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- % calculation of metric tensor (3-dim space-time)
- off echo;
- on revpri;
- n:=3;
- operator x$
- x(0):=t; x(1):=lambda0; x(2):=lambda1;
- % rules
- trig1:={sin(~x)^2=>(1-cos(x)^2)}$ let trig1$
- % procedures
- procedure scalprod(a,b); begin
- integer n; n:=first(length(a))-1;
- result:=for i:=0:n-1 sum a(i)*b(i);
- return result end;
- procedure showmatrix(mm);begin integer m,n;l:=length(mm);m:=first(l)-1;n:=second(l)-1;matrix hm(m,n);for i:=0:m-1 do for j:=0:n-1 do hm(i+1,j+1):=mm(i,j); write hm end;
- procedure showvector(vv);begin scalar n;n:=first(length(vv))-1;matrix hv(n,1);for i:=0:n-1 do hv(i+1,1):=vv(i);write hv end;
- array f(n+1), dfdt(n+1), dfdl0(n+1), dfdl1(n+1)$
- % current radius
- a:=a0*sqrt(1-t^2);
- % surface of hyper sphere in t and lambda
- f(0):=a*cos(lambda0)*cos(lambda1);
- f(1):=a*cos(lambda0)*sin(lambda1);
- f(2):=a*sin(lambda0);
- f(3):=a0*t;
- for i:=0:n do dfdt(i):=df(f(i),x(0))$
- for i:=0:n do dfdl0(i):=df(f(i),x(1))$
- for i:=0:n do dfdl1(i):=df(f(i),x(2))$
- array g(n,n)$
- g(0,0):=scalprod(dfdt,dfdt)$
- g(0,1):=scalprod(dfdt,dfdl0)$
- g(0,2):=scalprod(dfdt,dfdl1)$
- g(1,0):=scalprod(dfdl0,dfdt)$
- g(1,1):=scalprod(dfdl0,dfdl0)$
- g(1,2):=scalprod(dfdl0,dfdl1)$
- g(2,0):=scalprod(dfdl1,dfdt)$
- g(2,1):=scalprod(dfdl1,dfdl0)$
- g(2,2):=scalprod(dfdl1,dfdl1)$
- write "f = "; showvector(f);
- write "df/dt = "; showvector(dfdt);
- write "g = "; showmatrix(g);
- off revpri;
- on echo;
- end;
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