Extracting a vector from NDSolve solution

I made an NDSolveValue to get the variable ff[t]

sol = NDSolveValue[{l1[t] == 0, l3[t] == 0, l2[t] == 0, ic}, ff , {t, tmin,tmax}]

But ff[t] equals $$X_{i\bar{j}}~ \partial_t{f}^i (t) ~ \partial_t{f}^\bar{j} (t)$$ , where i,j = 1,2,3 and the $$\bar{}$$ is complex coordinates.

The question now can I extract from the solution of NDSolveValue the vector points of $$\dot{f}^i (t)$$ or $$\dot{f}^{\bar{j}} (t)$$

Also then can these vectors plotted ?

EDIT:

Here are the complete equations:

 l1[t_] = 3*(D[a[t], {t, 1}]^2/a[t]^2 + (D[a[t], {t, 1}]*D[b[t], {t, 1}])/(a[t]*b[t])) - 1/a[t]^4- ff[t];

l2[t_] = -(2*(D[a[t], {t, 2}]/a[t]) + D[a[t], {t, 1}]^2/a[t]^2 + D[b[t], {t, 2}]/b[t] + 2*((D[a[t], {t, 1}]*D[b[t], {t, 1}])/(a[t]*b[t]))) - (1/(3 a[t]^4))-ff[t];

l3[t_] := -3*(D[a[t], {t, 2}]/a[t] + D[a[t], {t, 1}]^2/a[t]^2)-ff[t]

with ic = {a[0] == 0.1, a'[0] == 0, b[0] == 0.1, b'[0] == 0};

Many thanks.

• If the code for your problem is small enough that it could be pasted into your post, not as a screenshot or a latex, but in a form that could be scraped off the screen and pasted into a Mathematica notebook and run, then readers might be able to try ideas and see if they could get you the matrix that you want. Next, if it isn't obvious to everyone else from the form of your matrix, can you describe what kind of 3d plot you expect from a 3x3 complex matrix of functions?
– Bill
Aug 8 '19 at 19:36
• @Bill , '' can you describe what kind of 3d plot you expect from a 3x3 complex matrix of functions" : well , I have made an edit for the question, to try to make a general plot for this matrix .. Aug 8 '19 at 20:09
• @Dr.phy If a and b are scalars, then how can ff be a matrix? In the form in which the problem is formulated, ff is a scalar that is uniquely determined from the solution of the system of equations. Aug 9 '19 at 4:18
• @AlexTrounev , Yeah you are absolutely right . $X_{ij} \dot{f}^i(t) \dot{f}^j (t)$ is as a whole a scalar, may be I need to modify the question, cause I still have a problem, I need to extract the vector $\dot{f}^i (t)$ Aug 9 '19 at 12:21
• @Dr.phy Using sol we have $X_{ij}\dot {f}^i\dot {f}^j=ff$. Now the question is this one equation (summation rule used) or 9? If one, then the equations are not enough to determine the vector. If there are 9 equations, then only 3 of them should be independent. This is a restriction on the matrix (for example, diagonal). Aug 9 '19 at 12:42