\documentclass{article}

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\usepackage{graphicx}
\usepackage{color}
\usepackage{amsmath}

\definecolor{labelcolor}{RGB}{100,0,0}

\begin{document}

\pagebreak{}
{\Huge {\sc Victor T.Toth}}
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\section{http://www.vttoth.com/CMS/projects/61-the-maxima-computer-algebra-system}


Example of using Maxima for derivation of Schwartzschild metric from Hilbert Variational principle.

Below L is Lambda-term.

Slightly corrected by S.Blinnikov on 21st October 2012 (Novosibirsk): Friedmann lines are removed.

\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i1)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
if get('ctensor,'version)=false then load(ctensor);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o1) }
/usr/share/maxima/5.24.0/share/tensor/ctensor.mac
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i2)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
if get('itensor,'version)=false then load(itensor);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o2) }
/usr/share/maxima/5.24.0/share/tensor/itensor.lisp
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i3)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
remsym(g,2,0);
remsym(g,0,2);
remsym(gg,2,0);
remsym(gg,0,2);
remcomps(gg);
imetric(gg);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
% \begin{math}\displaystyle
No symmetries have been declared for this tensor.
% \end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o3) }
done
\end{math}

No symmetries have been declared for this tensor.

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o4) }
done
\end{math}

No symmetries have been declared for this tensor.

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o5) }
done
\end{math}

No symmetries have been declared for this tensor.

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o6) }
done
\end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o7) }
done
\end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o8) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i9)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
icurvature([a,b,c],[e])*gg([d,e],[])$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i10)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(expand(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i11)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
%,ichr2$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i12)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(expand(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i13)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
canform(%)$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i14)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(expand(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i15)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
components(gg([a,b],[]),kdels([a,b],[u,v])*g([u,v],[])/2);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o15) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i16)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
components(gg([],[a,b]),kdels([u,v],[a,b])*g([],[u,v])/2);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o16) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i17)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
%th(4),gg$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i18)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(expand(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i19)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(canform(%))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i20)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
imetric(g);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o20) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i21)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(expand(%th(2))))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i22)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
remcomps(R);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o22) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i23)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
 components(R([a,b,c,d],[]),%th(2));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o23) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i24)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
g([],[a,b])*R([a,b,c,d])*g([],[c,d])$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i25)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(canform(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i26)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
contract(rename(canform(%)))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i27)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
components(R([],[]),%);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o27) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i28)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
decsym(g,2,0,[sym(all)],[]);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o28) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i29)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
decsym(g,0,2,[],[sym(all)]);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o29) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i30)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ishow(1/(16*%pi*G)*((2*L+'R([],[])))*sqrt(-determinant(g)))$
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%t30) }
\frac{\sqrt{-\mathrm{determinant}\left( g\right) }\,\left( 2\,L+R\right) }{16\,\pi \,G}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i31)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
L0:%,R$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i32)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
canform(contract(canform(rename(contract(expand(diff(L0,g([],[m,n]))-
    idiff(diff(L0,g([],[m,n],k)),k)+idiff(rename(idiff(contract(
    diff(L0,g([],[m,n],k,l))),k),1000),l)))))))$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i33)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ishow(e([m,n],[])=canform(%*16*%pi/sqrt(-determinant(g))))$
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%t33) }
{e}_{m n}=-\frac{{g}_{m n}\,L}{G}+\frac{{g}_{,\%3}^{\%1 \%2}\,{g}^{\%3 \%5}\,{g}_{\%1 \%2}\,{ichr2}_{\%4 \%5}^{\%4}\,{g}_{m n}}{2\,G}+\frac{{g}_{,\%3}^{\%1 \%2}\,{g}^{\%3 \%6}\,{g}_{,\%6}^{\%4 \%5}\,{g}_{\%1 \%2}\,{g}_{\%4 \%5}\,{g}_{m n}}{8\,G}+\frac{{g}^{\%2 \%4}\,{ichr2}_{\%1 \%2}^{\%1}\,{ichr2}_{\%3 \%4}^{\%3}\,{g}_{m n}}{G}+\frac{{g}_{,\%2}^{\%1 \%2}\,{g}_{,\%1}^{\%3 \%4}\,{g}_{\%3 \%4}\,{g}_{m n}}{8\,G}-\frac{3\,{g}_{,\%3}^{\%1 \%2}\,{g}^{\%3 \%6}\,{g}_{,\%6}^{\%4 \%5}\,{g}_{\%1 \%4}\,{g}_{\%2 \%5}\,{g}_{m n}}{16\,G}+\frac{5\,{g}_{,\%3}^{\%1 \%2}\,{g}^{\%3 \%6}\,{g}_{,\%6}^{\%4 \%5}\,{g}_{\%1 \%5}\,{g}_{\%2 \%4}\,{g}_{m n}}{16\,G}-\frac{{g}_{,\%3}^{\%1 \%2}\,{g}_{,\%1}^{\%3 \%4}\,{g}_{\%2 \%4}\,{g}_{m n}}{4\,G}+\frac{{g}_{,\%1}^{\%1 \%2}\,{ichr2}_{\%2 \%3}^{\%3}\,{g}_{m n}}{2\,G}+\frac{{g}_{,\%2}^{\%1 \%2}\,{ichr2}_{\%1 \%3}^{\%3}\,{g}_{m n}}{2\,G}+\frac{{g}^{\%2 \%3}\,{ichr2}_{\%1 \%2,\%3}^{\%1}\,{g}_{m n}}{G}-\frac{{g}_{,\%3}^{\%1 \%2}\,{g}_{,\%4}^{\%3 \%4}\,{g}_{\%1 \%2}\,{g}_{m n}}{8\,G}+\frac{{g}_{
,\%1 \%2}^{\%1 \%2}\,{g}_{m n}}{2\,G}+\frac{{g}_{,\%3}^{\%1 \%2}\,{g}^{\%3 \%5}\,{g}_{\%1 n}\,{g}_{\%2 m}\,{ichr2}_{\%4 \%5}^{\%4}}{2\,G}-\frac{{g}_{,n}^{\%1 \%2}\,{g}_{\%1 \%2}\,{ichr2}_{\%3 m}^{\%3}}{2\,G}+\frac{{g}_{,n}^{\%1 \%2}\,{g}_{,\%1}^{\%3 \%4}\,{g}_{\%2 \%4}\,{g}_{\%3 m}}{4\,G}+\frac{{g}_{,n}^{\%1 \%2}\,{g}_{,\%2}^{\%3 \%4}\,{g}_{\%1 \%4}\,{g}_{\%3 m}}{4\,G}-\frac{{g}_{,\%6}^{\%1 \%5}\,{g}_{,\%4}^{\%2 \%3}\,{g}^{\%4 \%6}\,{g}_{\%1 n}\,{g}_{\%2 m}\,{g}_{\%3 \%5}}{2\,G}-\frac{{g}_{,n}^{\%1 \%2}\,{g}_{,m}^{\%3 \%4}\,{g}_{\%1 \%2}\,{g}_{\%3 \%4}}{4\,G}-\frac{{ichr2}_{\%1 n}^{\%1}\,{ichr2}_{\%2 m}^{\%2}}{G}+\frac{9\,{g}_{,\%3}^{\%1 \%2}\,{g}_{,\%4}^{\%3 \%4}\,{g}_{\%1 n}\,{g}_{\%2 m}}{16\,G}-\frac{{g}_{,\%4}^{\%1 \%2}\,{g}_{,\%3}^{\%3 \%4}\,{g}_{\%1 n}\,{g}_{\%2 m}}{16\,G}+\frac{{g}_{,\%3 \%4}^{\%1 \%2}\,{g}^{\%3 \%4}\,{g}_{\%1 n}\,{g}_{\%2 m}}{2\,G}-\frac{{g}_{,\%3}^{\%1 \%4}\,{g}_{,\%4}^{\%2 \%3}\,{g}_{\%1 n}\,{g}_{\%2 m}}{2\,G}-\frac{{g}_{,n}^{\%1 \%2}\,{ichr2}_{\%1 \%3}^{\%3}\,{g}_{\%2 m}}{4\,G}-\
frac{{g}_{,\%1 n}^{\%1 \%2}\,{g}_{\%2 m}}{4\,G}+\frac{{g}_{,\%3}^{\%1 \%4}\,{g}_{,m}^{\%2 \%3}\,{g}_{\%1 n}\,{g}_{\%2 \%4}}{2\,G}+\frac{3\,{g}_{,n}^{\%1 \%2}\,{g}_{,m}^{\%3 \%4}\,{g}_{\%1 \%3}\,{g}_{\%2 \%4}}{8\,G}-\frac{{g}_{,m}^{\%1 \%2}\,{g}_{\%1 n}\,{ichr2}_{\%2 \%3}^{\%3}}{2\,G}-\frac{{g}_{,n}^{\%1 \%2}\,{g}_{\%1 m}\,{ichr2}_{\%2 \%3}^{\%3}}{4\,G}-\frac{{g}_{,m}^{\%2 \%3}\,{ichr2}_{\%1 n}^{\%1}\,{g}_{\%2 \%3}}{2\,G}-\frac{5\,{g}_{,n}^{\%1 \%2}\,{g}_{,m}^{\%3 \%4}\,{g}_{\%1 \%4}\,{g}_{\%2 \%3}}{8\,G}-\frac{{g}_{,\%2 m}^{\%1 \%2}\,{g}_{\%1 n}}{2\,G}-\frac{{ichr2}_{\%1 m,n}^{\%1}}{G}-\frac{{g}_{,\%2 n}^{\%1 \%2}\,{g}_{\%1 m}}{4\,G}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i34)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
EQ:ic_convert(%)$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i35)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ct_coords:[t,r,u,v];
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o35) }
[t,r,u,v]
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i36)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
lg:ident(4);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o36) }
\begin{pmatrix}1 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0 & 1 & 0\cr 0 & 0 & 0 & 1\end{pmatrix}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i37)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
lg[1,1]:B;
lg[2,2]:-A;
lg[3,3]:-r^2;
lg[4,4]:-r^2*sin(u)^2;
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o37) }
B
\end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o38) }
-A
\end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o39) }
-{r}^{2}
\end{math}

\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o40) }
-{r}^{2}\,{\mathrm{sin}\left( u\right) }^{2}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i41)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
kill(dependencies);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o41) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i42)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
dependencies(A(r),B(r));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o42) }
[\mathrm{A}\left( r\right) ,\mathrm{B}\left( r\right) ]
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i43)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
cmetric();
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o43) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i44)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
christof(false);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o44) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i45)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
e:zeromatrix(4,4);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o45) }
\begin{pmatrix}0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\end{pmatrix}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i46)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ev(EQ);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o46) }
done
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i47)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
E:expand(radcan(ug.e));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o47) }
\begin{pmatrix}-\frac{L}{G}+\frac{\frac{d}{d\,r}\,A}{r\,{A}^{2}\,G}-\frac{1}{{r}^{2}\,A\,G}+\frac{1}{{r}^{2}\,G} & 0 & 0 & 0\cr 0 & -\frac{L}{G}-\frac{\frac{d}{d\,r}\,B}{r\,A\,B\,G}-\frac{1}{{r}^{2}\,A\,G}+\frac{1}{{r}^{2}\,G} & 0 & 0\cr 0 & 0 & -\frac{L}{G}-\frac{\frac{{d}^{2}}{d\,{r}^{2}}\,B}{2\,A\,B\,G}+\frac{{\left( \frac{d}{d\,r}\,B\right) }^{2}}{4\,A\,{B}^{2}\,G}+\frac{\left( \frac{d}{d\,r}\,A\right) \,\left( \frac{d}{d\,r}\,B\right) }{4\,{A}^{2}\,B\,G}-\frac{\frac{d}{d\,r}\,B}{2\,r\,A\,B\,G}+\frac{\frac{d}{d\,r}\,A}{2\,r\,{A}^{2}\,G} & 0\cr 0 & 0 & 0 & -\frac{L}{G}-\frac{\frac{{d}^{2}}{d\,{r}^{2}}\,B}{2\,A\,B\,G}+\frac{{\left( \frac{d}{d\,r}\,B\right) }^{2}}{4\,A\,{B}^{2}\,G}+\frac{\left( \frac{d}{d\,r}\,A\right) \,\left( \frac{d}{d\,r}\,B\right) }{4\,{A}^{2}\,B\,G}-\frac{\frac{d}{d\,r}\,B}{2\,r\,A\,B\,G}+\frac{\frac{d}{d\,r}\,A}{2\,r\,{A}^{2}\,G}\end{pmatrix}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i48)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
exp:findde(E,2);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o48) }
[{r}^{2}\,{A}^{2}\,L-r\,\left( \frac{d}{d\,r}\,A\right) -{A}^{2}+A,{r}^{2}\,A\,B\,L+r\,\left( \frac{d}{d\,r}\,B\right) -A\,B+B,4\,r\,{A}^{2}\,{B}^{2}\,L+2\,r\,A\,B\,\left( \frac{{d}^{2}}{d\,{r}^{2}}\,B\right) -r\,A\,{\left( \frac{d}{d\,r}\,B\right) }^{2}-r\,\left( \frac{d}{d\,r}\,A\right) \,B\,\left( \frac{d}{d\,r}\,B\right) +2\,A\,B\,\left( \frac{d}{d\,r}\,B\right) -2\,\left( \frac{d}{d\,r}\,A\right) \,{B}^{2}]
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i49)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
solve(ode2(exp[1],A,r),A);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o49) }
[A=-\frac{3\,r}{{r}^{3}\,L-3\,r-3\,\%c}]
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i50)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
%,%c=-2*M;
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o50) }
[A=-\frac{3\,r}{6\,M+{r}^{3}\,L-3\,r}]
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i51)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
a:%[1],%c=-2*M;
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o51) }
A=-\frac{3\,r}{6\,M+{r}^{3}\,L-3\,r}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i52)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ode2(ev(exp[2],a),B,r);
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o52) }
B=\frac{\%c\,\left( 6\,M+{r}^{3}\,L-3\,r\right) }{r}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i53)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
b:ev(%,%c=rhs(solve(rhs(%)*rhs(a)=1,%c)[1]));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o53) }
B=-\frac{6\,M+{r}^{3}\,L-3\,r}{3\,r}
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i54)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
factor(ev(ev(exp[3],a,b),diff));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%o54) }
0
\end{math}
%%%%%%%%%%%%%%%


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i55)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
lg:ev(lg,a,b),L=0$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i56)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
ug:invert(lg)$
\end{verbatim}}
\end{minipage}


\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{8ex}{\color{red}\bf
\begin{verbatim}
(%i57)
\end{verbatim}}
\end{minipage}
\begin{minipage}[t]{\textwidth}{\color{blue}
\begin{verbatim}
block([title: "Schwarzschild Potential for Mass M=2",M:2.],
    wxplot3d([r*cos(th),r*sin(th),1-ug[1,1]],[r,5.,50.],[th,-%pi,%pi],
    ['grid,20,30],['z,-2,0],[psfile],['legend,title]));
\end{verbatim}}
\end{minipage}
%%% OUTPUT:
\begin{math}\displaystyle
\parbox{8ex}{\color{labelcolor}(\%t57) }
\includegraphics[width=9cm]{schwMaximaDerive_1.png}
\end{math}

% \begin{math}\displaystyle
% \parbox{8ex}{\color{labelcolor}(\%o57) }
%
% \end{math}
%%%%%%%%%%%%%%%

\end{document}
