Covariant derivative of a vector

I have a code that gives me the Christoffel symbols of a metric. How do I take the covariant derivative of a vector?

It does not necessarily have to build upon my code, but this is what I have used so far that gives me the affine connactions:

n = 4 
coord = {T, R, [Theta], [Phi]}

metric = {{-1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, R^2, 0}, {0, 0, 0, R^2 Sin[[Theta]]^2}}

inversemetric = Simplify[Inverse[metric]]

affine := affine = Simplify[Table[(1/2)*Sum[(inversemetric[[i, s]])*
       (D[metric[[s, j]], coord[[k]] ] +
         D[metric[[s, k]], coord[[j]] ] - 
         D[metric[[j, k]], coord[[s]] ]), {s, 1, n}],
    {i, 1, n}, {j, 1, n}, {k, 1, n}] ]

listaffine := 
 Table[If[UnsameQ[affine[[i, j, k]], 
    0], {ToString[[CapitalGamma][i, j, k]], affine[[i, j, k]]}] , {i,
    1, n}, {j, 1, n}, {k, 1, j}]

TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2], 
 TableSpacing -> {2, 2}]```