-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity...](limitten1.gif)
limitten.mws
The special case v(theta)=-1 , with constraints,
Notation diff(v(thtea),theta$n)=v[n]
| > | restart: |
the following substitution is made.
| > | P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t); |
| > | grtw(); |
Scalar invariant library.
Last modified 25 March 1997.
`Differential Invariants`
`Last modified Jan. 20, 1995`
`Basis/tetrad related object definitions`
`Last modified 23 January 2001`
`Last built 27 May, 1999`
`Last built 27 May, 1999`
| > | qload(gowdy); |
![`x `^a = vector([t, theta, x1, x2])](limitten12.gif){kind=small}
| > | grcalc(WeylSq); |
| > | gralter(_,13,6,7); |
Component simplification of a GRTensorII object:
Applying routine `Apply constraints repeatedly` to object WeylSq
Applying routine expand to object WeylSq
Applying routine factor to object WeylSq
| > | grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`); |
Applying routine subs to WeylSq
Keep all derivatives of Q[infinity](theta)
| > | grmap(_,subs,diff(Q[infinity](theta),theta$4)=Q[4],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(Q[infinity](theta),theta$3)=Q[3],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(Q[infinity](theta),theta)=Q[1],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(Q[infinity](theta),theta$2)=0,`x`); |
Applying routine subs to WeylSq
Keep all derivatives of v(theta)
| > | grmap(_,subs,diff(v(theta),theta$4)=v[4],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(v(theta),theta$2)=v[2],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(v(theta),theta$1)=v[1],`x`): |
Applying routine subs to WeylSq
Keep all derivatives of psi[Q](theta)
| > | grmap(_,subs,diff(psi[Q](theta),`$`(theta,2))=psi[Q2],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(psi[Q](theta),`$`(theta,3))=psi[Q3],`x`); |
Applying routine subs to WeylSq
| > | grmap(_,subs,diff(psi[Q](theta),`$`(theta,4))=psi[Q4],`x`); |
Applying routine subs to WeylSq
Apply constraints
| > | grmap(_,subs,diff(psi[Q](theta),theta)=0,`x`): |
Applying routine subs to WeylSq
| > | grmap(_,subs,psi[Q](theta)=0,`x`): |
Applying routine subs to WeylSq
| > | grmap(_,subs,v(theta)=-1,`x`): |
Applying routine subs to WeylSq
| > | gralter(_,6,7); |
Component simplification of a GRTensorII object:
Applying routine expand to object WeylSq
Applying routine factor to object WeylSq
| > | denom(grcomponent(WeylSq,[])); |
| > | factor(subs(t=0,numer(grcomponent(WeylSq,[])))); |
)^2*exp(0)^2*Q[3]^2*(Q[3]^4*exp(0)^4*exp(P[infinity](theta))^4*ln(0)^6+256*ln(0)^2-256*ln(0)+256-32*Q[3]^2*exp(0)^2*exp(P[infinity](theta))^2*ln(0)^3-96*Q[3]^2*exp(0)^2*exp(P[...](limitten21.gif)
| > | core:=limit(factor(t^3*grcomponent(WeylSq,[])/(log(t)^6*exp(gamma(t,theta)))),t=0); |
)^6*Q[3]^6](limitten22.gif)
| > | Ww:=core*log(t)^6*exp(gamma(t,theta))/t^3; |
)^6*Q[3]^6*ln(t)^6*exp(gamma(t,theta))/t^3](limitten23.gif)
| > | subs(Q[3]=diff(Q[infinity](theta),theta$3),Ww); |
)^6*diff(Q[infinity](theta),`$`(theta,3))^6*ln(t)^6*exp(gamma(t,theta))/t^3](limitten24.gif)
| > | latex(%); |
-{\frac {1}{256}}\,{\frac { \left( {e^{P_{{\infty }} \left( \theta
\right) }} \right) ^{6} \left( {\frac {d^{3}}{d{\theta}^{3}}}Q_{{
\infty }} \left( \theta \right) \right) ^{6} \left( \ln \left( t
\right) \right) ^{6}{e^{\gamma \left( t,\theta \right) }}}{{t}^{3}}}
| > |