DifferentialGeometry[DGsolve] - solve a list of tensor equations for an unknown list of tensors
DGsolve(Eq, T, options)
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.
a vector, differential form, or tensor, depending upon a number of arbitrary parameters or functions; or a list of such
(optional) a keyword argument to specify a set of auxiliary equations, to be solved in conjunction with the equations specified by the first argument
(optional) list of parameters and functions, explicitly specifying the unknowns to be solved for.
(optional) a Maple procedure which will be used to solve the equations
(optional) additional arguments to be passed to the procedure used the solve the equations
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 ℰ=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.
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.
frame name: M
g â‰” evalDG⁡dx &t dx+dy &t dy+du &s dv+f⁡x,u⁢du &t du
Here are the metrics of the form (4.2) with vanishing Einstein tensor.
Here are the metrics of the form (4.2) with vanishing Bach tensor.
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.
α â‰” evalDG⁡dx &w dy+r⁢dx &w du+s⁢dy &w dv:
DGsolve⁡α &wedge α,α,r,s
We define a connection Γ and calculate the parallel transport of a vector Xt along a curve Ct.
Gamma â‰” Connection⁡−D_x &t dx &t dy+D_y &t dy &t dx
C â‰” cos⁡t,sin⁡t
X â‰” evalDG⁡A⁡t⁢D_x+B⁡t⁢D_y
We can use the keyword argument auxiliaryequations to specify an initial position for the vector X.
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.
g â‰” evalDG⁡dx &t dx+dy &t dy+dz &t dz−dt &t dt
F â‰” evalDG⁡A⁡x,y,z,t⁢dx &w dy+B⁡x,y,z,t⁢dx &w dt
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