Questions on conditions for calculating flux integrals?

[Gridvvk]

Conditions for calculating flux integrals? [Figured it out]

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.

EDIT:
Nvm most of my questions are answered when I realized div F = 0 => F = curl G, so ques 1: Yes b/c of the aforementioned observation, ques 2: div F = 0 => flux of F = 0 by div. thm. , and ques 3 is just that observation div F = 0 => F = curl G.

 

[jvicens]

Hello,

Thank you for your post and for sharing your thought process. It seems like you have a good understanding of the concepts involved in calculating flux integrals. To answer your question, yes, div(G) = 0 is a sufficient condition for the flux integral to be independent of the surface, as long as they share the same boundary. This is because the divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the enclosed volume. So if div(G) = 0, then the flux of G through any closed surface will be zero.

You are correct that for a closed surface, the flux of a conservative field is always zero. This is because a conservative field can be written as the gradient of a scalar potential function, and the flux of a gradient field through a closed surface is always zero.

As for the relation between divergence and conservative fields, you are on the right track with your observation that div F = 0 implies that F = curl G. This means that the field is irrotational, or has no curl, which is a necessary condition for a field to be conservative. However, it is not a sufficient condition. There are other conditions that must be satisfied for a field to be conservative, such as the path independence of line integrals.

I hope this helps clarify some of your questions. Keep up the good work in understanding flux integrals and their conditions!