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

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

Calling Sequences

     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

(2.1)

g:=evalDGdx &t dx+dy &t dy+du &s dv+fx,udu &t du

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

(2.2)

 

Here are the metrics of the form (2.2) with vanishing Einstein tensor.

M > 

DGsolveEinsteinTensorg,g

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

(2.3)

 

Here are the metrics of the form (2.2) with vanishing Bach tensor.

M > 

DGsolveBachTensorg,g

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

(2.4)

 

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

(2.5)
M > 

α:=evalDGdx &w dy+rdx &w du+sdy &w dv:

M > 

DGsolveα &wedge α,α,r,s

dxdy+rdxdu,dxdy+sdydv

(2.6)

 

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

(2.7)
M > 

Γ:=ConnectionD_x &t dx &t dy+D_y &t dy &t dx

Γ:=D_xdxdy+D_ydydx

(2.8)
M > 

C:=cost,sint

C:=cost,sint

(2.9)
M > 

X:=evalDGAtD_x+BtD_y

X:=AtD_x+BtD_y

(2.10)
M > 

DGsolveParallelTransportEquationsC,X,Γ,t,X

_C2ⅇsintD_x+_C1ⅇcostD_y

(2.11)

 

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

M > 

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

ⅇsintD_x

(2.12)

 

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

(2.13)
M > 

g:=evalDGdx &t dx+dy &t dy+dz &t dzdt &t dt

g:=dxdx+dydy+dzdzdtdt

(2.14)
M > 

F:=evalDGAx,y,z,tdx &w dy+Bx,y,z,tdx &w dt

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

(2.15)
M > 

DGsolveExteriorDerivativeF,ExteriorDerivativeHodgeStarg,F,detmetric=1,F

_F1t+y+_F2tydxdy+_F1t+y_F2ty+_C1dxdt

(2.16)

See Also

DifferentialGeometry, DGNullSpace


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