Define Matrix Function in a For loop

Question

Let $g\in\mathcal R^{3\times 3}$ be a given known rank 2 tensor function, $gInv$ its inverse, and $dg\in\mathcal R^{3\times 3\times 3}$ its gradiant:

g[x_,y_,z_] = {{g11[x,y,z],g12[x,y,z],g13[x,y,z]},
               {g21[x,y,z],g22[x,y,z],g23[x,y,z]},
               {g31[x,y,z],g32[x,y,z],g33[x,y,z]}};
dg[x_,y_,z_] = Grad[g[x, y, z], {x, y, z}];

I want to define a rank 3 tensor function componentwise by looping over each entry. Ideally like this:

G[x_, y_, z_][[k, i, j]] = 0;
For[i = 1, i <= 3, i++, For[j = 1, j <= 3, j++, For[k = 1, k <= 3, k++, For[d = 1, d <= 3, d++,
   G[x_, y_, z_][[k, i, j]] +=  0.5 gInv[x, y, z][[k, d]]
   (dg[x, y, z][[i, d, j]] + dg[x, y, z][[d, j, i]] - dg[x, y, z][[i, j, d]]);
]]]]

or maybe like this if the += is not suitable for a function definition:

For[i = 1, i <= 3, i++, For[j = 1, j <= 3, j++, For[k = 1, k <= 3, k++,
   G[x_, y_, z_][[k, i, j]] = 
 0.5 gInv[x, y, z][[k, 1]](dg[x, y, z][[i, 1, j]] + dg[x, y, z][[1, j, i]] - dg[x, y, z][[i, j, 1]])
+0.5 gInv[x, y, z][[k, 2]](dg[x, y, z][[i, 2, j]] + dg[x, y, z][[2, j, i]] - dg[x, y, z][[i, j, 2]])
+0.5 gInv[x, y, z][[k, 3]](dg[x, y, z][[i, 3, j]] + dg[x, y, z][[3, j, i]] - dg[x, y, z][[i, j, 3]]);
]]]

However this notation does not work and unrolling all 4 For loops will most centainly introduce typos.

I am rather new to mathematica. Maybe there is a completely different solution which is much better. Feel free correct my approach or advice me something completely different. I might learn something.

Physics Background:

G are the Christoffel symbols from general relativity/differential geometry, and g the spatial 3-metric of a spatial hypersurface.

Answer 1

With the help of ebanb's comment I managed to solve my issue. Here is the solution for anybody who has a similar problem. By reversing the brakets and basicly defining one function for each tensor entry it it possible to loop over them the way I want to,

For[i = 1, i <= 3, i++, For[j = 1, j <= 3, j++, For[k = 1, k <= 3, k++,
   G[k, i, j][x_, y_, z_] = 
 0.5 gInv[k, 1][x, y, z](dg[i, 1, j][x, y, z] + dg[1, j, i][x, y, z] - dg[i, j, 1][x, y, z])
+0.5 gInv[k, 2][x, y, z](dg[i, 2, j][x, y, z] + dg[2, j, i][x, y, z] - dg[i, j, 2][x, y, z])
+0.5 gInv[k, 3][x, y, z](dg[i, 3, j][x, y, z] + dg[3, j, i][x, y, z] - dg[i, j, 3][x, y, z]);
]]]

This way also gInv and dg need to be defined in that manner. Finally I still need to be able to do Matrix operations on some of these tensors (e.g. inverting the 3x3 metric). For this I just create a Matrix and give it the corresponding entries,

g[x_,y_,z_] =
{{g[1,1][x,y,z],g[1,2][x,y,z],g[1,3][x,y,z]},
 {g[2,1][x,y,z],g[2,2][x,y,z],g[2,3][x,y,z]},
 {g[3,1][x,y,z],g[3,2][x,y,z],g[3,3][x,y,z]}};
gInv[x_,y_,z_] = Inverse[g[x,y,z]];

and then reextract them if needed for the loop.

For[i = 1, i <= 3, i++, For[j = 1, j <= 3, j++,
  gInv'[i, j][x_, y_, z_] = gInv[x, y, z][[i, j]];
  ]]