DifferentialGeometry:-Tools[&MatrixMinus, &MatrixMult, &MatrixPlus, &MatrixWedge] - Maple Help

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DifferentialGeometry:-Tools[&MatrixMinus, &MatrixMult, &MatrixPlus, &MatrixWedge]

Calling Sequence

A &MatrixMinus B - subtract two Matrices/Vectors of vectors, differential forms or tensors

A &MatrixMult C - multiply a Matrix/Vector A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars

C &MatrixMult A - multiply a Matrix A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars

A &MatrixPlus B - add two Matrices/Vectors of vectors, differential forms or tensors

E &MatrixWedge F - calculate the Matrix wedge product of two Matrices/Vectors of differential forms.

Parameters

A, B

-

two Matrices/Vectors of vectors, differential forms or tensors

C

-

a scalar or a Matrix/Vector of scalars

E, F

-

two Matrices/Vectors of differential forms

Description

• 

These commands provide, within the DifferentialGeometry environment, the basic arithmetical operations for Matrices or Vectors of: vectors, differential forms, or tensors.  They are particularly useful for curvature calculations for connections on principle bundles of matrix groups.

• 

These commands are part of the DifferentialGeometry:-Tools package, and so can be used in the form described above only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.

Examples

withDifferentialGeometry:withTools:

 

Define a 3-dimensional manifold M with coordinates [x, y, z].

DGsetupx,y,z,M:

 

Example 1 

Define two column Vectors of 1 forms A, B; a 2x2 matrix C of scalars; a row Vector of 1 forms E and a 2x2 Matrix of 1 forms F.

A:=VectorevalDGdxdy,dy+dx

A:=`*`dx`*`dy`*`dx+`*`dy

(1)

B:=VectorevalDGdx+2dy,dx+3dy

B:=`*`dx+2dy`*`dx+3dy

(2)

C:=Matrix1,2,3,4

C:=1234

(3)

E:=LinearAlgebra:-TransposeA

E:=`*`dx`*`dy`*`dx+`*`dy

(4)

F:=MatrixevalDGdxdz,dy,dz,dx+dy+3dz

F:=`*`dx`*`dz`*`dy`*`dz`*`dx+`*`dy+3dz

(5)

Perform various arithmetic operations with the quantities A, B, C, E, F.

A &MatrixPlus B

2dx+`*`dy2dx+4dy

(6)

A &MatrixMinus B

3dy2dy

(7)

a &MatrixMult A

adxadyadx+ady

(8)

C &MatrixMult A

3dx+`*`dy7dx+`*`dy

(9)

E &MatrixMult C

4dx+2dy6dx+2dy

(10)

E &MatrixWedge B

5dxdy

(11)

C &MatrixMult F

`*`dx+`*`dz2dx+3dy+6dz3dx+`*`dz4dx+7dy+12dz

(12)

F &MatrixWedge A

2dxdy+`*`dxdz`*`dydz4dxdz2dydz

(13)

F &MatrixWedge F

`*`dydz4dydz`*`dydz`*`dydz

(14)
M > 

See Also

DifferentialGeometry, LinearAlgebra, AlgebraicOperations, evalDG, DGzip, Matrix, Vector


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