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	Jet Bundles
Jet Figure 4 ←	↑	→ Jet LUDecomposition

Jet Figure 5

Edit detail for Jet Figure 5 revision 1 of 6

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Editor: 127.0.0.1 Time: 2007/11/11 19:07:57 GMT-8
Note: transferred from axiom-developer

changed:
-
\begin{axiom}
)lib JBC JBFC JBC- JBFC- BFC BFC- JB JBX JBLF DE SEM  DIFF JBE IJB CK
\end{axiom}

\begin{axiom}
jb:=IJB('x,'u,'p,4,3);
jbe:=JBE jb;
de:=DE(jb,jbe);
ck:=CK(jb,jbe);
\end{axiom}

\begin{axiom}
eq1:jbe := P(1,[4]) + U(1)*P(1,[1]) + U(2)*P(1,[2]) + U(3)*P(1,[3])
eq2:jbe := P(2,[4]) + U(1)*P(2,[1]) + U(2)*P(2,[2]) + U(3)*P(2,[3])
eq3:jbe := P(3,[4]) + U(1)*P(3,[1]) + U(2)*P(3,[2]) + U(3)*P(3,[3])
eq4:jbe := P(1,[1]) + P(2,[2]) + P(3,[3])
\end{axiom}

\begin{axiom}
euler:de := generate [eq1,eq2,eq3,eq4]
\end{axiom}

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

\begin{axiom}
complete euler
\end{axiom}

axiom)lib JBC JBFC JBC- JBFC- BFC BFC- JB JBX JBLF DE SEM  DIFF JBE IJB 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
   IndexedJetBundle is now explicitly exposed in frame initial 
   IndexedJetBundle will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/IJB.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:=IJB('x,'u,'p,4,3);
Type: Domain
axiom
jbe:=JBE jb;
Type: Domain
axiom
de:=DE(jb,jbe);
Type: Domain
axiom
ck:=CK(jb,jbe);
Type: Domain

axiom
eq1:jbe := P(1,[4]) + U(1)*P(1,[1]) + U(2)*P(1,[2]) + U(3)*P(1,[3])
\begin{equation}
\label{eq1}{p \sb {{4}} \sp {1}}+{{u \sp {3}} \  {p \sb {{3}} \sp {1}}}+{{u \sp {2}} \  
{p \sb {{2}} \sp {1}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {1}}} 
\end{equation}
Type: JetBundleExpression? IndexedJetBundle?(x,u,p,4,3)
axiom
eq2:jbe := P(2,[4]) + U(1)*P(2,[1]) + U(2)*P(2,[2]) + U(3)*P(2,[3])
\begin{equation}
\label{eq2}{p \sb {{4}} \sp {2}}+{{u \sp {3}} \  {p \sb {{3}} \sp {2}}}+{{u \sp {2}} \  
{p \sb {{2}} \sp {2}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {2}}} 
\end{equation}
Type: JetBundleExpression? IndexedJetBundle?(x,u,p,4,3)
axiom
eq3:jbe := P(3,[4]) + U(1)*P(3,[1]) + U(2)*P(3,[2]) + U(3)*P(3,[3])
\begin{equation}
\label{eq3}{p \sb {{4}} \sp {3}}+{{u \sp {3}} \  {p \sb {{3}} \sp {3}}}+{{u \sp {2}} \  
{p \sb {{2}} \sp {3}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {3}}} 
\end{equation}
Type: JetBundleExpression? IndexedJetBundle?(x,u,p,4,3)
axiom
eq4:jbe := P(1,[1]) + P(2,[2]) + P(3,[3])
\begin{equation}
\label{eq4}{p \sb {{3}} \sp {3}}+{p \sb {{2}} \sp {2}}+{p \sb {{1}} \sp {1}} 
\end{equation}
Type: JetBundleExpression? IndexedJetBundle?(x,u,p,4,3)

axiom
euler:de := generate [eq1,eq2,eq3,eq4]
\begin{equation}
\label{eq5}\begin{array}{c}{\ 
}\{{{p \sb {{4}} \sp {1}}+{{u \sp {3}} \  {p \sb {{3}} \sp {1}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {1}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {1}}}}=0}\{\ 
}\{{{p \sb {{4}} \sp {2}}+{{u \sp {3}} \  {p \sb {{3}} \sp {2}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {2}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {2}}}}=0}\{\ 
}\{{{p \sb {{4}} \sp {3}}+{{u \sp {3}} \  {p \sb {{3}} \sp {3}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {3}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {3}}}}=0}\{\ 
}\{{{p \sb {{3}} \sp {3}}+{p \sb {{2}} \sp {2}}+{p \sb {{1}} \sp 
{1}}}=0}\\end{array} 
\end{equation}
Type: DifferentialEquation?(IndexedJetBundle?(x,u,p,4,3),JetBundleExpression? IndexedJetBundle?(x,u,p,4,3))

axiom
setOutMode(14)$ck
\begin{equation}
\label{eq6}0 
\end{equation}
Type: NonNegativeInteger?
axiom
setSimpMode(1)$ck
\begin{equation}
\label{eq7}0 
\end{equation}
Type: NonNegativeInteger?

axiom
complete euler
\begin{equation}
\label{eq8}\end{equation}
\begin{equation}
\label{eq9}\mbox{\rm Symbol } {M \sb {1}} \mbox{\rm involutive! } \mbox{\rm Dimension: } 
8\end{equation}
\begin{equation}
\label{eq10}\end{equation}
\begin{equation}
\label{eq11}\mbox{\rm Equation } {R \sb {1}} \mbox{\rm not involutive! } \mbox{\rm 
Dimension: } {11}\end{equation}
\begin{equation}
\label{eq12}\end{equation}
\begin{equation}
\label{eq13}\mbox{\rm ======= } 1 \mbox{\rm . Projection =======}\end{equation}
\begin{equation}
\label{eq14}\mbox{\rm Integrability condition(s)}\end{equation}
\begin{equation}
\label{eq15}\begin{array}{c}{\ 
}\{{{2 \  {p \sb {{2}} \sp {3}} \  {p \sb {{3}} \sp {2}}}+{2 \  {p \sb {{1}} 
\sp {3}} \  {p \sb {{3}} \sp {1}}}+{2 \  {{p \sb {{2}} \sp {2}} \sp 2}}+{2 \  
{p \sb {{1}} \sp {1}} \  {p \sb {{2}} \sp {2}}}+{2 \  {p \sb {{1}} \sp {2}} \  
{p \sb {{2}} \sp {1}}}+{2 \  {{p \sb {{1}} \sp {1}} \sp 2}}}=0}\\end{array}\end{equation}
\begin{equation}
\label{eq16}=============================\end{equation}
\begin{equation}
\label{eq17}\end{equation}
\begin{equation}
\label{eq18}\mbox{\rm Symbol } {ALTSUPERSUB 
\left(
{M, \: 1, \: {\left( 1 
\right)}}
\right)}
\mbox{\rm involutive! } \mbox{\rm Dimension: } 7\end{equation}
\begin{equation}
\label{eq19}\end{equation}
\begin{equation}
\label{eq20}\mbox{\rm ************ Final Result ************}\end{equation}
\begin{equation}
\label{eq21}\end{equation}
\begin{equation}
\label{eq22}\mbox{\rm Equation } {ALTSUPERSUB 
\left(
{R, \: 1, \: {\left( 1 
\right)}}
\right)}
\mbox{\rm involutive!}\end{equation}
\begin{equation}
\label{eq23}\mbox{\rm System without prolonged equations. Dimension: } {14}\end{equation}
\begin{equation}
\label{eq24}\begin{array}{c}{\ 
}\{{{p \sb {{4}} \sp {3}}+{{u \sp {3}} \  {p \sb {{3}} \sp {3}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {3}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {3}}}}=0}\{\ 
}\{{{p \sb {{4}} \sp {2}}+{{u \sp {3}} \  {p \sb {{3}} \sp {2}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {2}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {2}}}}=0}\{\ 
}\{{{p \sb {{4}} \sp {1}}+{{u \sp {3}} \  {p \sb {{3}} \sp {1}}}+{{u \sp 
{2}} \  {p \sb {{2}} \sp {1}}}+{{u \sp {1}} \  {p \sb {{1}} \sp {1}}}}=0}\{\ 
}\{{{p \sb {{3}} \sp {3}}+{p \sb {{2}} \sp {2}}+{p \sb {{1}} \sp 
{1}}}=0}\{\ }\{{{2 \  {p \sb {{2}} \sp {3}} \  {p \sb {{3}} \sp {2}}}+{2 \  
{p \sb {{1}} \sp {3}} \  {p \sb {{3}} \sp {1}}}+{2 \  {{p \sb {{2}} \sp {2}} 
\sp 2}}+{2 \  {p \sb {{1}} \sp {1}} \  {p \sb {{2}} \sp {2}}}+{2 \  {p \sb 
{{1}} \sp {2}} \  {p \sb {{2}} \sp {1}}}+{2 \  {{p \sb {{1}} \sp {1}} \sp 
2}}}=0}\\end{array}\end{equation}
\begin{equation}
\label{eq25}\end{equation}
\begin{equation}
\label{eq26}\mbox{\rm Cartan characters: } {3, \: 3, \: 1, \: 0}\end{equation}
Type: Void

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### bottom level
(see the transcript file for additional information)
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Transcript written on 3000372303235824751-18px.log.