find the complex conjugate of a vector, tensor or differential form; find the conjugate of a quaternion or octonion - Maple Help

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DifferentialGeometry[DGconjugate] - find the complex conjugate of a vector, tensor or differential form; find the conjugate of a quaternion or octonion

DifferentialGeometry[DGRe] - find the real part of a vector, tensor or differential form; find the real part of a quaternion or octonion

DifferentialGeometry[DGIm] - find the  imaginary part of a vector, a tensor or differential form; find the imaginary part of a quaternion or octonion

Calling Sequences

   DGconjugate(T, option)

   DGconjugate(X)

   DGRe(T, option)

   DGRe(X)

   DGIm(T, option)

   DGIm(X)

Parameters

   T       - a tensor, differential form or vector defined on a manifold with complex coordinates

   X       - a quaternion or octonion

   option  - the keyword argument complexconjugatepairs = [[a1, a2], [b1, b2], ...] where [a1, a2], ... are Maple expressions (appearing the coefficients of T) which are to be interchanged under conjugation

 

 

Description

Examples

Description

• 

 The calling sequences DGconjugate(T, option), DGRe(T, option), DGIm(T, option) compute the complex conjugate, real part, and imaginary part of a tensor or differential form T. The coordinate variables are assumed to be real unless explicitly declared to be complex by using the keyword argument complexconjugatepairs as part of the calling sequence to DGsetup.

• 

The calling sequences DGconjugate(X), DGRe(X), DGIm(X) compute the complex conjugate, real part, and imaginary part of a quaternion or octonion X.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Define a 4-dimensional manifold with coordinates x, y, u, v, where x and y are real coordinates, u and v are complex coordinates and the complex conjugate of u is v.

 

DGsetupx,y,u,v,M,complexconjugatepairs=u,v

frame name: M

(2.1)

 

Calculate the complex conjugate of some vectors on M.

M > 

X1 ≔ evalDGD_x+D_u

X1:=D_x+D_u

(2.2)
M > 

DGconjugateX1

D_x+D_v

(2.3)
M > 

X2 ≔ evalDGID_x+D_uD_v

X2:=ID_x+D_uD_v

(2.4)
M > 

DGconjugateX2

ID_xD_u+D_v

(2.5)

 

Calculate the complex conjugate of a vector depending upon parameters α and β. First assume α and β are real.

M > 

X3 ≔ evalDGαD_x+uD_vβ

X3:=αD_x+uD_vβ

(2.6)
M > 

DGconjugateX3

αD_x+vD_uβ

(2.7)

 

Now suppose that α is complex and that the complex conjugate of α is β.

M > 

DGconjugateX3,complexconjugatepairs=α,β

βD_x+vD_uα

(2.8)

 

Calculate the complex conjugate of a rank 2 tensor:

M > 

T1 ≔ evalDGudx &t dy+ydx &t dv+u2+v2du &t dv

T1:=udxdy+ydxdv+u2+v2dudv

(2.9)
M > 

DGconjugateT1

vdxdy+ydxdu+u2+v2dvdu

(2.10)

 

Calculate the complex conjugate of a rank 4 differential form

M > 

ω1 ≔ evalDGIdx &w dy &w du &w dv

ω1:=Idxdydudv

(2.11)
M > 

DGconjugateω1

Idxdydudv

(2.12)

 

Example 2.

Calculate the real and imaginary parts of the vectors, tensors and differential forms defined in Example 1.

 

M > 

X1,DGReX1,DGImX1

D_x+D_u,D_x+12D_u+12D_v,12ID_u+12ID_v

(2.13)
M > 

X2,DGReX2,DGImX2

ID_x+D_uD_v,0D_x,D_xID_u+ID_v

(2.14)
M > 

T1

udxdy+ydxdv+u2+v2dudv

(2.15)
M > 

DGReT1

12v+12udxdy+12ydxdu+12ydxdv+12u2+12v2dudv+12u2+12v2dvdu

(2.16)
M > 

DGImT1

12Iv+udxdy+12Iydxdu12Iydxdv12Iu2+v2dudv+12Iu2+v2dvdu

(2.17)
M > 

ω1,DGReω1,DGImω1

Idxdydudv,Idxdydudv,0dxdydudv

(2.18)

 

Example 3.

The command DGconjugate works with anholonomic frames. To check this, first define an anholonomic frame and initialize it..

 

alg > 

FD ≔ FrameDataD_x,uD_y,uD_u+vD_v,vD_uuD_v,N

FD:=E2,E3=E2,E2,E4=vE2u

(2.19)
M > 

DGsetupFD

frame name: N

(2.20)
M > 

DGconjugateE1,E2,E3,E4

E1,vE2u,E3,E4

(2.21)

 

Example 4.

Find the conjugate of a quaternion. First use the command AlgebraData to obtain the structure equations for the quaternions.

AD ≔ AlgebraLibraryDataQuaternions,alg

AD:=e12=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

(2.22)

 

The labels for the vectors and dual 1-forms can be specified upon initialization of the algebra. We will use the standard e,i,j,k for the quaternion basis vectors, and α,β,δ,ε for the dual 1-forms.

 

DGsetupAD,'e','i','j','k','α','β','δ','ε'

algebra name: alg

(2.23)
alg > 

MultiplicationTable

| eijk---- ---- ---- ---- ---- e| eijki| iekjj| jkeik| kjie

(2.24)

 

Define a quaternion.

M > 

X ≔ evalDG3e+2i3j+4k

X:=3e+2i3j+4k

(2.25)
alg > 

DGconjugateX

3e2i+3j4k

(2.26)

 

Example 5.

Find the conjugate of an octonian. Use the command AlgebraData to obtain the structure equations for the octonions.

AD ≔ AlgebraLibraryDataOctonions,alg

AD:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e1.e5=e5,e1.e6=e6,e1.e7=e7,e1.e8=e8,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e2.e5=e6,e2.e6=e5,e2.e7=e8,e2.e8=e7,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e3.e5=e7,e3.e6=e8,e3.e7=e5,e3.e8=e6,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1,e4.e5=e8,e4.e6=e7,e4.e7=e6,e4.e8=e5,e5.e1=e5,e5.e2=e6,e5.e3=e7,e5.e4=e8,e52=e1,e5.e6=e2,e5.e7=e3,e5.e8=e4,e6.e1=e6,e6.e2=e5,e6.e3=e8,e6.e4=e7,e6.e5=e2,e62=e1,e6.e7=e4,e6.e8=e3,e7.e1=e7,e7.e2=e8,e7.e3=e5,e7.e4=e6,e7.e5=e3,e7.e6=e4,e72=e1,e7.e8=e2,e8.e1=e8,e8.e2=e7,e8.e3=e6,e8.e4=e5,e8.e5=e4,e8.e6=e3,e8.e7=e2,e82=e1

(2.27)
alg > 

DGsetupAD

algebra name: alg

(2.28)

 

Define an octonion.

X ≔ evalDG3e1+2e33e6+4e8

X:=3e1+2e33e6+4e8

(2.29)
alg > 

DGconjugateX

3e12e3+3e64e8

(2.30)

 

See Also

DifferentialGeometry

LieAlgebras

AlgebraData

AlgebraInverse

 AlgebraNorm

 


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