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Surface level and surface integral

  1. Dec 4, 2013 #1
    I have some questions, all associated. So, first, if a curve level is defined as:
    [tex]f(x,y)=k[/tex]
    or vectorially as:
    [tex]f(c(t))=k[/tex]
    and its curve integral associated as:
    [tex]\bigtriangledown f(c(t))\cdot c'_{t}(t)=k[/tex]

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

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

    And more, all this above make I think if is possible to extend the gradient's theorem (that is specific to line integral):
    [tex]\int_{t_0}^{t_1} \bigtriangledown f\cdot \hat{t}\;ds=\Delta f[/tex]
    to surface integral...?
     
  2. jcsd
  3. Dec 10, 2013 #2
    Last edited: Dec 10, 2013
  4. Dec 25, 2013 #3
    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!?
     
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