This is the solution to the problem in 366f18P.pdf done in Matheamtica
As stated in this old take home exam problem lots of repetative calculation is required
You should be able to do any of the simple 1-element calculations by hand and check with
these geenral results (e.g., find one Christoffel symbol rather than all non-zero)


gij=DiagonalMatrix[{1, Sin[x]^2, Sin[x]^2 Sin[t]^2}]


gijk=ConstantArray[0,{3,3,3}];
Do[ Do[ Do[
gijk[[i,j,k]]=Switch[k,1,D[gij[[i,j]],x],2,D[gij[[i,j]],t],3,D[gij[[i,j]],p]],{k,3}],{j,3}],{i,3}]

gIJ=Inverse[gij]
Glik=ConstantArray[0,{3,3,3}];
Do[ Do[ Do[
sum=Sum[gIJ[[l,j]] (gijk[[i,j,k]]+gijk[[j,k,i]]-gijk[[k,i,j]]),{j,3}];
Glik[[l,i,k]]=FullSimplify[sum/2],{k,3}],{l,3}],{i,3}]

Glik[[1,All,All]]

                                                                          2
Out[7]= {{0, 0, 0}, {0, -(Cos[x] Sin[x]), 0}, {0, 0, -(Cos[x] Sin[t]  Sin[x])}}


Glik[[2,All,All]]   
Out[8]= {{0, Cot[x], 0}, {Cot[x], 0, 0}, {0, 0, -(Cos[t] Sin[t])}}

Glik[[3,All,All]]
                1                          1
Out[9]= {{0, 0, -, 0}, {0, 0, Cot[t], 0}, {-, Cot[t], 0, 0}, {0, 0, 0, 0}}
                r                          r

Glik[[3,All,All]]

Out[9]= {{0, 0, Cot[x]}, {0, 0, Cot[t]}, {Cot[x], Cot[t], 0}}

Glijk=ConstantArray[0,{3,3,3,3}];
Do[ Do[ Do[ Do [
Glijk[[l,i,j,k]]=Switch[k,1,D[Glik[[l,i,j]],x],2,D[Glik[[l,i,j]],t],3,D[Glik[[l,i,j]],p],3,0],{k,3}],{l,3}],{j,3}],{i,3}]

RDabc=ConstantArray[0,{3,3,3,3}];

Do[ Do[ Do[ Do [
sum1=Sum[Glik[[j,a1,b1]] Glik[[d1,c1,j]]-Glik[[j,a1,c1]] Glik[[d1,b1,j]],{j,3}];
RDabc[[d1,a1,b1,c1]]=FullSimplify[sum1+Glijk[[d1,a1,b1,c1]]-Glijk[[d1,a1,c1,b1]] ],{a1,3}],{b1,3}],{c1,3}],{d1,3}]
Out[14]= {{{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
 
                  2             2
>     {{0, -Sin[x] , 0}, {Sin[x] , 0, 0}, {0, 0, 0}}, 
 
                      2       2                      2       2
>     {{0, 0, -(Sin[t]  Sin[x] )}, {0, 0, 0}, {Sin[t]  Sin[x] , 0, 0}}}, 
 
>    {{{0, 1, 0}, {-1, 0, 0}, {0, 0, 0}}, {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
 
                                 2       2              2       2
>     {{0, 0, 0}, {0, 0, -(Sin[t]  Sin[x] )}, {0, Sin[t]  Sin[x] , 0}}}, 
 
>    {{{0, 0, 1}, {0, 0, 0}, {-1, 0, 0}}, 
 
                               2              2
>     {{0, 0, 0}, {0, 0, Sin[x] }, {0, -Sin[x] , 0}}, 
 
>     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}}}

Rab=ConstantArray[0,{3,3}];
Do[ Do[ 
Rab[[a1,b1]]=Sum[RDabc[[d1,a1,b1,d1]],{d1,3}],{a1,3}],{b1,3}]
Rab=Simplify[Rab]

                                 2                     2       2
Out[17]= {{2, 0, 0}, {0, 2 Sin[x] , 0}, {0, 0, 2 Sin[t]  Sin[x] }}

RAb=ConstantArray[0,{3,3}];
Do[ Do[ 
RAb[[a1,b1]]=Sum[Rab[[d1,b1]] gIJ[[d1,a1]],{d1,3}],{a1,3}],{b1,3}]
RAb=Simplify[RAb]

Out[21]= {{2, 0, 0}, {0, 2, 0}, {0, 0, 2}}

R=Tr[RAb]
Out[22]= 6

Rdabc=ConstantArray[0,{3,3,3,3}];
Do[ Do[ Do[ Do [
Rdabc[[d1,a1,b1,c1]]=Sum[gij[[d1,j]] RDabc[[j,a1,b1,c1]],{j,3}],{a1,3}],{b1,3}],{c1,3}],{d1,3}]
Out[25]= {{{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
 
                  2             2
>     {{0, -Sin[x] , 0}, {Sin[x] , 0, 0}, {0, 0, 0}}, 
 
                      2       2                      2       2
>     {{0, 0, -(Sin[t]  Sin[x] )}, {0, 0, 0}, {Sin[t]  Sin[x] , 0, 0}}}, 
 
                 2              2
>    {{{0, Sin[x] , 0}, {-Sin[x] , 0, 0}, {0, 0, 0}}, 
 
>     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 
 
                                 2       4              2       4
>     {{0, 0, 0}, {0, 0, -(Sin[t]  Sin[x] )}, {0, Sin[t]  Sin[x] , 0}}}, 
 
                    2       2                       2       2
>    {{{0, 0, Sin[t]  Sin[x] }, {0, 0, 0}, {-(Sin[t]  Sin[x] ), 0, 0}}, 
 
                               2       4               2       4
>     {{0, 0, 0}, {0, 0, Sin[t]  Sin[x] }, {0, -(Sin[t]  Sin[x] ), 0}}, 
 
>     {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}}}


