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When I was going through "The Feynman's Lecture on physics" Volume-2 , I found the line

"It is useful to speak of the flux not only through a completely closed surface, but through any bounded Surface"

My question is isn't The BOUNDED AND CLOSED surface same?...but if they are not then why there is such a line Feynman used?

enter image description here

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One meaning of "closed" refers to whether the set "joins up" with itself, and encloses a space (there is also a topological meaning, but that probably is not what is meant here). See https://en.wikipedia.org/wiki/Surface_(topology)#Closed_surfaces So, for instance, a balloon would be closed in this sense. A cup would not be closed, because it's open on the top.

"Bounded" means there is some sphere with finite radius that completely encloses the surface. Both of these surfaces would be bounded; unless the balloon/cup is infinitely large, there is some sphere that contains them.

The electric flux through any closed surface is equal to the electric charge contained inside the surface (see "Gaussian surface"). If we have a non-closed surface, the electric flux can't be calculated by taking the charge enclosed inside the surface, because there is no "inside". But there are still methods that we can use to calculate the electric flux through this surface.

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