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DifferentialGeometry

  

Hook

  

form the interior product of a vector or a list of vectors with a differential form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Hook(X, omega)

Hook(Y, T, indexlist)

Parameters

X

-

a vector or a list of vectors

omega

-

a differential p-form

Y

-

a list of vectors or differential 1-forms

T

-

a tensor

indexlist

-

(optional) a list of positive integers labeling various arguments of the tensor T 

Description

• 

If  X is a vector and omega a differential p-form, then the interior product or hook of  X and omega is the differential (p - 1)-form theta = Hook(X, omega) defined by theta(Y1, Y2, ..., Yq) = omega(X, Y1, Y2, ..., Yq), where q = p - 1 and Y1, Y2, ...,  Yq are arbitrary vectors.

• 

More generally, given a list of vectors [X1, X2, ... , Xr] and a differential p-form omega then theta = Hook([X1, X2, ..., Xr], omega) is the differential form of degree p - r defined by theta(Y1, Y2, ..., Yq) = omega(X1, X2, ..., Xr, Y1, Y2, ..., Yq), where q = p - r and Y1, Y2, ...,Yq are arbitrary vectors.

• 

If  Y = [X1, X2, alpha1, ..., Xr] is a list of  vectors or differential 1-forms and T is a tensor of total rank s = r, then the second calling sequence evaluates the scalar T(X1, X2, alpha1, ... , Xr).  If s > r and  indexlist = [i_1, i_2, ..., i_r], then Hook(Y, T, indexlist) calculates the rank s - r tensor obtained by evaluating the i_k-th argument of the tensor T on the k-th element of the list Y, for k = 1, 2, ... r.

• 

This command is part of the DifferentialGeometry package, and so can be used in the form Hook(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Hook.

Examples

withDifferentialGeometry:

DGsetupx,y,z,M:

 

Example 1.

Define vectors X1, X2, X3.

X1evalDGaD_x+bD_y+cD_z

X1aD_x+bD_y+cD_z

(1)

X2evalDG2D_yD_z

X22D_yD_z

(2)

X3evalDGD_x+D_z

X3D_x+D_z

(3)

Define a 1-form alpha, a 2-form beta and a 3-form omega.

αevalDG3dx+4dy7dz

α3dx+4dy7dz

(4)

βevalDGdx &w dy3dx &w dz

βdxdy3dxdz

(5)

ωevalDGdx &w dy &w dz

ωdxdydz

(6)

Compute various interior products.  Note that Hook(Y, Hook(X, omega) = Hook([X, Y], omega).

fHookX1,α

f7c+4b+3a

(7)

θHookX1,β

θ3c+bdx+ady3adz

(8)

HookX2,θ

5a

(9)

HookX1,X2,β

5a

(10)

HookX2,X1,β

5a

(11)

HookX1,X2,X3,ω

2ab2c

(12)

 

Example 2.

Evaluate a  type (1,3) tensor T on various vectors and differential forms.

TevalDG2dx &t dy &t dz &t D_y3dy &t dx &t dz &t D_z

T2dxdydzD_y3dydxdzD_z

(13)

HookD_x,D_y,D_z,dy,T

2

(14)

HookX1,X3,T

3bdzD_z

(15)

HookX1,X2,X3,α,T

16a

(16)

HookD_y,dz,T,1,4

3dxdz

(17)

HookX1,X2,T,1,3

3bdxD_z2adyD_y

(18)

 

Example 3. 

The interior product can be calculated for abstract forms.

DGsetupω1,ω2,ω3,β1=dgform2,dω1=ω2 &w ω3,dω2=β1,M3

frame name: M3

(19)

HookD_omega1,ω1;HookD_omega1,β1

1

ι1β1

(20)

Structure equations for interior products can be specified.

DGsetupM3,,hookD_omega1,β1=ω3

updated frame: M3

(21)

HookD_omega1,β1

ω3

(22)
M3 > 

See Also

DifferentialGeometry

ContractIndices

 


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