DifferentialGeometry:-Tools[DGbiform, DGform, DGtensor, DGvector] - Maple Programming Help

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DifferentialGeometry:-Tools[DGbiform, DGform, DGtensor, DGvector]

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DGbiform(x, M)

DGform(x, M)

DGtensor(x, indexType, M)

DGvector(y, M)

Parameters

x

-

a positive integer, a list of positive integers, a coordinate variable, or a list of coordinate variables

M

-

(optional) the name of defined frame

indexType

-

specifying the index type of the tensor

y

-

a positive integer or a coordinate variable

Description

• 

The command DGform will create a single term differential form.  Let Theta = [theta_1, theta_2, theta_3, ...] denote the coframe for the current frame or, if the optional argument M is given, the frame M.  The list Theta can be obtained from the command DGinfo with the keyword "frameBaseForms" or "frameJetForms".  Let V = [x_1, x_2, x_3, ...] denote the local coordinates for the current frame or, if the optional argument M is given, the frame M.  The list V can be obtained from the command DGinfo with the keyword "frameIndependentVariables" or "frameJetVariables". If the integer i or coordinate x_i is given, the command returns the corresponding 1-form theta_i.  If a list of p integers [i, j, k, ...] or coordinates [x_i, x_j, x_k, ...] is given, the command returns the p-form  theta_i &w theta_j &w theta_k...

• 

The commands DGbiform, DGtensor, and DGvector work in a similar fashion.

• 

The command DGform is part of the DifferentialGeometry:-Tools package and so can be used in the form DGform(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGform.  DGbiform, DGtensor, and DGvector work in the same way.

Examples

withDifferentialGeometry:withTools:

Example 1.

Define a manifold M with coordinates [x, y, z, w].

DGsetupx,y,z,w,M:

DGvectorx

D_x

(1)

DGvector3

D_z

(2)

DGformy

dy

(3)

DGform4

dw

(4)

DGformx,y

dxdy

(5)

DGform1,2,3,4

dxdydzdw

(6)

DGtensorx,con_bas

D_x

(7)

DGtensor2,cov_bas

dy

(8)

DGtensor1,1,1,cov_bas,con_bas,cov_bas,

dxD_xdx

(9)

 

Example 2.

Define a rank 3 vector bundle E with coordinates [x, y, u, v, w] over a two dimensional base with coordinates [x, y].

DGsetupx,y,u,v,w,E

frame name: E

(10)

DGtensorx,v,v,con_bas,cov_bas,con_bas,

D_xdvD_v

(11)

 

Example 2.

Define the jet space J^2(R^2, R^2) for two functions u and v of 2 independent variables x and y.

DGsetupx,y,u,v,J,2:

ωDGbiformu1,1

ωCu1,1

(12)

convertω,DGform

u1,1,1dxu1,1,2dy+du1,1

(13)
J > 

See Also

DifferentialGeometry

Tools

evalDG

DGinfo

DGzip

 


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