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convert routines of the DifferentialGeometry package

 Calling Sequence convert(A, DGArray) convert(B, DGvector) convert(C, DGform) convert(C, DGspinor) convert(E, DGtensor) convert(F, DGjet) convert(F, DGdiff) convert(G, DGbiform, bidegree)

Parameters

 A - a tensor B - a Matrix, a tensor, or the infinitesimal symmetry generators for a system of differential equations, as calculated by PDEtools:-Infinitesimals C - a differential form or a differential biform E - an Array, a vector, a differential form, or a spinor-tensor F - a Maple expression G - a differential form

Description

 • convert(A, DGArray) converts a rank r tensor A to an r-dimensional array.  For example, if A= a * dx1 &t dx2 &t dx3 and T = convert(A, DGArray), then T[1, 2, 3] = a and all other entries of T are 0.
 • If B is a square matrix, then convert(B, DGvector) returns a linear vector field.
 • If B is a a rank 1 contravariant tensor, then convert(B, DGvector) returns the corresponding vector field.
 • If B is a list of infinitesimal symmetry generators for a system of differential equations, as calculated by PDEtools:-Infinitesimals, then convert(B, DGvector) returns a list of vector fields.
 • If C is a rank 1 covariant tensor, then convert(B, DGform) returns a differential 1-form.
 • If C is a differential biform, that is, a form expressed in terms of the contact forms on a jet space, then convert(C, DGform) returns the differential form on jet space obtained by replacing the contact forms by their coordinate formulas.
 • If E is a spin-tensor, then convert(E, DGspinor, sigma, indexlist) converts tensorial indices of E to pairs of spinorial indices.
 • If E is an Array, then convert(E, DGtensor, indextype) converts E to an r-dimensional array.
 • If E is a vector field, then convert(E, DGtensor) converts E to a rank 1 contravariant tensor.
 • If E is a differential p-form, then convert(E, DGtensor) converts E to a rank p contravariant tensor.
 • If E is a spinor-tensor, convert(E, DGtensor, sigma, indexlist) converts pairs of spinorial indices of E to tensorial indices.
 • convert(F, DGjet) converts a Maple expression F involving the Maple command diff, applied to functions u(x, y, z, ...), v(x, y, z, ...), ... to the standard indexed notation for derivatives, for example, diff(u(x, y, z), x) -> u[1], diff(u(x, y, z), x, y, z, z) -> u[1, 2, 3, 3] etc.
 • convert(F, DGdiff) performs the inverse to the conversion command convert/DGjet: it converts expressions F involving the indexed notation for derivatives to expressions containing the Maple diff command.
 • convert(G, DGbiform) converts a differential p-form G, defined on a jet space J^k(M, N) , into a list of (p + 1)-biforms, Theta = [theta_0, theta_1, theta_2, .. theta_p], where theta_k has contact degree k, and G = theta_0 + theta_1 + theta_2 + ... + theta_p.

Examples

convert/DGArray

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

Example 1.

We define a manifold M with coordinates [x, y, z]. On M we define two tensors T1 and T2.  We convert these tensors to a Maple Array

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 > $\mathrm{T1}≔\mathrm{evalDG}\left(\mathrm{D_z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_x}+\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right)$
 ${\mathrm{T1}}{≔}{\mathrm{D_x}}{}{\mathrm{D_y}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}$ (1)
 > $\mathrm{T2}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\left(\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{T2}}{≔}{2}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}$ (2)
 > $\mathrm{A1}≔\mathrm{convert}\left(\mathrm{T1},\mathrm{DGArray}\right)$
 ${\mathrm{A1}}{≔}\left[\right]$ (3)
 > $\mathrm{A2}≔\mathrm{convert}\left(\mathrm{T2},\mathrm{DGArray}\right)$
 > $\mathrm{A2}\left[1,1,2\right]$
 ${2}$ (4)

convert/DGvector

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

Example 1.

We define a manifold M with coordinates [x, y, z].  We convert a matrix to a linear vector field.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,0,1\right],\left[0,2,0\right],\left[0,3,0\right]\right]\right)$
 ${A}{≔}\left[\right]$ (5)
 > $\mathrm{convert}\left(A,\mathrm{DGvector}\right)$
 $\left({x}{+}{z}\right){}{\mathrm{D_x}}{+}{2}{}{y}{}{\mathrm{D_y}}{+}{3}{}{y}{}{\mathrm{D_z}}$ (6)

Example 2.

We define a manifold M with coordinates [x, y, z].  On M we define two tensors T1 and T2 and contract the 1st and 3rd indices of T1 against the 1st and second indices of T2.  The result is a contravariant rank 1 tensor S on M.  We then convert S to a vector field X.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 > $\mathrm{T1}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\left(\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{T1}}{≔}{2}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}$ (7)
 > $\mathrm{T2}≔\mathrm{evalDG}\left(\mathrm{D_z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_x}+\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right)$
 ${\mathrm{T2}}{≔}{\mathrm{D_x}}{}{\mathrm{D_y}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}$ (8)
 > $S≔\mathrm{ContractIndices}\left(\mathrm{T1},\mathrm{T2},\left[\left[1,1\right],\left[3,2\right]\right]\right)$
 ${S}{≔}{2}{}{\mathrm{D_x}}{+}{\mathrm{D_z}}$ (9)
 > $X≔\mathrm{convert}\left(S,\mathrm{DGvector}\right)$
 ${X}{≔}{2}{}{\mathrm{D_x}}{+}{\mathrm{D_z}}$ (10)
 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(S,"ObjectType"\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(X,"ObjectType"\right)$
 ${"tensor"}{,}{"vector"}$ (11)
 > $\mathrm{lprint}\left(S\right),\mathrm{lprint}\left(X\right)$
 _DG([["tensor", M, [["con_bas"], []]], [[[1], 2], [[3], 1]]]) _DG([["vector", M, []], [[[1], 2], [[3], 1]]])

Example 3.

We use the PDEtools:-Infinitesimals program to find the infinitesimals symmetries of a system of differential equations.  We convert the output of this program, which lists of the components of the infinitesimal symmetries, to a list of vector fields on the manifold of independent and dependent variables.

 > $\mathrm{DE}≔\left[\mathrm{diff}\left(u\left(x,y\right),x,y\right)={v\left(x,y\right)}^{2},\mathrm{diff}\left(v\left(x,y\right),\mathrm{}\left(x,2\right)\right)=u\left(x,y\right)\right]$
 ${\mathrm{DE}}{≔}\left[\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){=}{{v}{}\left({x}{,}{y}\right)}^{{2}}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({x}{,}{y}\right){=}{u}{}\left({x}{,}{y}\right)\right]$ (12)
 > $\mathrm{Sym}≔\mathrm{PDEtools}:-\mathrm{Infinitesimals}\left(\mathrm{DE},\left[u,v\right]\right)$
 ${\mathrm{Sym}}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{1}{,}{{\mathrm{_η}}}_{{u}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_η}}}_{{v}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{1}{,}{{\mathrm{_ξ}}}_{{y}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_η}}}_{{u}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_η}}}_{{v}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_ξ}}}_{{y}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{y}{,}{{\mathrm{_η}}}_{{u}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{-}{u}{,}{{\mathrm{_η}}}_{{v}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{-}{v}\right]{,}\left[{{\mathrm{_ξ}}}_{{x}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{x}{,}{{\mathrm{_ξ}}}_{{y}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{0}{,}{{\mathrm{_η}}}_{{u}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{-}{5}{}{u}{,}{{\mathrm{_η}}}_{{v}}{}\left({x}{,}{y}{,}{u}{,}{v}\right){=}{-}{3}{}{v}\right]$ (13)
 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],J,2\right)$
 ${\mathrm{frame name: J}}$ (14)
 > $\mathrm{convert}\left(\left[\mathrm{Sym}\right],\mathrm{DGvector}\right)$
 $\left[{\mathrm{D_y}}{,}{\mathrm{D_x}}{,}{y}{}{\mathrm{D_y}}{-}{u}\left[\right]{}{\mathrm{D_u}}\left[\right]{-}{v}\left[\right]{}{\mathrm{D_v}}\left[\right]{,}{x}{}{\mathrm{D_x}}{-}{5}{}{u}\left[\right]{}{\mathrm{D_u}}\left[\right]{-}{3}{}{v}\left[\right]{}{\mathrm{D_v}}\left[\right]\right]$ (15)

convert/DGform

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

Example 1.

We define a manifold M with coordinates [x, y, z].  On M we define a tensor T and contract the 1st and 3rd indices of T.  The result is a covariant rank 1 tensor S on M.  We then convert S to a differential 1-form alpha.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 > $\mathrm{T1}≔\mathrm{evalDG}\left(2\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\left(\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{T1}}{≔}{2}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}$ (16)
 > $S≔\mathrm{ContractIndices}\left(\mathrm{T1},\left[\left[1,2\right]\right]\right)$
 ${S}{≔}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}$ (17)
 > $X≔\mathrm{convert}\left(S,\mathrm{DGform}\right)$
 ${X}{≔}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}$ (18)

The tensor S and the 1 form alpha both have the same external displays but are have different internal representations.  We can see this using DGinfo or lprint.

 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(S,"ObjectType"\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(X,"ObjectType"\right)$
 ${"tensor"}{,}{"form"}$ (19)
 > $\mathrm{lprint}\left(S\right),\mathrm{lprint}\left(X\right)$
 _DG([["tensor", M, [["cov_bas"], []]], [[[1], 1], [[2], 2]]]) _DG([["form", M, 1], [[[1], 1], [[2], 2]]])

Example 2.

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{Jet},2\right):$
 > $\mathrm{θ1},\mathrm{θ2},\mathrm{θ3}≔\mathrm{Cu}\left[\right],\mathrm{Cu}\left[1\right],\mathrm{Cu}\left[2\right]$
 ${\mathrm{θ1}}{,}{\mathrm{θ2}}{,}{\mathrm{θ3}}{≔}{\mathrm{Cu}}\left[\right]{,}{{\mathrm{Cu}}}_{{1}}{,}{{\mathrm{Cu}}}_{{2}}$ (20)
 > $\mathrm{convert}\left(\mathrm{θ1},\mathrm{DGform}\right)$
 ${-}{{u}}_{{1}}{}{\mathrm{dx}}{-}{{u}}_{{2}}{}{\mathrm{dy}}{+}{\mathrm{du}}\left[\right]$ (21)
 > $\mathrm{convert}\left(\mathrm{θ2},\mathrm{DGform}\right)$
 ${-}{{u}}_{{1}{,}{1}}{}{\mathrm{dx}}{-}{{u}}_{{1}{,}{2}}{}{\mathrm{dy}}{+}{{\mathrm{du}}}_{{1}}$ (22)
 > $\mathrm{convert}\left(\mathrm{θ3},\mathrm{DGform}\right)$
 ${-}{{u}}_{{1}{,}{2}}{}{\mathrm{dx}}{-}{{u}}_{{2}{,}{2}}{}{\mathrm{dy}}{+}{{\mathrm{du}}}_{{2}}$ (23)

convert/DGspinor

The solder form sigma is the fundamental spin-tensor in spinor algebra.  It provides for a 1-1 correspondence between vectors and rank 2 Hermitian spinors. The convert/DGspinor uses a user-specified solder form to convert tensor indices of a given spinor-tensor to pairs of spinor indices.

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

To illustrate this convert procedure we first construct a solder form. Define a vector bundle over M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (24)

Define a spacetime metric g on M. Our spinor conventions assume the metric has signature [1, -1, -1, -1].

 > $g≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{≔}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (25)

Define an orthonormal frame on M with respect to the metric g.

 > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{≔}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (26)

Calculate the solder form sigma from the frame F.  See Spinor, SolderForm.

 > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{\sigma }}{≔}\frac{\sqrt{{2}}{}{\mathrm{dt}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}{}{\mathrm{dt}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}{}{\mathrm{dx}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}{}{\mathrm{dx}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}{}{\mathrm{dz}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{\sqrt{{2}}{}{\mathrm{dz}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (27)

Example 1.

Define a vector X and convert it to a contravariant rank 2 spinor psi.  The convert/DGspinor command requires the solder form as a 3rd argument and a list of tensorial indices to be converted to spinorial indices as a 4th argument.

 > $X≔\mathrm{evalDG}\left(a\mathrm{D_t}+b\mathrm{D_y}\right)$
 ${X}{≔}{a}{}{\mathrm{D_t}}{+}{b}{}{\mathrm{D_y}}$ (28)
 > $\mathrm{\psi }≔\mathrm{convert}\left(X,\mathrm{DGspinor},\mathrm{\sigma },\left[1\right]\right)$
 ${\mathrm{\psi }}{≔}\frac{\sqrt{{2}}{}{a}{}{\mathrm{D_z1}}}{{2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{b}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{b}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}{}{a}{}{\mathrm{D_z2}}}{{2}}{}{\mathrm{D_w2}}$ (29)

Example 2.

Define a rank 3 tensor and convert the 1st and 3rd indices to pairs of spinor indices to obtain a rank 5 spin-tensor chi. Note that the spinorial indices are moved to the right.

 > $T≔\mathrm{evalDG}\left(\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${T}{≔}{\mathrm{D_x}}{}{\mathrm{D_y}}{}{\mathrm{dz}}$ (30)
 > $\mathrm{\chi }≔\mathrm{convert}\left(T,\mathrm{DGspinor},\mathrm{\sigma },\left[1,3\right]\right)$
 ${\mathrm{\chi }}{≔}\frac{{\mathrm{D_y}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{-}\frac{{\mathrm{D_y}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{+}\frac{{\mathrm{D_y}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{-}\frac{{\mathrm{D_y}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}$ (31)

Example 3.

With the convert/DGspinor command, the spinor psi. can be converted back to the vector X and the spin-tensor chi back to the tensor T.

 > $\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[1,2\right]\right]\right)$
 ${a}{}{\mathrm{D_t}}{+}{b}{}{\mathrm{D_y}}$ (32)
 > $\mathrm{convert}\left(\mathrm{\chi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[2,3\right],\left[4,5\right]\right]\right)$
 ${\mathrm{D_y}}{}{\mathrm{D_x}}{}{\mathrm{dz}}$ (33)

convert/DGtensor

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

We define a manifold M with coordinates [x, y, z].

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

Example 1.

We define a Matrix' A1 and convert it to a rank 2 tensor in 4 different ways:

 > $\mathrm{A1}≔\mathrm{Matrix}\left(\left[\left[1,0,2\right],\left[0,2,4\right],\left[1,0,0\right]\right]\right)$
 ${\mathrm{A1}}{≔}\left[\right]$ (34)
 > $\mathrm{convert}\left(\mathrm{A1},\mathrm{DGtensor},\left[\left["con_bas","con_bas"\right],\left[\right]\right]\right)$
 ${\mathrm{D_x}}{}{\mathrm{D_x}}{+}{2}{}{\mathrm{D_x}}{}{\mathrm{D_z}}{+}{2}{}{\mathrm{D_y}}{}{\mathrm{D_y}}{+}{4}{}{\mathrm{D_y}}{}{\mathrm{D_z}}{+}{\mathrm{D_z}}{}{\mathrm{D_x}}$ (35)
 > $\mathrm{convert}\left(\mathrm{A1},\mathrm{DGtensor},\left[\left["cov_bas","con_bas"\right],\left[\right]\right]\right)$
 ${\mathrm{dx}}{}{\mathrm{D_x}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{D_z}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{D_y}}{+}{4}{}{\mathrm{dy}}{}{\mathrm{D_z}}{+}{\mathrm{dz}}{}{\mathrm{D_x}}$ (36)
 > $\mathrm{convert}\left(\mathrm{A1},\mathrm{DGtensor},\left[\left["con_bas","cov_bas"\right],\left[\right]\right]\right)$
 ${\mathrm{D_x}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{D_x}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{D_y}}{}{\mathrm{dy}}{+}{4}{}{\mathrm{D_y}}{}{\mathrm{dz}}{+}{\mathrm{D_z}}{}{\mathrm{dx}}$ (37)
 > $\mathrm{convert}\left(\mathrm{A1},\mathrm{DGtensor},\left[\left["cov_bas","cov_bas"\right],\left[\right]\right]\right)$
 ${\mathrm{dx}}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{2}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{4}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{dx}}$ (38)

Example 2.

We define a four-dimensional array and convert it to a rank 4 covariant tensor.

 > $\mathrm{A2}≔\mathrm{Array}\left(1..3,1..3,1..3,1..3\right):$
 > $\mathrm{A2}\left[1,2,2,3\right]≔m$
 ${{\mathrm{A2}}}_{{1}{,}{2}{,}{2}{,}{3}}{≔}{m}$ (39)
 > $\mathrm{A2}\left[1,3,2,1\right]≔n$
 ${{\mathrm{A2}}}_{{1}{,}{3}{,}{2}{,}{1}}{≔}{n}$ (40)
 > $\mathrm{convert}\left(\mathrm{A2},\mathrm{DGtensor},\left[\left["con_bas","cov_bas","cov_bas","cov_bas"\right],\left[\right]\right]\right)$
 ${m}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{n}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (41)

Example 3.

We define a vector field and convert it to a rank 1 contravariant tensor field.

 > $X≔\mathrm{evalDG}\left({x}^{2}\mathrm{D_y}-{y}^{2}\mathrm{D_z}\right)$
 ${X}{≔}{{x}}^{{2}}{}{\mathrm{D_y}}{-}{{y}}^{{2}}{}{\mathrm{D_z}}$ (42)
 > $T≔\mathrm{convert}\left(X,\mathrm{DGtensor}\right)$
 ${T}{≔}{{x}}^{{2}}{}{\mathrm{D_y}}{-}{{y}}^{{2}}{}{\mathrm{D_z}}$ (43)

The tensor T and vector X both have the same external displays but have different internal representations.  We can see this using DGinfo or lprint.

 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(X,"ObjectType"\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(T,"ObjectType"\right)$
 ${"vector"}{,}{"tensor"}$ (44)
 > $\mathrm{lprint}\left(X\right),\mathrm{lprint}\left(T\right)$
 _DG([["vector", M, []], [[[2], x^2], [[3], -y^2]]]) _DG([["tensor", M, [["con_bas"], []]], [[[2], x^2], [[3], -y^2]]])

Example 4.

We define a differential 2-form and convert it to a rank 2 covariant tensor field.

 > $\mathrm{\omega }≔\mathrm{evalDG}\left(y\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-z\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${\mathrm{\omega }}{≔}{y}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{-}{z}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (45)
 > $\mathrm{convert}\left(\mathrm{\omega },\mathrm{DGtensor}\right)$
 ${y}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{y}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}{z}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}{z}{}{\mathrm{dz}}{}{\mathrm{dy}}$ (46)

Example 5.

A spinor or spinor-tensor can be converted to a tensor using a solder form sigma. To work with spinors, define a vector bundle over M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (47)

Define a spacetime metric g on M. Our spinor convention require that the metric has signature [1, -1,-1,-1].

 > $g≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{≔}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (48)

Define an orthonormal frame on M with respect to the metric g.

 > $F≔\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{≔}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (49)

Calculate the solder form sigma from the frame F.  See Spinor, SolderForm.

 > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{\sigma }}{≔}\frac{\sqrt{{2}}{}{\mathrm{dt}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}{}{\mathrm{dt}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}{}{\mathrm{dx}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}{}{\mathrm{dx}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}{}{\mathrm{dz}}}{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{\sqrt{{2}}{}{\mathrm{dz}}}{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (50)

Define a contravariant rank 2 spinor psi and convert it to a rank 1 contravariant tensor X.  In this situation, convert/DGtensor command requires the solder form as a 3rd argument and a list of pairs of spinorial indices to be converted to tensorial indices as a 4th argument.

 > $\mathrm{\psi }≔\mathrm{evalDG}\left(-\frac{1}{2}I{2}^{\frac{1}{2}}\mathrm{D_z1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w2}+\frac{1}{2}I{2}^{\frac{1}{2}}\mathrm{D_z2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w1}\right)$
 ${\mathrm{\psi }}{≔}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}$ (51)
 > $X≔\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[1,2\right]\right]\right)$
 ${X}{≔}{\mathrm{D_y}}$ (52)

Example 6.

Define a covariant rank 2 spinor psi and convert it to a rank 1 covariant tensor omega.

 > $\mathrm{\psi }≔\mathrm{evalDG}\left({2}^{\frac{1}{2}}\mathrm{dz2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)$
 ${\mathrm{\psi }}{≔}\sqrt{{2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}$ (53)
 > $\mathrm{\omega }≔\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[1,2\right]\right]\right)$
 ${\mathrm{\omega }}{≔}{\mathrm{dt}}{-}{\mathrm{dz}}$ (54)

Example 7.

Define a covariant rank 4 spinor psi and convert it to a rank 2 covariant tensor G.

 > $\mathrm{\psi }≔\mathrm{evalDG}\left(\left(\left(\mathrm{dz1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}-\left(\left(\mathrm{dz1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}-\left(\left(\mathrm{dz2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}+\left(\left(\mathrm{dz2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dw1}\right)$
 ${\mathrm{\psi }}{≔}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{-}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{-}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{+}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}$ (55)
 > $G≔\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[1,2\right],\left[3,4\right]\right]\right)$
 ${G}{≔}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (56)

Example 8.

Define a mixed contravariant rank 5 spin-tensor psi and convert it to a rank 3 contravariant tensor in 2 different ways. (The tensors T1 and T2 are complex because psi is not Hermitian.)

 > $\mathrm{\psi }≔\mathrm{evalDG}\left(-\frac{1}{2}I{2}^{\frac{1}{2}}\left(\left(\mathrm{D_t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(&t\left(\mathrm{D_z1}\right)\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w2}+\frac{1}{2}I{2}^{\frac{1}{2}}\left(\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_w1}\right)$
 ${\mathrm{\psi }}{≔}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}$ (57)
 > $\mathrm{T1}≔\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[2,4\right]\right]\right)$
 ${\mathrm{T1}}{≔}{-}\frac{{I}}{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{+}\frac{{\mathrm{D_t}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{+}\frac{{I}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{+}\frac{{\mathrm{D_x}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}$ (58)
 > $\mathrm{T2}≔\mathrm{convert}\left(\mathrm{\psi },\mathrm{DGtensor},\mathrm{\sigma },\left[\left[3,4\right]\right]\right)$
 ${\mathrm{T2}}{≔}{-}\frac{{I}}{{2}}{}{\mathrm{D_t}}{}{\mathrm{D_x}}{}{\mathrm{D_z1}}{+}\frac{{\mathrm{D_t}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z1}}{+}\frac{{I}}{{2}}{}{\mathrm{D_x}}{}{\mathrm{D_x}}{}{\mathrm{D_z2}}{+}\frac{{\mathrm{D_x}}}{{2}}{}{\mathrm{D_y}}{}{\mathrm{D_z2}}$ (59)

convert/DGjet, convert/DGdiff

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

Example 1.

We define the 2nd order jet space J^2(R^2, R) for one function u of two independent variables [x, y].

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

We convert the Laplace of a function, defined using the Maple diff command into a standard index notation for coordinates on jet spaces.

 > $L≔\mathrm{diff}\left(u\left(x,y\right),x,x\right)+\mathrm{diff}\left(u\left(x,y\right),y,y\right)$
 ${L}{≔}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{y}\right)$ (60)
 > $\mathrm{convert}\left(L,\mathrm{DGjet}\right)$
 ${{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}$ (61)

Example 2.

 • We retrieve a 4th order ODE from Kamke and rewrite it in standard index notation for coordinates on jet spaces.
 > $\mathrm{DGsetup}\left(\left[x\right],\left[y\right],J,4\right)$
 ${\mathrm{frame name: J}}$ (62)
 > $\mathrm{ODE}≔\mathrm{Retrieve}\left("Kamke",1,\left[4,8\right],\mathrm{variables}=\left[x,y\right]\right)$
 ${\mathrm{ODE}}{≔}\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\left({a}{}{{x}}^{{2}}{+}{b}{}{\mathrm{\lambda }}{+}{c}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({\mathrm{\alpha }}{}{{x}}^{{2}}{+}{\mathrm{\beta }}{}{\mathrm{\lambda }}{+}{\mathrm{\gamma }}\right){}{y}{}\left({x}\right)$ (63)
 > $\mathrm{convert}\left(\mathrm{ODE},\mathrm{DGjet}\right)$
 ${{y}}_{{1}{,}{1}{,}{1}{,}{1}}{+}\left({a}{}{{x}}^{{2}}{+}{b}{}{\mathrm{\lambda }}{+}{c}\right){}{{y}}_{{1}{,}{1}}{+}\left({\mathrm{\alpha }}{}{{x}}^{{2}}{+}{\mathrm{\beta }}{}{\mathrm{\lambda }}{+}{\mathrm{\gamma }}\right){}{y}\left[\right]$ (64)

Example 3.

Define a 2-form on J^2(R^2, R). We convert the KdV equation, given in jet notation, to ordinary derivative notation.

 > $\mathrm{DGsetup}\left(\left[t,x\right],\left[u\right],J,3\right):$
 > $\mathrm{\Delta }≔u\left[1\right]=u\left[2,2,2\right]+u\left[\right]u\left[2\right]$
 ${\mathrm{\Delta }}{≔}{{u}}_{{1}}{=}{u}\left[\right]{}{{u}}_{{2}}{+}{{u}}_{{2}{,}{2}{,}{2}}$ (65)
 > $\mathrm{convert}\left(\mathrm{\Delta },\mathrm{DGdiff}\right)$
 $\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right){=}{u}{}\left({t}{,}{x}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right)\right){+}\frac{{{\partial }}^{{3}}}{{\partial }{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}{,}{x}\right)$ (66)

convert/DGbiform

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

Define a 2-form on J^2(R^2, R). We define the 2nd order jet space J^2(R^2, R) for one function u of two independent variables [x, y].

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

Define a 2-form on J^2(R^2, R).

 > $\mathrm{\omega }≔\mathrm{evalDG}\left(\mathrm{du}\left[1\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\left[2\right]\right)$
 ${\mathrm{\omega }}{≔}{{\mathrm{du}}}_{{1}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}}$ (67)

Define a 2-form on J^2(R^2, R). To obtain the representation of omega in terms of the contact forms Cu[1] = du[1] - u[1, 1]*dx - u[1, 2]*dy and Cu[2] = du[2] - u[1, 2]*dx - u[2, 2]*dy   (*) we solve the equations (*) for du[1] and du[2], substitute for du[1] and du[2] into omega, and then grade the terms according to their contact degree -- [2, 0] : not contact forms; [1,1] : linear in contact forms; [0, 2] : quadratic in contact forms.

 > $\mathrm{convert}\left(\mathrm{\omega },\mathrm{DGbiform},\left[2,0\right]\right)$
 $\left({{u}}_{{1}{,}{1}}{}{{u}}_{{2}{,}{2}}{-}{{u}}_{{1}{,}{2}}^{{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}$ (68)
 > $\mathrm{convert}\left(\mathrm{\omega },\mathrm{DGbiform},\left[1,1\right]\right)$
 ${-}{{u}}_{{1}{,}{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{{u}}_{{1}{,}{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{-}{{u}}_{{2}{,}{2}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{{u}}_{{1}{,}{2}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}$ (69)
 > $\mathrm{convert}\left(\mathrm{\omega },\mathrm{DGbiform},\left[0,2\right]\right)$
 ${{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}$ (70)
 > $\mathrm{convert}\left(\mathrm{\omega },\mathrm{DGbiform}\right)$
 $\left[{{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{,}{-}{{u}}_{{1}{,}{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{{u}}_{{1}{,}{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{-}{{u}}_{{2}{,}{2}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{{u}}_{{1}{,}{2}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{,}\left({{u}}_{{1}{,}{1}}{}{{u}}_{{2}{,}{2}}{-}{{u}}_{{1}{,}{2}}^{{2}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}\right]$ (71)

 See Also