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DifferentialGeometry[algebraic operations]

addition, subtraction, scalar multiplication, wedge product, tensor product

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

A &plus B - add two vectors, differential forms or tensors

A &minus B- subtract one vector, differential form or tensor from another

A &mult B - multiply a Maple expression by a vector, differential form or tensor

A &wedge B- form the wedge (or skew) product of a pair of differential forms or multi-vectors

A &tensor B- form the tensor product of a pair of tensors

A &algmult B - multiply two vectors in an algebra  

Parameters

A, B

-

Maple expressions, differential forms or tensors

Description

• 

In the DifferentialGeometry package the wedge product of  1-forms is defined in terms of the tensor product by αβ=αββα.

• 

When using these commands together within a single Maple expression, it is important to use parentheses to insure that the operations are executed in the correct order.

• 

In an interactive Maple session, it is usually more convenient to use the commands evalDG and DGzip to perform these basic algebraic operations.

• 

Here are the precise lists of admissible arguments for these commands.

• 

A &plus B, A &minus B -- A and B: Maple expressions, vectors, differential forms of the same degree, differential biforms of the same bidegree, tensors with the same index type and density weights. A and B must be defined on the same frame.

• 

A &mult B -- A: a Maple expression; B: a Maple expression, vector, differential form, differential biform, tensor. A and B must be defined on the same frame.

• 

A &wedge B -- A and B: Maple expressions or differential forms, differential biforms.  If A and B are forms, then the sum of their degrees cannot exceed the dimension of  the frame on which they are defined. If A and B are bi-forms, then the sum of their horizontal degrees cannot exceed the dimension of the base manifold on which they are defined.  A and B must be defined on the same frame.

• 

A &tensor B -- A and B: Maple expressions, vectors, differential 1-forms, tensors.  A and B must be defined on the same frame.

• 

These commands are part of the DifferentialGeometry package, and so can be used in the forms given above only after executing the command with(DifferentialGeometry).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Use DGsetup to define a 3-dimensional manifold M with coordinates [x, y, z].

DGsetupx,y,z,M,verbose

The following coordinates have been protected:

x,y,z

The following vector fields have been defined and protected:

D_x,D_y,D_z

The following differential 1-forms have been defined and protected:

dx,dy,dz

frame name: M

(1)

 

Example 1.

Create linear combinations of vector fields and differential 1-forms using &plus and &mult.

X1D_x &plus D_z

X1D_x+D_z

(2)

X23z &mult D_x &plus 2y &mult D_y

X23zD_x2yD_y

(3)

X3X2 &minus 3z &mult X1

X32yD_y3zD_z

(4)

α1sinz &mult dx &minus cosy &mult dz

α1sinzdxcosydz

(5)

α2cosx &mult dy &plus cosz &mult dz

α2cosxdy+coszdz

(6)

 

Example 2.

Create differential 2-forms using &plus and &mult and &wedge.

α32 &mult dx &wedge dy &plus 5 &mult dy &wedge dz

α32dxdy+5dydz

(7)

α4α1 &wedge α2

α4sinzcosxdxdy+sinzcoszdxdz+cosycosxdydz

(8)

α5α1 &wedge α2 &minus α3

α52+sinzcosxdxdy+sinzcoszdxdz+5+cosycosxdydz

(9)

α6α1 &wedge α3

α62cosy5sinzdxdydz

(10)

 

Example 3.

Create various tensors using &plus, &mult and &tensor.

T1X1 &tensor X1

T1D_xD_x+D_xD_z+D_zD_x+D_zD_z

(11)

T2X1 &tensor α1

T2sinzD_xdxcosyD_xdz+sinzD_zdxcosyD_zdz

(12)

T31 &tensor dx &wedge dy

T3dxdydydx

(13)

T4dx &tensor dx &tensor D_y &tensor D_z &tensor dz

T4dxdxD_yD_zdz

(14)

T51y2 &mult dx &t dx+dy &t dy

T5dxy2dx+dyy2dy

(15)

 

Example 4.

Create a multi-vector using &plus, &mult and &tensor.

V12 &mult D_x &wedge D_y &plus 3 &mult D_y &wedge D_z

V12D_xD_y+3D_yD_z

(16)

Example 5.

Use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.

LAAlgebraLibraryDataQuaternions,Q

LAe12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1

(17)

Initialize.

DGsetupLA,e,i,j,k,θ

algebra name: Q

(18)

Calculate some simple sums and products of quaternions.

Q1i &algmult j

Q1k

(19)

Q2e &plus i &plus j &plus k &algmult e &minus i &plus j &plus k

Q24e

(20)
Q > 

See Also

DifferentialGeometry

DGzip

evalDG

 


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