solve a list of tensor equations for an unknown list of tensors - Maple Programming Help

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DifferentialGeometry[DGsolve] - solve a list of tensor equations for an unknown list of tensors

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DGsolve(Eq, T, options)

Parameters

Eq

-

a vector, differential form or tensor constructed from the objects in the 2nd argument; or list of such. The vanishing of these tensors defines the equations to be solved.

T

-

a vector, differential form, or tensor, depending upon a number of arbitrary parameters or functions; or a list of such

auxiliaryequations

-

(optional) a keyword argument to specify a set of auxiliary equations, to be solved in conjunction with the equations specified by the first argument

unknowns

-

(optional) list of parameters and functions, explicitly specifying the unknowns to be solved for.

method

-

(optional) a Maple procedure which will be used to solve the equations

other

-

(optional) additional arguments to be passed to the procedure used the solve the equations

Description

• 

 Let T  be a vector, a differential form, or a tensor which depends upon a number of parameters f1, f2 ..., fn . These parameters may be constants or functions. Now let ℰ be a differential-geometric construction depending upon T which can be implemented in Maple by a sequence of commands in the DifferentialGeometry package. For example, T could be a metric tensor and ℰ the Einstein tensor constructed from g. The command DGsolve will solve the equations obtained by setting to zero all the components of ℰ for the unknowns f1, f2 ..., fn. The output is a set containing those T solving =0 (obtainable by Maple).

• 

 Additional constraints (for example, initial conditions or inequalities) can be imposed upon the unknowns f1, f2 ..., fn  with the keyword argument auxiliaryequations.

• 

The command DGsolve uses the general purpose solver PDEtools:-Solve to solve the system ℰ =0 for the unknowns f1, f2 ..., fn. The keyword argument method can be used to specify a particular Maple solver (for example, solve, pdsolve, dsolve) or a customized solver created by the user.

• 

If the equations defined by ℰ =0 are homogenous linear algebraic equations, then the command DGNullSpace can also be used.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Let M  be a 4-dimensional space. We define a metric tensor depending upon an arbitrary function. We find the metrics which have vanishing Einstein tensor, and vanishing Bach tensor.

 

DGsetupx,y,u,v,M

frame name: M

(4.1)

gevalDGdx&tdx+dy&tdy+du&sdv+fx,udu&tdu

g:=dxdx+dydy+fx,ududu+12dudv+12dvdu

(4.2)

 

Here are the metrics of the form (4.2) with vanishing Einstein tensor.

M > 

DGsolveEinsteinTensorg,g

dxdx+dydy+_F1ux+_F2ududu+12dudv+12dvdu

(4.3)

 

Here are the metrics of the form (4.2) with vanishing Bach tensor.

M > 

DGsolveBachTensorg,g

dxdx+dydy+16_F1ux3+12_F2ux2+_F3ux+_F4ududu+12dudv+12dvdu

(4.4)

 

Example 2.

In this example we define a 2-form α which depends upon parameters r, s. We find those values of the parameters for which α α = 0.

M > 

DGsetupx,y,u,v,M

frame name: M

(4.5)
M > 

αevalDGdx&wdy+rdx&wdu+sdy&wdv:

M > 

DGsolveα&wedgeα,α,r,s

dxdy+rdxdu,dxdy+sdydv

(4.6)

 

Example 3.

We define a connection Γ and calculate the parallel transport of a vector Xt along a curve Ct.

M > 

DGsetupx,y,M

frame name: M

(4.7)
M > 

GammaConnectionD_x&tdx&tdy+D_y&tdy&tdx

Γ:=D_xdxdy+D_ydydx

(4.8)
M > 

Ccost,sint

C:=cost,sint

(4.9)
M > 

XevalDGAtD_x+BtD_y

X:=AtD_x+BtD_y

(4.10)
M > 

DGsolveParallelTransportEquationsC,X,Gamma,t,X

_C2ⅇsintD_x+_C1ⅇcostD_y

(4.11)

 

We can use the keyword argument auxiliaryequations to specify an initial position for the vector X.

M > 

DGsolveParallelTransportEquationsC,X,Gamma,t,X,auxiliaryequations=A0=1,B0=0

ⅇsintD_x

(4.12)

 

Example 4.

The source-free Maxwell equations may be expressed in terms of a 2-form F by the equations dF =0 and d*F =0, where d is the exterior derivative and * is the Hodge star operator. In this example we define a 2-form F depending on 2 functions of 4 variables and solve the Maxwell equations for F.

 

M > 

DGsetupx,y,z,t,M

frame name: M

(4.13)
M > 

gevalDGdx&tdx+dy&tdy+dz&tdzdt&tdt

g:=dxdx+dydy+dzdzdtdt

(4.14)
M > 

FevalDGAx,y,z,tdx&wdy+Bx,y,z,tdx&wdt

F:=Ax,y,z,tdxdy+Bx,y,z,tdxdt

(4.15)
M > 

DGsolveExteriorDerivativeF,ExteriorDerivativeHodgeStarg,F,detmetric=1,F

_F1t+y+_F2tydxdy+_F1t+y_F2ty+_C1dxdt

(4.16)

See Also

DifferentialGeometry

DGNullSpace