Surface level and surface integral

[Jhenrique]

I have some questions, all associated. So, first, if a curve level is defined as:
$$f(x,y)=k$$
or vectorially as:
$$f(c(t))=k$$
and its curve integral associated as:
$$\bigtriangledown f(c(t))\cdot c'_{t}(t)=k$$

Then, how is the equation of a surface integral associated to surface level:
$$f(x,y,z)=k$$
$$f(S(t,s))=k$$

Would be this?
$$\bigtriangledown f(S(t,s))\cdot (S'_{t}(t,s)\times S'_{s}(t,s))=k$$

And more, all this above make I think if is possible to extend the gradient's theorem (that is specific to line integral):
$$\int_{t_0}^{t_1} \bigtriangledown f\cdot \hat{t}\;ds=\Delta f$$
to surface integral...?

 

[Jhenrique]

Do you understood what I wanted ask? I wanted, mainly, to extend the concept of "Conjuntos de nivel y gradientes" (http://es.wikipedia.org/wiki/Conjunto_de_nivel) to the case of 2 parameters...

 

[Jhenrique]

I think I found the answer for my question. In 2D, we have a level curve given by the equation f(s(t))=k, and the integral curves that cross it is given by ∇f·ds/dt=0. In 3D, we have a level surface given by the equation f(S(t,s))=k, and the integral curves that cross this surface are probably given by two equations ∇f·dS/dt=0 and ∇f·dS/ds=0. Someone understood what I wanted say!?