I have some questions, all associated. So, first, if a curve level is defined as:(adsbygoogle = window.adsbygoogle || []).push({});

[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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# Surface level and surface integral

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