Some moderately interesting Maxima examples.

First, this is how we can prove that the covariant derivative of the metric vanishes (but only if the metric is symmetric!)

load(itensor); imetric(g); ishow(covdiff(g([],[i,j]),k))$ %,ichr2$ ishow(contract(canform(contract(canform(rename(expand(%)))))))$ ishow(covdiff(g([i,j],[]),k))$ %,ichr2$ ishow(canform(contract(rename(expand(%)))))$ decsym(g,2,0,[sym(all)],[]); decsym(g,0,2,[],[sym(all)]); ishow(covdiff(g([],[i,j]),k))$ %,ichr2$ ishow(contract(canform(contract(canform(rename(expand(%)))))))$ ishow(covdiff(g([i,j],[]),k))$ %,ichr2$ ishow(canform(contract(rename(expand(%)))))$

Next, the equation of motion for a perfect fluid:

load(itensor); imetric(g); decsym(g,2,0,[sym(all)],[]); decsym(g,0,2,[],[sym(all)]); defcon(v,v,u); components(u([],[]),1); components(T([],[i,j]),(rho([],[])+p([],[]))*v([],[i])*v([],[j]) -p([],[])*g([],[i,j])); ishow(covdiff(T([],[i,j]),i))$ ishow(canform(%))$ ishow(canform(rename(contract(expand(%)))))$ %,ichr2$ canform(%)$ ishow(canform(rename(contract(expand(%)))))$

Finally, the equation of motion in the spherically symmetric, static case:

load(ctensor); load(itensor); K:J([i],[])=covdiff(T([i],[j]),j); E:ic_convert(K); ct_coords:[t,r,u,v]; lg:ident(4); lg[1,1]:B; lg[2,2]:-A; lg[3,3]:-r^2; lg[4,4]:-r^2*sin(u)^2; depends([A,B,T,rho,p],[r]); derivabbrev:true; cmetric(); christof(mcs); J:[0,0,0,0]; ev(E); T:ident(4); T[1,1]:rho; T[2,2]:T[3,3]:T[4,4]:p; J,ev;

These examples are probably not profound enough to include with Maxima, but are useful to remember.