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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

>	$with (DifferentialGeometry) &colon;$

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

>	$DGsetup ([x, y, z], M) &colon;$

>	$T1 ≔ evalDG (D_z &t D_x + D_x &t D_y)$

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

(1)

>	$T2 ≔ evalDG (2 (dx &t D_x) &t dy + (dz &t D_z) &t dx)$

$_DG ([[tensor, M, [[cov_bas, con_bas, cov_bas], []]], [[[1, 1, 2], 2], [[3, 3, 1], 1]]])$

(2)

>	$A1 ≔ convert (T1, DGArray)$

$[\begin{array}{r} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}]$

(3)

>	$A2 ≔ convert (T2, DGArray)$

>	${A2}_{1, 1, 2}$

$2$

(4)

convert/DGvector

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

Example 1.

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

>	$DGsetup ([x, y, z], M) &colon;$

>	$A ≔ Matrix ([[1, 0, 1], [0, 2, 0], [0, 3, 0]])$

$[\begin{array}{r} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 3 & 0 \end{array}]$

(5)

>	$convert (A, DGvector)$

$_DG ([[vector, M, []], [[[1], x + z], [[2], 2 y], [[3], 3 y]]])$

(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.

>	$DGsetup ([x, y, z], M) &colon;$

>	$T1 ≔ evalDG (2 (dx &t D_x) &t dy + (dz &t D_z) &t dx)$

$_DG ([[tensor, M, [[cov_bas, con_bas, cov_bas], []]], [[[1, 1, 2], 2], [[3, 3, 1], 1]]])$

(7)

>	$T2 ≔ evalDG (D_z &t D_x + D_x &t D_y)$

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

(8)

>	$S ≔ ContractIndices (T1, T2, [[1, 1], [3, 2]])$

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

(9)

>	$X ≔ convert (S, DGvector)$

$_DG ([[vector, M, []], [[[1], 2], [[3], 1]]])$

(10)

>	$Tools :- DGinfo (S, ObjectType), Tools :- DGinfo (X, ObjectType)$

$tensor, vector$

(11)

>	$lprint (S), lprint (X)$

_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.

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

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

(12)

>	$Sym ≔ PDEtools :- Infinitesimals (DE, [u, v])$

$Sym ≔ [{_ξ}_{x} (x, y, u, v) = 0, {_ξ}_{y} (x, y, u, v) = 1, {_η}_{u} (x, y, u, v) = 0, {_η}_{v} (x, y, u, v) = 0], [{_ξ}_{x} (x, y, u, v) = 1, {_ξ}_{y} (x, y, u, v) = 0, {_η}_{u} (x, y, u, v) = 0, {_η}_{v} (x, y, u, v) = 0], [{_ξ}_{x} (x, y, u, v) = 0, {_ξ}_{y} (x, y, u, v) = y, {_η}_{u} (x, y, u, v) = - u, {_η}_{v} (x, y, u, v) = - v], [{_ξ}_{x} (x, y, u, v) = x, {_ξ}_{y} (x, y, u, v) = 0, {_η}_{u} (x, y, u, v) = - 5 u, {_η}_{v} (x, y, u, v) = - 3 v]$

(13)

>	$DGsetup ([x, y], [u, v], J, 2)$

$frame name: J$

(14)

>	$convert ([Sym], DGvector)$

$[_DG ([[vector, J, []], [[[2], 1]]]),_DG ([[vector, J, []], [[[1], 1]]]),_DG ([[vector, J, []], [[[2], y], [[3], - u_{[]}], [[4], - v_{[]}]]]),_DG ([[vector, J, []], [[[1], x], [[3], - 5 u_{[]}], [[4], - 3 v_{[]}]]])]$

(15)

convert/DGform

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

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.

>	$DGsetup ([x, y, z], M) &colon;$

>	$T1 ≔ evalDG (2 (dx &t D_x) &t dy + (dz &t D_z) &t dx)$

$_DG ([[tensor, M, [[cov_bas, con_bas, cov_bas], []]], [[[1, 1, 2], 2], [[3, 3, 1], 1]]])$

(16)

>	$S ≔ ContractIndices (T1, [[1, 2]])$

$_DG ([[tensor, M, [[cov_bas], []]], [[[1], 1], [[2], 2]]])$

(17)

>	$X ≔ convert (S, DGform)$

$_DG ([[form, M, 1], [[[1], 1], [[2], 2]]])$

(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.

>	$Tools :- DGinfo (S, ObjectType), Tools :- DGinfo (X, ObjectType)$

$tensor, form$

(19)

>	$lprint (S), lprint (X)$

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

Example 2.

>	$DGsetup ([x, y], [u], Jet, 2) &colon;$

>	$θ1, θ2, θ3 ≔ {Cu}_{[]}, {Cu}_{1}, {Cu}_{2}$

$θ1, θ2, θ3 ≔_DG ([[biform, Jet, [0, 1]], [[[3], 1]]]),_DG ([[biform, Jet, [0, 1]], [[[4], 1]]]),_DG ([[biform, Jet, [0, 1]], [[[5], 1]]])$

(20)

>	$convert (θ1, DGform)$

$_DG ([[form, Jet, 1], [[[1], - u_{1}], [[2], - u_{2}], [[3], 1]]])$

(21)

>	$convert (θ2, DGform)$

$_DG ([[form, Jet, 1], [[[1], - u_{1, 1}], [[2], - u_{1, 2}], [[4], 1]]])$

(22)

>	$convert (θ3, DGform)$

$_DG ([[form, Jet, 1], [[[1], - u_{1, 2}], [[2], - u_{2, 2}], [[5], 1]]])$

(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.

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

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].

>	$DGsetup ([t, x, y, z], [z1, z2, w1, w2], M)$

$frame name: M$

(24)

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

>	$g ≔ evalDG (dt &t dt - dx &t dx - dy &t dy - dz &t dz)$

$_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], - 1], [[3, 3], - 1], [[4, 4], - 1]]])$

(25)

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

>	$F ≔ [D_t, D_x, D_y, D_z]$

$[_DG ([[vector, M, []], [[[1], 1]]]),_DG ([[vector, M, []], [[[2], 1]]]),_DG ([[vector, M, []], [[[3], 1]]]),_DG ([[vector, M, []], [[[4], 1]]])]$

(26)

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

>	$σ ≔ SolderForm (F)$

$_DG ([[tensor, M, [[cov_bas, con_vrt, con_vrt], []]], [[[1, 5, 7], \frac{1}{2} \sqrt{2}], [[1, 6, 8], \frac{1}{2} \sqrt{2}], [[2, 5, 8], \frac{1}{2} \sqrt{2}], [[2, 6, 7], \frac{1}{2} \sqrt{2}], [[3, 5, 8], - \frac{1}{2} I \sqrt{2}], [[3, 6, 7], \frac{1}{2} I \sqrt{2}], [[4, 5, 7], \frac{1}{2} \sqrt{2}], [[4, 6, 8], - \frac{1}{2} \sqrt{2}]]])$

(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.

>	$X ≔ evalDG (a D_t + b D_y)$

$_DG ([[vector, M, []], [[[1], a], [[3], b]]])$

(28)

>	$ψ ≔ convert (X, DGspinor, σ, [1])$

$_DG ([[tensor, M, [[con_vrt, con_vrt], []]], [[[5, 7], \frac{1}{2} \sqrt{2} a], [[5, 8], - \frac{1}{2} I \sqrt{2} b], [[6, 7], \frac{1}{2} I \sqrt{2} b], [[6, 8], \frac{1}{2} \sqrt{2} a]]])$

(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.

>	$T ≔ evalDG ((D_x &t D_y) &t dz)$

$_DG ([[tensor, M, [[con_bas, con_bas, cov_bas], []]], [[[2, 3, 4], 1]]])$

(30)

>	$χ ≔ convert (T, DGspinor, σ, [1, 3])$

$_DG ([[tensor, M, [[con_bas, con_vrt, con_vrt, cov_vrt, cov_vrt], []]], [[[3, 5, 8, 5, 7], \frac{1}{2}], [[3, 5, 8, 6, 8], - \frac{1}{2}], [[3, 6, 7, 5, 7], \frac{1}{2}], [[3, 6, 7, 6, 8], - \frac{1}{2}]]])$

(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.

>	$convert (ψ, DGtensor, σ, [[1, 2]])$

$_DG ([[tensor, M, [[con_bas], []]], [[[1], a], [[3], b]]])$

(32)

>	$convert (χ, DGtensor, σ, [[2, 3], [4, 5]])$

$_DG ([[tensor, M, [[con_bas, con_bas, cov_bas], []]], [[[3, 2, 4], 1]]])$

(33)

convert/DGtensor

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

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

>	$DGsetup ([x, y, z], M) &colon;$

Example 1.

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

>	$A1 ≔ Matrix ([[1, 0, 2], [0, 2, 4], [1, 0, 0]])$

$[\begin{array}{r} 1 & 0 & 2 \\ 0 & 2 & 4 \\ 1 & 0 & 0 \end{array}]$

(34)

>	$convert (A1, DGtensor, [[con_bas, con_bas], []])$

$_DG ([[tensor, M, [[con_bas, con_bas], []]], [[[1, 1], 1], [[1, 3], 2], [[2, 2], 2], [[2, 3], 4], [[3, 1], 1]]])$

(35)

>	$convert (A1, DGtensor, [[cov_bas, con_bas], []])$

$_DG ([[tensor, M, [[cov_bas, con_bas], []]], [[[1, 1], 1], [[1, 3], 2], [[2, 2], 2], [[2, 3], 4], [[3, 1], 1]]])$

(36)

>	$convert (A1, DGtensor, [[con_bas, cov_bas], []])$

$_DG ([[tensor, M, [[con_bas, cov_bas], []]], [[[1, 1], 1], [[1, 3], 2], [[2, 2], 2], [[2, 3], 4], [[3, 1], 1]]])$

(37)

>	$convert (A1, DGtensor, [[cov_bas, cov_bas], []])$

$_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[1, 3], 2], [[2, 2], 2], [[2, 3], 4], [[3, 1], 1]]])$

(38)

Example 2.

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

>	$A2 ≔ Array (1 .. 3, 1 .. 3, 1 .. 3, 1 .. 3) &colon;$

>	${A2}_{1, 2, 2, 3} ≔ m$

${A2}_{1, 2, 2, 3} ≔ m$

(39)

>	${A2}_{1, 3, 2, 1} ≔ n$

${A2}_{1, 3, 2, 1} ≔ n$

(40)

>	$convert (A2, DGtensor, [[con_bas, cov_bas, cov_bas, cov_bas], []])$

$_DG ([[tensor, M, [[con_bas, cov_bas, cov_bas, cov_bas], []]], [[[1, 2, 2, 3], m], [[1, 3, 2, 1], n]]])$

(41)

Example 3.

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

>	$X ≔ evalDG (x^{2} D_y - y^{2} D_z)$

$_DG ([[vector, M, []], [[[2], x^{2}], [[3], - y^{2}]]])$

(42)

>	$T ≔ convert (X, DGtensor)$

$_DG ([[tensor, M, [[con_bas], []]], [[[2], x^{2}], [[3], - y^{2}]]])$

(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.

>	$Tools :- DGinfo (X, ObjectType), Tools :- DGinfo (T, ObjectType)$

$vector, tensor$

(44)

>	$lprint (X), lprint (T)$

_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.

>	$ω ≔ evalDG (y dx &w dy - z dy &w dz)$

$_DG ([[form, M, 2], [[[1, 2], y], [[2, 3], - z]]])$

(45)

>	$convert (ω, DGtensor)$

$_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 2], y], [[2, 1], - y], [[2, 3], - z], [[3, 2], z]]])$

(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].

>	$DGsetup ([t, x, y, z], [z1, z2, w1, w2], M)$

$frame name: M$

(47)

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

>	$g ≔ evalDG (dt &t dt - dx &t dx - dy &t dy - dz &t dz)$

$_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], - 1], [[3, 3], - 1], [[4, 4], - 1]]])$

(48)

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

>	$F ≔ [D_t, D_x, D_y, D_z]$

$[_DG ([[vector, M, []], [[[1], 1]]]),_DG ([[vector, M, []], [[[2], 1]]]),_DG ([[vector, M, []], [[[3], 1]]]),_DG ([[vector, M, []], [[[4], 1]]])]$

(49)

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

>	$σ ≔ SolderForm (F)$

(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.

>	$ψ ≔ evalDG (- \frac{1 I 2^{\frac{1}{2}} D_z1 &t D_w2}{2} + \frac{1 I 2^{\frac{1}{2}} D_z2 &t D_w1}{2})$

$_DG ([[tensor, M, [[con_vrt, con_vrt], []]], [[[5, 8], - \frac{1}{2} I \sqrt{2}], [[6, 7], \frac{1}{2} I \sqrt{2}]]])$

(51)

>	$X ≔ convert (ψ, DGtensor, σ, [[1, 2]])$

$_DG ([[tensor, M, [[con_bas], []]], [[[3], 1]]])$

(52)

Example 6.

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

>	$ψ ≔ evalDG (2^{\frac{1}{2}} dz2 &t dw2)$

$_DG ([[tensor, M, [[cov_vrt, cov_vrt], []]], [[[6, 8], \sqrt{2}]]])$

(53)

>	$ω ≔ convert (ψ, DGtensor, σ, [[1, 2]])$

$_DG ([[tensor, M, [[cov_bas], []]], [[[1], 1], [[4], - 1]]])$

(54)

Example 7.

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

>	$ψ ≔ evalDG (((dz1 &t dw1) &t dz2) &t dw2 - ((dz1 &t dw2) &t dz2) &t dw1 - ((dz2 &t dw1) &t dz1) &t dw2 + ((dz2 &t dw2) &t dz1) &t dw1)$

$_DG ([[tensor, M, [[cov_vrt, cov_vrt, cov_vrt, cov_vrt], []]], [[[5, 7, 6, 8], 1], [[5, 8, 6, 7], - 1], [[6, 7, 5, 8], - 1], [[6, 8, 5, 7], 1]]])$

(55)

>	$G ≔ convert (ψ, DGtensor, σ, [[1, 2], [3, 4]])$

$_DG ([[tensor, M, [[cov_bas, cov_bas], []]], [[[1, 1], 1], [[2, 2], - 1], [[3, 3], - 1], [[4, 4], - 1]]])$

(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.)

>	$ψ ≔ evalDG (- \frac{1 I 2^{\frac{1}{2}} ((D_t &t D_z1) &t (`&t` (D_z1))) &t D_w2}{2} + \frac{1 I 2^{\frac{1}{2}} ((D_x &t D_z2) &t D_z2) &t D_w1}{2})$

$_DG ([[tensor, M, [[con_bas, con_vrt, con_vrt, con_vrt], []]], [[[1, 5, 5, 8], - \frac{1}{2} I \sqrt{2}], [[2, 6, 6, 7], \frac{1}{2} I \sqrt{2}]]])$

(57)

>	$T1 ≔ convert (ψ, DGtensor, σ, [[2, 4]])$

$_DG ([[tensor, M, [[con_bas, con_bas, con_vrt], []]], [[[1, 2, 5], - \frac{1}{2} I], [[1, 3, 5], \frac{1}{2}], [[2, 2, 6], \frac{1}{2} I], [[2, 3, 6], \frac{1}{2}]]])$

(58)

>	$T2 ≔ convert (ψ, DGtensor, σ, [[3, 4]])$

$_DG ([[tensor, M, [[con_bas, con_bas, con_vrt], []]], [[[1, 2, 5], - \frac{1}{2} I], [[1, 3, 5], \frac{1}{2}], [[2, 2, 6], \frac{1}{2} I], [[2, 3, 6], \frac{1}{2}]]])$

(59)

convert/DGjet, convert/DGdiff

>	$with (DifferentialGeometry) &colon;$ $with (Library) &colon;$

Example 1.

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

>	$DGsetup ([x, y], [u], Jet, 2) &colon;$

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

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

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

(60)

>	$convert (L, DGjet)$

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

(61)

Example 2.

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

>	$DGsetup ([x], [y], J, 4)$

$frame name: J$

(62)

>	$ODE ≔ Retrieve (Kamke, 1, [4, 8], variables = [x, y])$

$ODE ≔ \frac{{&DifferentialD;}^{4}}{&DifferentialD; x^{4}} y (x) + (a x^{2} + b λ + c) (\frac{{&DifferentialD;}^{2}}{&DifferentialD; x^{2}} y (x)) + (α x^{2} + β λ + γ) y (x)$

(63)

>	$convert (ODE, DGjet)$

$y_{1, 1, 1, 1} + (a x^{2} + b λ + c) y_{1, 1} + (α x^{2} + β λ + γ) y []$

(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.

>	$DGsetup ([t, x], [u], J, 3) &colon;$

>	$Δ ≔ u_{1} = u_{2, 2, 2} + u_{[]} u_{2}$

$Δ ≔ u_{1} = u [] u_{2} + u_{2, 2, 2}$

(65)

>	$convert (Δ, DGdiff)$

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

(66)

convert/DGbiform

>	$with (DifferentialGeometry) &colon;$

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].

>	$DGsetup ([x, y], [u], Jet, 2) &colon;$

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

>	$ω ≔ evalDG (({du}_{1}) &w ({du}_{2}))$

$_DG ([[form, Jet, 2], [[[4, 5], 1]]])$

(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.

>	$convert (ω, DGbiform, [2, 0])$

$_DG ([[biform, Jet, [2, 0]], [[[1, 2], u_{1, 1} u_{2, 2} - u_{1, 2}^{2}]]])$

(68)

>	$convert (ω, DGbiform, [1, 1])$

$_DG ([[biform, Jet, [1, 1]], [[[1, 4], - u_{1, 2}], [[1, 5], u_{1, 1}], [[2, 4], - u_{2, 2}], [[2, 5], u_{1, 2}]]])$

(69)

>	$convert (ω, DGbiform, [0, 2])$

$_DG ([[biform, Jet, [0, 2]], [[[4, 5], 1]]])$

(70)

>	$convert (ω, DGbiform)$

$[_DG ([[biform, Jet, [0, 2]], [[[4, 5], 1]]]),_DG ([[biform, Jet, [1, 1]], [[[1, 4], - u_{1, 2}], [[1, 5], u_{1, 1}], [[2, 4], - u_{2, 2}], [[2, 5], u_{1, 2}]]]),_DG ([[biform, Jet, [2, 0]], [[[1, 2], u_{1, 1} u_{2, 2} - u_{1, 2}^{2}]]])]$

(71)

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