Thank you so much again!

If I really need to find the abbreviated formula for the given numeric result,

What other computer program can do this job other than Mathematica..?

Could you pleae recommend any of them?

I really do appreciate your help! 

Thank you Bill!

You are quite a huge helper!

Hope you have successes for your future careers as well!

Thanks to Nasser as well!

To be honest, my final step towards the problem will be having the simplest expression of the

result that I can use in practice. For this reason, something like including the expression like

sigma (for summation not for standard deviation) or the most convenient way to understand the

result will be quite useful for making the extenuating practical test for this result.

The final result that I want to make into the simplest expression to be used in practice and extended useages

will be generated from the following codes:

as = Variance[Table[u[i, t] - u[i, t - 1], {t, 1, 3}, {i, 1, 2}]];
aa = Total[Flatten[FullSimplify[as, Element[u[_, _], Reals]]]];
cv = Covariance[Table[u[i, t] - u[i, t - 1], {t, 1, 3}, {i, 1, 2}]];
m = FullSimplify[cv, Element[u[_, _], Reals]];
bb = Total[Flatten[FullSimplify[m - IdentityMatrix[2]*Diagonal]]];
D[Simplify[((aa + 2*bb)^(1/2))*F, Element[u[_, _], Reals]], u[1, 2]]

-> here, the u[1,2] in the last part of the last line can vary. This is the reason why I am trying to derive

a generalized expression for future usages.

Is there a way to do so?

Again..Thank you very much for you sincere helps and hope all successes for your future endeavors!

Trying a very small example, perhaps 3x3 like I did, to verify if that is working can help you track down problems you can't see in larger problems.

Being extremely careful to not make small mistakes, like leaving the [ m ] off the end of Diagonal as you did, is very important in Mathematica.
Mathematica is not like a bright graduate student who can see all your typographical errors and understand what you meant and give you the
correct answer anyway. There is no "do what I mean" button which can do that in Mathematica yet and I do not expect one any time soon.
REALLY understanding the "Mathematica way of thinking" and mastering all the details of the first 200 Mathematica functions that you will use
every day will be of great assistance to you in the future and make your life much easier.

Does this example help solve part of your problem?

as Bill pointed out, MatrixForm changes the head of the expression. Two common ways to show the matrix as MatrixForm and still be able to use the matrix as List, is to protect the matrix, like this:



Another way, but not recommended is to do this:
What this does is keep the MatrixForm head in there, attached to the data but uses First@mat to jump into the data inside it. But again, this is not a good way to handle it. Better to use one of the above two methods.