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

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

 

Calling Sequence

Parameters

Description

Examples

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.

AVectorevalDGdxdy,dy+dx

_DGform,M,1,1,1,2,1_DGform,M,1,1,1,2,1

(1)

BVectorevalDGdx+2dy,dx+3dy

_DGform,M,1,1,1,2,2_DGform,M,1,1,1,2,3

(2)

CMatrix1,2,3,4

1234

(3)

ELinearAlgebra:-TransposeA

_DGform,M,1,1,1,2,1_DGform,M,1,1,1,2,1

(4)

FMatrixevalDGdxdz,dy,dz,dx+dy+3dz

_DGform,M,1,1,1,3,1_DGform,M,1,2,1_DGform,M,1,3,1_DGform,M,1,1,1,2,1,3,3

(5)

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

A&MatrixPlusB

_DGform,M,1,1,2,2,1_DGform,M,1,1,2,2,4

(6)

A&MatrixMinusB

_DGform,M,1,2,3_DGform,M,1,2,2

(7)

a&MatrixMultA

_DGform,M,1,1,a,2,a_DGform,M,1,1,a,2,a

(8)

C&MatrixMultA

_DGform,M,1,1,3,2,1_DGform,M,1,1,7,2,1

(9)

E&MatrixMultC

_DGform,M,1,1,4,2,2_DGform,M,1,1,6,2,2

(10)

E&MatrixWedgeB

_DGform,M,2,1,2,5

(11)

C&MatrixMultF

_DGform,M,1,1,1,3,1_DGform,M,1,1,2,2,3,3,6_DGform,M,1,1,3,3,1_DGform,M,1,1,4,2,7,3,12

(12)

F&MatrixWedgeA

_DGform,M,2,1,2,2,1,3,1,2,3,1_DGform,M,2,1,3,4,2,3,2

(13)

F&MatrixWedgeF

_DGform,M,2,2,3,1_DGform,M,2,2,3,4_DGform,M,2,2,3,1_DGform,M,2,2,3,1

(14)
M > 

See Also

DifferentialGeometry

LinearAlgebra

AlgebraicOperations

evalDG

DGzip

Matrix

Vector