$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{T1}≔\mathrm{evalDG}\left(\mathrm{D\_z}\&t\mathrm{D\_x}+\mathrm{D\_x}\&t\mathrm{D\_y}\right)$

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

$\mathrm{T2}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\&t\mathrm{D\_x}\right)\&t\mathrm{dy}+\left(\mathrm{dz}\&t\mathrm{D\_z}\right)\&t\mathrm{dx}\right)$

$\mathrm{T2}≔2\mathrm{dx}\mathrm{D\_x}\mathrm{dy}+\mathrm{dz}\mathrm{D\_z}\mathrm{dx}$

$\mathrm{A1}≔\mathrm{convert}\left(\mathrm{T1}\,\mathrm{DGArray}\right)$

$\mathrm{A1}≔\left[\right]$

$\mathrm{A2}≔\mathrm{convert}\left(\mathrm{T2}\,\mathrm{DGArray}\right)$

$\mathrm{A2}\left[1,1,2\right]$

$2$

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,0\,1\right]\,\left[0\,2\,0\right]\,\left[0\,3\,0\right]\right]\right)$

$A≔\left[\right]$

$\mathrm{convert}\left(A\,\mathrm{DGvector}\right)$

$\left(x+z\right)\mathrm{D\_x}+2y\mathrm{D\_y}+3y\mathrm{D\_z}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{T1}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\&t\mathrm{D\_x}\right)\&t\mathrm{dy}+\left(\mathrm{dz}\&t\mathrm{D\_z}\right)\&t\mathrm{dx}\right)$

$\mathrm{T1}≔2\mathrm{dx}\mathrm{D\_x}\mathrm{dy}+\mathrm{dz}\mathrm{D\_z}\mathrm{dx}$

$\mathrm{T2}≔\mathrm{evalDG}\left(\mathrm{D\_z}\&t\mathrm{D\_x}+\mathrm{D\_x}\&t\mathrm{D\_y}\right)$

$\mathrm{T2}≔\mathrm{D\_x}\mathrm{D\_y}+\mathrm{D\_z}\mathrm{D\_x}$

$S≔\mathrm{ContractIndices}\left(\mathrm{T1}\,\mathrm{T2}\,\left[\left[1\,1\right]\,\left[3\,2\right]\right]\right)$

$S≔2\mathrm{D\_x}+\mathrm{D\_z}$

$X≔\mathrm{convert}\left(S\,\mathrm{DGvector}\right)$

$X≔2\mathrm{D\_x}+\mathrm{D\_z}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(S\,''ObjectType''\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(X\,''ObjectType''\right)$

$''tensor'',''vector''$

$\mathrm{lprint}\left(S\right),\mathrm{lprint}\left(X\right)$

_DG([["tensor", M, [["con_bas"], []]], [[[1], 2], [[3], 1]]])
_DG([["vector", M, []], [[[1], 2], [[3], 1]]])

$\mathrm{DE}≔\left[\mathrm{diff}\left(u\left(x\,y\right)\,x\,y\right)={v\left(x\,y\right)}^2\,\mathrm{diff}\left(v\left(x\,y\right)\,\mathrm{`\$`}\left(x\,2\right)\right)=u\left(x\,y\right)\right]$

$\mathrm{DE}≔\left[\frac{{\partial }^2}{\partial x\partial y}u\left(x\,y\right)={v\left(x\,y\right)}^2\,\frac{{\partial }^2}{\partial x^2}v\left(x\,y\right)=u\left(x\,y\right)\right]$

$\mathrm{Sym}≔\mathrm{PDEtools}:-\mathrm{Infinitesimals}\left(\mathrm{DE}\,\left[u\,v\right]\right)$

$\mathrm{Sym}≔\left[{\mathrm{\_\ξ}}_x\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\,v\right)=1\,{\mathrm{\_\η}}_u\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\η}}_v\left(x\,y\,u\,v\right)=0\right],\left[{\mathrm{\_\ξ}}_x\left(x\,y\,u\,v\right)=1\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\η}}_u\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\η}}_v\left(x\,y\,u\,v\right)=0\right],\left[{\mathrm{\_\ξ}}_x\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\,v\right)=y\,{\mathrm{\_\η}}_u\left(x\,y\,u\,v\right)=-u\,{\mathrm{\_\η}}_v\left(x\,y\,u\,v\right)=-v\right],\left[{\mathrm{\_\ξ}}_x\left(x\,y\,u\,v\right)=x\,{\mathrm{\_\ξ}}_y\left(x\,y\,u\,v\right)=0\,{\mathrm{\_\η}}_u\left(x\,y\,u\,v\right)=-5u\,{\mathrm{\_\η}}_v\left(x\,y\,u\,v\right)=-3v\right]$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\right]\,J\,2\right)$

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

$\mathrm{convert}\left(\left[\mathrm{Sym}\right]\,\mathrm{DGvector}\right)$

$\left[\mathrm{D\_y}\,\mathrm{D\_x}\,y\mathrm{D\_y}-u\left[\right]\mathrm{D\_u}\left[\right]-v\left[\right]\mathrm{D\_v}\left[\right]\,x\mathrm{D\_x}-5u\left[\right]\mathrm{D\_u}\left[\right]-3v\left[\right]\mathrm{D\_v}\left[\right]\right]$

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{T1}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\&t\mathrm{D\_x}\right)\&t\mathrm{dy}+\left(\mathrm{dz}\&t\mathrm{D\_z}\right)\&t\mathrm{dx}\right)$

$\mathrm{T1}≔2\mathrm{dx}\mathrm{D\_x}\mathrm{dy}+\mathrm{dz}\mathrm{D\_z}\mathrm{dx}$

$S≔\mathrm{ContractIndices}\left(\mathrm{T1}\,\left[\left[1\,2\right]\right]\right)$

$S≔\mathrm{dx}+2\mathrm{dy}$

$X≔\mathrm{convert}\left(S\,\mathrm{DGform}\right)$

$X≔\mathrm{dx}+2\mathrm{dy}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(S\,''ObjectType''\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(X\,''ObjectType''\right)$

$''tensor'',''form''$

$\mathrm{lprint}\left(S\right),\mathrm{lprint}\left(X\right)$

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{Jet}\,2\right)\:$

$\mathrm{\θ1},\mathrm{\θ2},\mathrm{\θ3}≔\mathrm{Cu}\left[\right],\mathrm{Cu}\left[1\right],\mathrm{Cu}\left[2\right]$

$\mathrm{\θ1},\mathrm{\θ2},\mathrm{\θ3}≔\mathrm{Cu}\left[\right],{\mathrm{Cu}}_1,{\mathrm{Cu}}_2$

$\mathrm{convert}\left(\mathrm{\θ1}\,\mathrm{DGform}\right)$

$-u_1\mathrm{dx}-u_2\mathrm{dy}+\mathrm{du}\left[\right]$

$\mathrm{convert}\left(\mathrm{\θ2}\,\mathrm{DGform}\right)$

$-u_{1,1}\mathrm{dx}-u_{1,2}\mathrm{dy}+{\mathrm{du}}_1$

$\mathrm{convert}\left(\mathrm{\θ3}\,\mathrm{DGform}\right)$

$-u_{1,2}\mathrm{dx}-u_{2,2}\mathrm{dy}+{\mathrm{du}}_2$

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

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

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

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$

$\mathrm{\sigma }≔\frac{\sqrt{2}\mathrm{dt}}{2}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{\sqrt{2}\mathrm{dt}}{2}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{\sqrt{2}\mathrm{dx}}{2}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}\mathrm{dx}}{2}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}\mathrm{dz}}{2}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{\sqrt{2}\mathrm{dz}}{2}\mathrm{D\_z2}\mathrm{D\_w2}$

$X≔\mathrm{evalDG}\left(a\mathrm{D\_t}+b\mathrm{D\_y}\right)$

$X≔a\mathrm{D\_t}+b\mathrm{D\_y}$

$\mathrm{\psi }≔\mathrm{convert}\left(X\,\mathrm{DGspinor}\,\mathrm{\sigma }\,\left[1\right]\right)$

$\mathrm{\psi }≔\frac{\sqrt{2}a\mathrm{D\_z1}}{2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}b\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}b\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}a\mathrm{D\_z2}}{2}\mathrm{D\_w2}$

$T≔\mathrm{evalDG}\left(\left(\mathrm{D\_x}\&t\mathrm{D\_y}\right)\&t\mathrm{dz}\right)$

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

$\mathrm{\chi }≔\mathrm{convert}\left(T\,\mathrm{DGspinor}\,\mathrm{\sigma }\,\left[1\,3\right]\right)$

$\mathrm{\chi }≔\frac{\mathrm{D\_y}}{2}\mathrm{D\_z1}\mathrm{D\_w2}\mathrm{dz1}\mathrm{dw1}-\frac{\mathrm{D\_y}}{2}\mathrm{D\_z1}\mathrm{D\_w2}\mathrm{dz2}\mathrm{dw2}+\frac{\mathrm{D\_y}}{2}\mathrm{D\_z2}\mathrm{D\_w1}\mathrm{dz1}\mathrm{dw1}-\frac{\mathrm{D\_y}}{2}\mathrm{D\_z2}\mathrm{D\_w1}\mathrm{dz2}\mathrm{dw2}$

$\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[1\,2\right]\right]\right)$

$a\mathrm{D\_t}+b\mathrm{D\_y}$

$\mathrm{convert}\left(\mathrm{\chi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[2\,3\right]\,\left[4\,5\right]\right]\right)$

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

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{A1}≔\mathrm{Matrix}\left(\left[\left[1\,0\,2\right]\,\left[0\,2\,4\right]\,\left[1\,0\,0\right]\right]\right)$

$\mathrm{A1}≔\left[\right]$

$\mathrm{convert}\left(\mathrm{A1}\,\mathrm{DGtensor}\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right)$

$\mathrm{D\_x}\mathrm{D\_x}+2\mathrm{D\_x}\mathrm{D\_z}+2\mathrm{D\_y}\mathrm{D\_y}+4\mathrm{D\_y}\mathrm{D\_z}+\mathrm{D\_z}\mathrm{D\_x}$

$\mathrm{convert}\left(\mathrm{A1}\,\mathrm{DGtensor}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right)$

$\mathrm{dx}\mathrm{D\_x}+2\mathrm{dx}\mathrm{D\_z}+2\mathrm{dy}\mathrm{D\_y}+4\mathrm{dy}\mathrm{D\_z}+\mathrm{dz}\mathrm{D\_x}$

$\mathrm{convert}\left(\mathrm{A1}\,\mathrm{DGtensor}\,\left[\left[''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right)$

$\mathrm{D\_x}\mathrm{dx}+2\mathrm{D\_x}\mathrm{dz}+2\mathrm{D\_y}\mathrm{dy}+4\mathrm{D\_y}\mathrm{dz}+\mathrm{D\_z}\mathrm{dx}$

$\mathrm{convert}\left(\mathrm{A1}\,\mathrm{DGtensor}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right)$

$\mathrm{dx}\mathrm{dx}+2\mathrm{dx}\mathrm{dz}+2\mathrm{dy}\mathrm{dy}+4\mathrm{dy}\mathrm{dz}+\mathrm{dz}\mathrm{dx}$

$\mathrm{A2}≔\mathrm{Array}\left(1..3\,1..3\,1..3\,1..3\right)\:$

$\mathrm{A2}\left[1,2,2,3\right]≔m$

${\mathrm{A2}}_{1,2,2,3}≔m$

$\mathrm{A2}\left[1,3,2,1\right]≔n$

${\mathrm{A2}}_{1,3,2,1}≔n$

$\mathrm{convert}\left(\mathrm{A2}\,\mathrm{DGtensor}\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right)$

$m\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\mathrm{dz}+n\mathrm{D\_x}\mathrm{dz}\mathrm{dy}\mathrm{dx}$

$X≔\mathrm{evalDG}\left(x^2\mathrm{D\_y}-y^2\mathrm{D\_z}\right)$

$X≔x^2\mathrm{D\_y}-y^2\mathrm{D\_z}$

$T≔\mathrm{convert}\left(X\,\mathrm{DGtensor}\right)$

$T≔x^2\mathrm{D\_y}-y^2\mathrm{D\_z}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(X\,''ObjectType''\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(T\,''ObjectType''\right)$

$''vector'',''tensor''$

$\mathrm{lprint}\left(X\right),\mathrm{lprint}\left(T\right)$

_DG([["vector", M, []], [[[2], x^2], [[3], -y^2]]])
_DG([["tensor", M, [["con_bas"], []]], [[[2], x^2], [[3], -y^2]]])

$\mathrm{\omega }≔\mathrm{evalDG}\left(y\mathrm{dx}\&w\mathrm{dy}-z\mathrm{dy}\&w\mathrm{dz}\right)$

$\mathrm{\omega }≔y\mathrm{dx}\bigwedge \mathrm{dy}-z\mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{convert}\left(\mathrm{\omega }\,\mathrm{DGtensor}\right)$

$y\mathrm{dx}\mathrm{dy}-y\mathrm{dy}\mathrm{dx}-z\mathrm{dy}\mathrm{dz}+z\mathrm{dz}\mathrm{dy}$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g≔\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

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

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

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$

$\mathrm{\sigma }≔\frac{\sqrt{2}\mathrm{dt}}{2}\mathrm{D\_z1}\mathrm{D\_w1}+\frac{\sqrt{2}\mathrm{dt}}{2}\mathrm{D\_z2}\mathrm{D\_w2}+\frac{\sqrt{2}\mathrm{dx}}{2}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\sqrt{2}\mathrm{dx}}{2}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}+\frac{\sqrt{2}\mathrm{dz}}{2}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{\sqrt{2}\mathrm{dz}}{2}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{\psi }≔\mathrm{evalDG}\left(-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_z1}\&t\mathrm{D\_w2}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\mathrm{D\_z2}\&t\mathrm{D\_w1}\right)$

$\mathrm{\psi }≔-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_z2}\mathrm{D\_w1}$

$X≔\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[1\,2\right]\right]\right)$

$X≔\mathrm{D\_y}$

$\mathrm{\psi }≔\mathrm{evalDG}\left(2^{\frac{1}{2}}\mathrm{dz2}\&t\mathrm{dw2}\right)$

$\mathrm{\psi }≔\sqrt{2}\mathrm{dz2}\mathrm{dw2}$

$\mathrm{\omega }≔\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[1\,2\right]\right]\right)$

$\mathrm{\omega }≔\mathrm{dt}-\mathrm{dz}$

$\mathrm{\psi }≔\mathrm{evalDG}\left(\left(\left(\mathrm{dz1}\&t\mathrm{dw1}\right)\&t\mathrm{dz2}\right)\&t\mathrm{dw2}-\left(\left(\mathrm{dz1}\&t\mathrm{dw2}\right)\&t\mathrm{dz2}\right)\&t\mathrm{dw1}-\left(\left(\mathrm{dz2}\&t\mathrm{dw1}\right)\&t\mathrm{dz1}\right)\&t\mathrm{dw2}+\left(\left(\mathrm{dz2}\&t\mathrm{dw2}\right)\&t\mathrm{dz1}\right)\&t\mathrm{dw1}\right)$

$\mathrm{\psi }≔\mathrm{dz1}\mathrm{dw1}\mathrm{dz2}\mathrm{dw2}-\mathrm{dz1}\mathrm{dw2}\mathrm{dz2}\mathrm{dw1}-\mathrm{dz2}\mathrm{dw1}\mathrm{dz1}\mathrm{dw2}+\mathrm{dz2}\mathrm{dw2}\mathrm{dz1}\mathrm{dw1}$

$G≔\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

$G≔\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{\psi }≔\mathrm{evalDG}\left(-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\left(\left(\mathrm{D\_t}\&t\mathrm{D\_z1}\right)\&t\left(\&t\left(\mathrm{D\_z1}\right)\right)\right)\&t\mathrm{D\_w2}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}\left(\left(\mathrm{D\_x}\&t\mathrm{D\_z2}\right)\&t\mathrm{D\_z2}\right)\&t\mathrm{D\_w1}\right)$

$\mathrm{\psi }≔-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_t}\mathrm{D\_z1}\mathrm{D\_z1}\mathrm{D\_w2}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_x}\mathrm{D\_z2}\mathrm{D\_z2}\mathrm{D\_w1}$

$\mathrm{T1}≔\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[2\,4\right]\right]\right)$

$\mathrm{T1}≔-\frac{\mathrm{I}}{2}\mathrm{D\_t}\mathrm{D\_x}\mathrm{D\_z1}+\frac{\mathrm{D\_t}}{2}\mathrm{D\_y}\mathrm{D\_z1}+\frac{\mathrm{I}}{2}\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_z2}+\frac{\mathrm{D\_x}}{2}\mathrm{D\_y}\mathrm{D\_z2}$

$\mathrm{T2}≔\mathrm{convert}\left(\mathrm{\psi }\,\mathrm{DGtensor}\,\mathrm{\sigma }\,\left[\left[3\,4\right]\right]\right)$

$\mathrm{T2}≔-\frac{\mathrm{I}}{2}\mathrm{D\_t}\mathrm{D\_x}\mathrm{D\_z1}+\frac{\mathrm{D\_t}}{2}\mathrm{D\_y}\mathrm{D\_z1}+\frac{\mathrm{I}}{2}\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_z2}+\frac{\mathrm{D\_x}}{2}\mathrm{D\_y}\mathrm{D\_z2}$

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Library}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{Jet}\,2\right)\:$

$L≔\mathrm{diff}\left(u\left(x\,y\right)\,x\,x\right)+\mathrm{diff}\left(u\left(x\,y\right)\,y\,y\right)$

$L≔\frac{{\partial }^2}{\partial x^2}u\left(x\,y\right)+\frac{{\partial }^2}{\partial y^2}u\left(x\,y\right)$

$\mathrm{convert}\left(L\,\mathrm{DGjet}\right)$

$u_{1,1}+u_{2,2}$

We retrieve a 4th order ODE from Kamke and rewrite it in standard index notation for coordinates on jet spaces.

$\mathrm{DGsetup}\left(\left[x\right]\,\left[y\right]\,J\,4\right)$

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

$\mathrm{ODE}≔\mathrm{Retrieve}\left(''Kamke''\,1\,\left[4\,8\right]\,\mathrm{variables}=\left[x\,y\right]\right)$

$\mathrm{ODE}≔\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)+\left(a{x}^2+b\mathrm{\lambda }+c\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(\mathrm{\alpha }{x}^2+\mathrm{\beta }\mathrm{\lambda }+\mathrm{\gamma }\right)y\left(x\right)$

$\mathrm{convert}\left(\mathrm{ODE}\,\mathrm{DGjet}\right)$

$y_{1,1,1,1}+\left(a{x}^2+b\mathrm{\lambda }+c\right)y_{1,1}+\left(\mathrm{\alpha }{x}^2+\mathrm{\beta }\mathrm{\lambda }+\mathrm{\gamma }\right)y\left[\right]$

$\mathrm{DGsetup}\left(\left[t\,x\right]\,\left[u\right]\,J\,3\right)\:$

$\mathrm{\Delta }≔u\left[1\right]=u\left[2,2,2\right]+u\left[\right]u\left[2\right]$

$\mathrm{\Delta }≔u_1=u\left[\right]u_2+u_{2,2,2}$

$\mathrm{convert}\left(\mathrm{\Delta }\,\mathrm{DGdiff}\right)$

$\frac{\partial }{\partial t}u\left(t\,x\right)=u\left(t\,x\right)\left(\frac{\partial }{\partial x}u\left(t\,x\right)\right)+\frac{{\partial }^3}{\partial x^3}u\left(t\,x\right)$

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{Jet}\,2\right)\:$

$\mathrm{\omega }≔\mathrm{evalDG}\left(\mathrm{du}\left[1\right]\&w\mathrm{du}\left[2\right]\right)$

$\mathrm{\omega }≔{\mathrm{du}}_1\bigwedge {\mathrm{du}}_2$

$\mathrm{convert}\left(\mathrm{\omega }\,\mathrm{DGbiform}\,\left[2\,0\right]\right)$

$\left(u_{1,1}{u}_{2,2}-u_{1,2}^2\right)\mathrm{Dx}\bigwedge \mathrm{Dy}$

$\mathrm{convert}\left(\mathrm{\omega }\,\mathrm{DGbiform}\,\left[1\,1\right]\right)$

$-u_{1,2}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1+u_{1,1}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_2-u_{2,2}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_1+u_{1,2}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_2$

$\mathrm{convert}\left(\mathrm{\omega }\,\mathrm{DGbiform}\,\left[0\,2\right]\right)$

${\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_2$

$\mathrm{convert}\left(\mathrm{\omega }\,\mathrm{DGbiform}\right)$

$\left[{\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_2\,-u_{1,2}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1+u_{1,1}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_2-u_{2,2}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_1+u_{1,2}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_2\,\left(u_{1,1}{u}_{2,2}-u_{1,2}^2\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\right]$