Also, since each of the x1, x2, x3 terms are functions dependent on variables, I might not have reached a solution.

Hi Meryem,

any progress in your problem?

Did you realize that with your gradf a function giving this gradient does not exist?

What about telling us the background of your problem (without the lengthy Sin and Cos terms)?

Greetings HD

Rename your Grad(f) given above to gf = ... your expression.... and try

D[ gf[[1]], x2] == D[ gf[[2]], x1]
D[ gf[[1]], x3] == D[ gf[[3]], x1]
D[ gf[[2]], x3] == D[ gf[[3]], x2]

That means the function you are looking for does not exist

Here {x1, x2, x3} are not components of x. I named the components of Grad(f) as x1, x2, x3, respectively. So I made a naming as "Grad(f) = {x1, x2, x3}". The "x" is just the name of the surface.

First I computed the N and H values for the x surface. Then I calculated Grad(f) from the equation "<N, Grad(f)> = H" given above.

Grad(f) is in the file I sent. I don't know the function f and my main goal is to find the function f. How can I find a function f using only Grad(f)?

I want to underline what the first responding poster said* a minimization problem (other than 0) requires a constraint which still hasn't been stated. Say, two curves and a condition that says "what is considered minimal" (the distance between them or along them? a mutual distance from the origin?)*

You've also said you want f, but I assume f( x ) is the surface you gave, that the surface is not the constraint. If the constraint is the surface and you gave us (grad f)={x1,x2,f[x1,x2]} then you've still withheld (grad f)={x1,x2,x3} meaning before you altered it.

I can only guess you mean the direction of greatest change is normal to the surface and of least change is tangent to surface as a condition. In that light you want the equation that is always tangent to the surface and the gradient is not necessary since the surface was given. If r[t,s] = surface given, then r'[t,s] (easily done by mathematica), is tangent to the surface at each point and so the direction of least change. If you have N you should already have T, the tangent vector.

(* the vector valued surface *)
In[109]:= D[(a + t Cos[b s] - g[t] Sin[b s]) {Cos[s], Sin[s], 0} + {0,
    0, t Sin[b s] + g[t] Cos[b s]}, t, s]

(* the direction of least change, r'[t,s] *)
Out[109]= {Cos[s] (-b Sin[b s] - b Cos[b s] Derivative[1][g][t]) - 
  Sin[s] (Cos[b s] - Sin[b s] Derivative[1][g][t]), 
 Sin[s] (-b Sin[b s] - b Cos[b s] Derivative[1][g][t]) + 
  Cos[s] (Cos[b s] - Sin[b s] Derivative[1][g][t]), 
 b Cos[b s] - b Sin[b s] Derivative[1][g][t]}

If you mean minimal curvature K, then since you have a wavy sin cos surface then you may only need the maximum points of the surface to find minimal K or the 0 points for greatest K (ie, it may be a trick question not requiring the kind of solution your looking for).

You should ask a teacher or look in the book's "solution guide". And you should ask people this kind of question without the problem FULLY TYPED OUT as the book had it (and page, and name) - which likely requires being in a different forum altogether.

There's allot of free math showing how to find f from the gradient of f. Also the lagrange multiplier method (as well as others) for minimization. And linear algebra algorithms as well.

It's against Community policy to "drop a problem you haven't attempted to solve" in the forum, nor should problems be contrived to be unsolvable by certain methods without being introducing as such.