is it posibile to find the region of integration using $∫∫F.n dr$ if the surface projected on $xy-plane$ extends with out limits like the surface $z=x-y$. what i meant is if we project ,for example,$z=x-y$ it would be x=y whose variable does not have a definite interval.So "is there a way to find the interval and do the integral for example say $F=xi+yj+zk$ or any simple vector field function?
2026-04-01 15:04:02.1775055842
Surface integrals on Vector field
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1
It doesn't work like that. $$\iint_S F\cdot \hat n\,dA$$is integrated over all points on the surface $S$. $F$ and $\hat n$ are functions of the surface points. In order to do the integration, you have to express the suface - at least locally - as a function of two variables. You cannot project into a one-dimensional set and pick up every point on the surface properly.
Now there are situations where you can convert the two-dimensional integration into a one-dimensional one, but this is because you effectively treat it as an iterated integral and use tricks to do one of the one dimensional integrals. But you are still accounting for the 2nd dimension that way.
In your description, you are trying to ignore one dimension altogether, and that will not work.