Surface level and surface integral

  • #1
Jhenrique
685
4
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...?
 
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  • #2
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  • #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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