If one uses Stokes' theorem and if two oriented surfaces S1 and S2 share a boundary ∂s then the flux integral of curl(F) across S1 equals the flux integral of curl(F) across S2. However, in general it won't be true that flux integral G across S1 equals the flux integral of G across S2. If you can show that div(G) = 0, then will the flux integral be independent of the surface (assuming they share ∂s?)
I was thinking if flux of G across S1 = flux of G across S2, when flux of G across S1 U S2 is 0 and then you can apply the divergence theorem, and div G must be 0. My question is, is Div(G) = 0 a sufficient condition, or only works sometimes?
Also if a surface S is closed (has no boundary) then we know flux of curl(F) through S is 0, but not necessarily flux of F through S. If F is a conservative field then given S is closed then flux of F through S is zero, but does div F = 0 also imply that flux of F through S, when S is closed, is 0?
Conservative fields have 0 curl, but is there any relation between divergence and conservative fields? What are the implications of no divergence, everywhere I look it says "under suitable conditions" you can do this and this, but it doesn't clarify what those suitable conditions are.
I was thinking if flux of G across S1 = flux of G across S2, when flux of G across S1 U S2 is 0 and then you can apply the divergence theorem, and div G must be 0. My question is, is Div(G) = 0 a sufficient condition, or only works sometimes?
Also if a surface S is closed (has no boundary) then we know flux of curl(F) through S is 0, but not necessarily flux of F through S. If F is a conservative field then given S is closed then flux of F through S is zero, but does div F = 0 also imply that flux of F through S, when S is closed, is 0?
Conservative fields have 0 curl, but is there any relation between divergence and conservative fields? What are the implications of no divergence, everywhere I look it says "under suitable conditions" you can do this and this, but it doesn't clarify what those suitable conditions are.
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