I have been reading proof of divergence theorem , but there has been a question that I just cannot solve for my self , and I searched a lot , but not much was out there , mathematically what is wrong with open surfaces in here? (why nobody even discussed this ? ) , I have a good understanding about proof , but as long as I don't answer this question to my self I just wont comprehend it fully.
For Example, we have a one sided open cylinder , and it is full of water , and we have a steady flow of water from closed side to open side it is obvious that ($h$ is flow of water) $$\int \nabla. h \, dv = 0$$ because there is no change in whole cylinder but $$\int h.da \neq 0$$
What is happening here?
according to DIVERGENCE THEOREM it should be : $$\int \nabla. h \, dv = \int h.da $$
well , to begin with an open surface doesn't contain any volume , so comparing the to integrals is not correct . for divergence theorem to work we need volume and for volume we need closed surface so that's it.