], E3):

gE3 := evalDG(y^2*dx &t dy + dx &s dz + x* dy &s dz);

$\mathrm{gE3}≔y^2\mathrm{dx}\mathrm{dy}\+\mathrm{dx}\mathrm{dz}\+x\mathrm{dy}\mathrm{dz}\+\mathrm{dz}\mathrm{dx}\+x\mathrm{dz}\mathrm{dy}$

T := Transformation(E3, E3, [x = x, y = y, z = z + epsilon]);

$T≔\left[x\=x\,y\=y\,z\=z\+\mathrm{\ε}\right]$

PushPullTensor(T, gE3) &minus gE3;

$0\mathrm{dx}\mathrm{dx}$

pi := Transformation(E3, E2, [x = x, y = y]);

$\mathrm{\π}≔\left[x\=x\,y\=y\right]$

sigma := Transformation(E2, E3, [x = x ,y = y, z = 0]);

$\mathrm{\σ}≔\left[x\=x\,y\=y\,z\=0\right]$

ComposeTransformations(pi, sigma);

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

gE2 := PushPullTensor(pi, sigma, gE3);

$\mathrm{gE2}≔y^2\mathrm{dx}\mathrm{dy}$