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Jet Bundles
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Jet Figure 4
  
last edited 10 years ago by test1

Edit detail for Jet Figure 4 revision 1 of 4
1 2 3 4
Editor: 127.0.0.1
Time: 2007/11/11 18:51:42 GMT-8
Note: transferred from axiom-developer 
changed:
-
\begin{axiom}
)lib JBC JBFC JBC- JBFC- BFC BFC- JB JBX   JBLF  DE SEM  DIFF JBE CK
\end{axiom}

\begin{axiom}
jb:=JB(['x,'y,'z],['u]);
jbx:=JBX jb;
jbl:=JBLF(jb,jbx);
de:=DE(jb,jbl);
ck:=CK(jb,jbl);
\end{axiom}

\begin{axiom}
eq1:jbl := D('u,['z,'z]) + 'y * D('u,['x,'x])
\end{axiom}
\begin{axiom}
eq2:jbl := D('u,['y,'y])
\end{axiom}
\begin{axiom}
printSys [eq1,eq2]
\end{axiom}

\begin{axiom}
janet:de := generate [eq1,eq2]
\end{axiom}

\begin{axiom}
setOutMode(14)$ck
setRedMode(1)$ck
\end{axiom}

\begin{axiom}
complete janet
\end{axiom}



axiom
)lib JBC JBFC JBC- JBFC- BFC BFC- JB JBX   JBLF  DE SEM  DIFF JBE CK
   JetBundleCategory is now explicitly exposed in frame initial 
   JetBundleCategory will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/JBC.NRLIB/code
   JetBundleFunctionCategory is now explicitly exposed in frame initial
   JetBundleFunctionCategory will be automatically loaded when needed 
      from /var/zope2/var/LatexWiki/JBFC.NRLIB/code
   JetBundleCategory& is now explicitly exposed in frame initial 
   JetBundleCategory& will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/JBC-.NRLIB/code
   JetBundleFunctionCategory& is now explicitly exposed in frame 
      initial 
   JetBundleFunctionCategory& will be automatically loaded when needed 
      from /var/zope2/var/LatexWiki/JBFC-.NRLIB/code
   BaseFunctionCategory is now explicitly exposed in frame initial 
   BaseFunctionCategory will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/BFC.NRLIB/code
   BaseFunctionCategory& is now explicitly exposed in frame initial 
   BaseFunctionCategory& will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/BFC-.NRLIB/code
   JetBundle is now explicitly exposed in frame initial 
   JetBundle will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/JB.NRLIB/code
   JetBundleXExpression is now explicitly exposed in frame initial 
   JetBundleXExpression will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/JBX.NRLIB/code
   JetBundleLinearFunction is now explicitly exposed in frame initial 
   JetBundleLinearFunction will be automatically loaded when needed 
      from /var/zope2/var/LatexWiki/JBLF.NRLIB/code
   DifferentialEquation is now explicitly exposed in frame initial 
   DifferentialEquation will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/DE.NRLIB/code
   SparseEchelonMatrix is now explicitly exposed in frame initial 
   SparseEchelonMatrix will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/SEM.NRLIB/code
   Differential is now explicitly exposed in frame initial 
   Differential will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/DIFF.NRLIB/code
   JetBundleExpression is now explicitly exposed in frame initial 
   JetBundleExpression will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/JBE.NRLIB/code
   CartanKuranishi is now explicitly exposed in frame initial 
   CartanKuranishi will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CK.NRLIB/code


axiom
jb:=JB(['x,'y,'z],['u]);


Type: Domain


axiom
jbx:=JBX jb;


Type: Domain


axiom
jbl:=JBLF(jb,jbx);


Type: Domain


axiom
de:=DE(jb,jbl);


Type: Domain


axiom
ck:=CK(jb,jbl);


Type: Domain


axiom
eq1:jbl := D('u,['z,'z]) + 'y * D('u,['x,'x])

\begin{equation}
\label{eq1}{u \sb {z, \: z}}+{y \  {u \sb {x, \: x}}} 
\end{equation}

Type: JetBundleLinearFunction?(JetBundle?([x,y,z]?,[u]?),JetBundleXExpression? JetBundle?([x,y,z]?,[u]?))


axiom
eq2:jbl := D('u,['y,'y])

\begin{equation}
\label{eq2}u \sb {y, \: y} 
\end{equation}

Type: JetBundleLinearFunction?(JetBundle?([x,y,z]?,[u]?),JetBundleXExpression? JetBundle?([x,y,z]?,[u]?))


axiom
printSys [eq1,eq2]

\begin{equation}
\label{eq3}\begin{array}{c}{\ 
}\{{{u \sb {z, \: z}}+{y \  {u \sb {x, \: x}}}}=0}\{\ }\{{u \sb {y, \: 
y}}=0}\\end{array} 
\end{equation}

Type: OutputForm?


axiom
janet:de := generate [eq1,eq2]

\begin{equation}
\label{eq4}\begin{array}{c}{\ 
}\{{{u \sb {z, \: z}}+{y \  {u \sb {x, \: x}}}}=0}\{\ }\{{u \sb {y, \: 
y}}=0}\\end{array} 
\end{equation}

Type: DifferentialEquation?(JetBundle?([x,y,z]?,[u]?),JetBundleLinearFunction?(JetBundle?([x,y,z]?,[u]?),JetBundleXExpression? JetBundle?([x,y,z]?,[u]?)))


axiom
setOutMode(14)$ck

\begin{equation}
\label{eq5}0 
\end{equation}

Type: NonNegativeInteger?


axiom
setRedMode(1)$ck

\begin{equation}
\label{eq6}0 
\end{equation}

Type: NonNegativeInteger?


axiom
complete janet

\begin{equation}
\label{eq7}\end{equation}
\begin{equation}
\label{eq8}\mbox{\rm Symbol } {M \sb {2}} \mbox{\rm not involutive! } \mbox{\rm 
Dimension: } 4\end{equation}
\begin{equation}
\label{eq9}\end{equation}
\begin{equation}
\label{eq10}\mbox{\rm Symbol } {M \sb {3}} \mbox{\rm involutive! } \mbox{\rm Dimension: } 
4\end{equation}
\begin{equation}
\label{eq11}\end{equation}
\begin{equation}
\label{eq12}\mbox{\rm Equation } {R \sb {3}} \mbox{\rm not involutive! } \mbox{\rm 
Dimension: } {12}\end{equation}
\begin{equation}
\label{eq13}\end{equation}
\begin{equation}
\label{eq14}\mbox{\rm ======= } 1 \mbox{\rm . Projection =======}\end{equation}
\begin{equation}
\label{eq15}\mbox{\rm Integrability condition(s)}\end{equation}
\begin{equation}
\label{eq16}\begin{array}{c}{\ 
}\{{u \sb {y, \: x, \: x}}=0}\\end{array}\end{equation}
\begin{equation}
\label{eq17}=============================\end{equation}
\begin{equation}
\label{eq18}\end{equation}
\begin{equation}
\label{eq19}\mbox{\rm Symbol } {ALTSUPERSUB 
\left(
{M, \: 3, \: {\left( 1 
\right)}}
\right)}
\mbox{\rm not involutive! } \mbox{\rm Dimension: } 3\end{equation}
\begin{equation}
\label{eq20}\end{equation}
\begin{equation}
\label{eq21}\mbox{\rm Symbol } {ALTSUPERSUB 
\left(
{M, \: 4, \: {\left( 1 
\right)}}
\right)}
\mbox{\rm involutive! } \mbox{\rm Dimension: } 2\end{equation}
\begin{equation}
\label{eq22}\end{equation}
\begin{equation}
\label{eq23}\mbox{\rm Equation } {ALTSUPERSUB 
\left(
{R, \: 4, \: {\left( 1 
\right)}}
\right)}
\mbox{\rm not involutive! } \mbox{\rm Dimension: } {13}\end{equation}
\begin{equation}
\label{eq24}\end{equation}
\begin{equation}
\label{eq25}\mbox{\rm ======= } 2 \mbox{\rm . Projection =======}\end{equation}
\begin{equation}
\label{eq26}\mbox{\rm Integrability condition(s)}\end{equation}
\begin{equation}
\label{eq27}\begin{array}{c}{\ 
}\{{u \sb {x, \: x, \: x, \: x}}=0}\\end{array}\end{equation}
\begin{equation}
\label{eq28}=============================\end{equation}
\begin{equation}
\label{eq29}\end{equation}
\begin{equation}
\label{eq30}\mbox{\rm Symbol } {ALTSUPERSUB 
\left(
{M, \: 4, \: {\left( 2 
\right)}}
\right)}
\mbox{\rm not involutive! } \mbox{\rm Dimension: } 1\end{equation}
\begin{equation}
\label{eq31}\end{equation}
\begin{equation}
\label{eq32}\mbox{\rm Symbol } {ALTSUPERSUB 
\left(
{M, \: 5, \: {\left( 2 
\right)}}
\right)}
\mbox{\rm involutive! } \mbox{\rm Dimension: } 0\end{equation}
\begin{equation}
\label{eq33}\end{equation}
\begin{equation}
\label{eq34}\mbox{\rm ************ Final Result ************}\end{equation}
\begin{equation}
\label{eq35}\end{equation}
\begin{equation}
\label{eq36}\mbox{\rm Equation } {ALTSUPERSUB 
\left(
{R, \: 5, \: {\left( 2 
\right)}}
\right)}
\mbox{\rm involutive!}\end{equation}
\begin{equation}
\label{eq37}\mbox{\rm System without prolonged equations. Dimension: } {12}\end{equation}
\begin{equation}
\label{eq38}\begin{array}{c}{\ 
}\{{u \sb {x, \: x, \: x, \: x}}=0}\{\ }\{{u \sb {y, \: x, \: x}}=0}\{\ 
}\{{{u \sb {z, \: z}}+{y \  {u \sb {x, \: x}}}}=0}\{\ }\{{u \sb {y, \: 
y}}=0}\\end{array}\end{equation}
\begin{equation}
\label{eq39}\end{equation}
\begin{equation}
\label{eq40}\mbox{\rm System of finite type.}\end{equation}

Type: Void
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