DGconjugate - Maple Help

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 Sequence 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

 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

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

 > $\mathrm{DGsetup}\left(\left[x,y,u,v\right],M,\mathrm{complexconjugatepairs}=\left[\left[u,v\right]\right]\right)$
 ${\mathrm{frame name: M}}$ (4.1)

Calculate the complex conjugate of some vectors on $M$.

 M > $\mathrm{X1}≔\mathrm{evalDG}\left(\mathrm{D_x}+\mathrm{D_u}\right)$
 ${\mathrm{X1}}{:=}{\mathrm{D_x}}{+}{\mathrm{D_u}}$ (4.2)
 M > $\mathrm{DGconjugate}\left(\mathrm{X1}\right)$
 ${\mathrm{D_x}}{+}{\mathrm{D_v}}$ (4.3)
 M > $\mathrm{X2}≔\mathrm{evalDG}\left(I\mathrm{D_x}+\mathrm{D_u}-\mathrm{D_v}\right)$
 ${\mathrm{X2}}{:=}{I}{}{\mathrm{D_x}}{+}{\mathrm{D_u}}{-}{\mathrm{D_v}}$ (4.4)
 M > $\mathrm{DGconjugate}\left(\mathrm{X2}\right)$
 ${-}{I}{}{\mathrm{D_x}}{-}{\mathrm{D_u}}{+}{\mathrm{D_v}}$ (4.5)

Calculate the complex conjugate of a vector depending upon parameters $\mathrm{\alpha }$ and $\mathrm{β}$. First assume and are real.

 M > $\mathrm{X3}≔\mathrm{evalDG}\left(\mathrm{\alpha }\mathrm{D_x}+\frac{u}{\mathrm{\beta }}\mathrm{D_v}\right)$
 ${\mathrm{X3}}{:=}{\mathrm{α}}{}{\mathrm{D_x}}{+}\frac{{u}{}{\mathrm{D_v}}}{{\mathrm{β}}}$ (4.6)
 M > $\mathrm{DGconjugate}\left(\mathrm{X3}\right)$
 ${\mathrm{α}}{}{\mathrm{D_x}}{+}\frac{{v}{}{\mathrm{D_u}}}{{\mathrm{β}}}$ (4.7)

Now suppose that is complex and that the complex conjugate of is $\mathrm{β}$.

 M > $\mathrm{DGconjugate}\left(\mathrm{X3},\mathrm{complexconjugatepairs}=\left[\left[\mathrm{\alpha },\mathrm{\beta }\right]\right]\right)$
 ${\mathrm{β}}{}{\mathrm{D_x}}{+}\frac{{v}{}{\mathrm{D_u}}}{{\mathrm{α}}}$ (4.8)

Calculate the complex conjugate of a rank 2 tensor:

 M > $\mathrm{T1}≔\mathrm{evalDG}\left(u\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+y\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}+\left({u}^{2}+{v}^{2}\right)\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}\right)$
 ${\mathrm{T1}}{:=}{u}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{y}{}{\mathrm{dx}}{}{\mathrm{dv}}{+}\left({{u}}^{{2}}{+}{{v}}^{{2}}\right){}{\mathrm{du}}{}{\mathrm{dv}}$ (4.9)
 M > $\mathrm{DGconjugate}\left(\mathrm{T1}\right)$
 ${v}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{y}{}{\mathrm{dx}}{}{\mathrm{du}}{+}\left({{u}}^{{2}}{+}{{v}}^{{2}}\right){}{\mathrm{dv}}{}{\mathrm{du}}$ (4.10)

Calculate the complex conjugate of a rank 4 differential form

 M > $\mathrm{ω1}≔\mathrm{evalDG}\left(I\left(\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}\right)$
 ${\mathrm{ω1}}{:=}{I}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{du}}{}{\bigwedge }{}{\mathrm{dv}}$ (4.11)
 M > $\mathrm{DGconjugate}\left(\mathrm{ω1}\right)$
 ${I}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{du}}{}{\bigwedge }{}{\mathrm{dv}}$ (4.12)

Example 2.

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

 M > $\mathrm{X1},\mathrm{DGRe}\left(\mathrm{X1}\right),\mathrm{DGIm}\left(\mathrm{X1}\right)$
 ${\mathrm{D_x}}{+}{\mathrm{D_u}}{,}{\mathrm{D_x}}{+}\frac{{1}}{{2}}{}{\mathrm{D_u}}{+}\frac{{1}}{{2}}{}{\mathrm{D_v}}{,}{-}\frac{{1}}{{2}}{}{I}{}{\mathrm{D_u}}{+}\frac{{1}}{{2}}{}{I}{}{\mathrm{D_v}}$ (4.13)
 M > $\mathrm{X2},\mathrm{DGRe}\left(\mathrm{X2}\right),\mathrm{DGIm}\left(\mathrm{X2}\right)$
 ${I}{}{\mathrm{D_x}}{+}{\mathrm{D_u}}{-}{\mathrm{D_v}}{,}{0}{}{\mathrm{D_x}}{,}{\mathrm{D_x}}{-}{I}{}{\mathrm{D_u}}{+}{I}{}{\mathrm{D_v}}$ (4.14)
 M > $\mathrm{T1}$
 ${u}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{y}{}{\mathrm{dx}}{}{\mathrm{dv}}{+}\left({{u}}^{{2}}{+}{{v}}^{{2}}\right){}{\mathrm{du}}{}{\mathrm{dv}}$ (4.15)
 M > $\mathrm{DGRe}\left(\mathrm{T1}\right)$
 $\left(\frac{{1}}{{2}}{}{v}{+}\frac{{1}}{{2}}{}{u}\right){}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{y}{}{\mathrm{dx}}{}{\mathrm{du}}{+}\frac{{1}}{{2}}{}{y}{}{\mathrm{dx}}{}{\mathrm{dv}}{+}\left(\frac{{1}}{{2}}{}{{u}}^{{2}}{+}\frac{{1}}{{2}}{}{{v}}^{{2}}\right){}{\mathrm{du}}{}{\mathrm{dv}}{+}\left(\frac{{1}}{{2}}{}{{u}}^{{2}}{+}\frac{{1}}{{2}}{}{{v}}^{{2}}\right){}{\mathrm{dv}}{}{\mathrm{du}}$ (4.16)
 M > $\mathrm{DGIm}\left(\mathrm{T1}\right)$
 ${-}\frac{{1}}{{2}}{}{I}{}\left({-}{v}{+}{u}\right){}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{I}{}{y}{}{\mathrm{dx}}{}{\mathrm{du}}{-}\frac{{1}}{{2}}{}{I}{}{y}{}{\mathrm{dx}}{}{\mathrm{dv}}{-}\frac{{1}}{{2}}{}{I}{}\left({{u}}^{{2}}{+}{{v}}^{{2}}\right){}{\mathrm{du}}{}{\mathrm{dv}}{+}\frac{{1}}{{2}}{}{I}{}\left({{u}}^{{2}}{+}{{v}}^{{2}}\right){}{\mathrm{dv}}{}{\mathrm{du}}$ (4.17)
 M > $\mathrm{ω1},\mathrm{DGRe}\left(\mathrm{ω1}\right),\mathrm{DGIm}\left(\mathrm{ω1}\right)$
 ${I}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{du}}{}{\bigwedge }{}{\mathrm{dv}}{,}{I}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{du}}{}{\bigwedge }{}{\mathrm{dv}}{,}{0}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{du}}{}{\bigwedge }{}{\mathrm{dv}}$ (4.18)

Example 3.

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

 alg > $\mathrm{FD}≔\mathrm{FrameData}\left(\left[\mathrm{D_x},u\mathrm{D_y},u\mathrm{D_u}+v\mathrm{D_v},v\mathrm{D_u}-u\mathrm{D_v}\right],N\right)$
 ${\mathrm{FD}}{:=}\left[\left[{\mathrm{E2}}{,}{\mathrm{E3}}\right]{=}{-}{\mathrm{E2}}{,}\left[{\mathrm{E2}}{,}{\mathrm{E4}}\right]{=}{-}\frac{{v}{}{\mathrm{E2}}}{{u}}\right]$ (4.19)
 M > $\mathrm{DGsetup}\left(\mathrm{FD}\right)$
 ${\mathrm{frame name: N}}$ (4.20)
 M > $\mathrm{DGconjugate}\left(\left[\mathrm{E1},\mathrm{E2},\mathrm{E3},\mathrm{E4}\right]\right)$
 $\left[{\mathrm{E1}}{,}\frac{{v}{}{\mathrm{E2}}}{{u}}{,}{\mathrm{E3}}{,}{-}{\mathrm{E4}}\right]$ (4.21)

Example 4.

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

 > $\mathrm{AD}≔\mathrm{AlgebraLibraryData}\left("Quaternions",\mathrm{alg}\right)$
 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (4.22)

The labels for the vectors and dual 1-forms can be specified upon initialization of the algebra. We will use the standard for the quaternion basis vectors, and for the dual 1-forms.

 > $\mathrm{DGsetup}\left(\mathrm{AD},\left['e','i','j','k'\right],\left['\mathrm{\alpha }','\mathrm{\beta }','\mathrm{\delta }','\mathrm{\epsilon }'\right]\right)$
 ${\mathrm{algebra name: alg}}$ (4.23)
 alg > $\mathrm{MultiplicationTable}\left(\right)$
 $\left[\begin{array}{cccccc}{}& {|}& {e}& {i}& {j}& {k}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {e}& {|}& {e}& {i}& {j}& {k}\\ {i}& {|}& {i}& {-}{e}& {k}& {-}{j}\\ {j}& {|}& {j}& {-}{k}& {-}{e}& {i}\\ {k}& {|}& {k}& {j}& {-}{i}& {-}{e}\end{array}\right]$ (4.24)

Define a quaternion.

 M > $X≔\mathrm{evalDG}\left(3e+2i-3j+4k\right)$
 ${X}{:=}{3}{}{e}{+}{2}{}{i}{-}{3}{}{j}{+}{4}{}{k}$ (4.25)
 alg > $\mathrm{DGconjugate}\left(X\right)$
 ${3}{}{e}{-}{2}{}{i}{+}{3}{}{j}{-}{4}{}{k}$ (4.26)

Example 5.

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

 > $\mathrm{AD}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{alg}\right)$
 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{e5}}{=}{\mathrm{e5}}{,}{\mathrm{e1}}{.}{\mathrm{e6}}{=}{\mathrm{e6}}{,}{\mathrm{e1}}{.}{\mathrm{e7}}{=}{\mathrm{e7}}{,}{\mathrm{e1}}{.}{\mathrm{e8}}{=}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e5}}{=}{\mathrm{e6}}{,}{\mathrm{e2}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e2}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e8}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e5}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e6}}{=}{\mathrm{e8}}{,}{\mathrm{e3}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e3}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e4}}{.}{\mathrm{e5}}{=}{\mathrm{e8}}{,}{\mathrm{e4}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{.}{\mathrm{e7}}{=}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e1}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e5}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e5}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e8}}{,}{{\mathrm{e5}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e5}}{.}{\mathrm{e6}}{=}{\mathrm{e2}}{,}{\mathrm{e5}}{.}{\mathrm{e7}}{=}{\mathrm{e3}}{,}{\mathrm{e5}}{.}{\mathrm{e8}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e1}}{=}{\mathrm{e6}}{,}{\mathrm{e6}}{.}{\mathrm{e2}}{=}{\mathrm{e5}}{,}{\mathrm{e6}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e6}}{.}{\mathrm{e4}}{=}{\mathrm{e7}}{,}{\mathrm{e6}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e6}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e6}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e8}}{=}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e1}}{=}{\mathrm{e7}}{,}{\mathrm{e7}}{.}{\mathrm{e2}}{=}{\mathrm{e8}}{,}{\mathrm{e7}}{.}{\mathrm{e3}}{=}{\mathrm{e5}}{,}{\mathrm{e7}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e7}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e6}}{=}{\mathrm{e4}}{,}{{\mathrm{e7}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e7}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e2}}{,}{\mathrm{e8}}{.}{\mathrm{e1}}{=}{\mathrm{e8}}{,}{\mathrm{e8}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e8}}{.}{\mathrm{e3}}{=}{\mathrm{e6}}{,}{\mathrm{e8}}{.}{\mathrm{e4}}{=}{\mathrm{e5}}{,}{\mathrm{e8}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e8}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e8}}{.}{\mathrm{e7}}{=}{\mathrm{e2}}{,}{{\mathrm{e8}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (4.27)
 alg > $\mathrm{DGsetup}\left(\mathrm{AD}\right)$
 ${\mathrm{algebra name: alg}}$ (4.28)

Define an octonion.

 > $X≔\mathrm{evalDG}\left(3\mathrm{e1}+2\mathrm{e3}-3\mathrm{e6}+4\mathrm{e8}\right)$
 ${X}{:=}{3}{}{\mathrm{e1}}{+}{2}{}{\mathrm{e3}}{-}{3}{}{\mathrm{e6}}{+}{4}{}{\mathrm{e8}}$ (4.29)
 alg > $\mathrm{DGconjugate}\left(X\right)$
 ${3}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e3}}{+}{3}{}{\mathrm{e6}}{-}{4}{}{\mathrm{e8}}$ (4.30)