Divergence theorem for a non-closed surface?

In summary, the use of divergence (Gauss') theorem for open surfaces with boundaries is limited. It can only be directly applied to calculate the flux through surfaces with boundaries, such as a paraboloid with an attached plane. However, in order to use it for non-closed surfaces, the flux through the plane must be subtracted from the total flux calculated using the theorem. It is important to note that the theorem is derived based on the flux through infinitesimally small cubes on a closed surface, so it may not make sense to use it for non-closed surfaces. Alternatively, there is a curl theorem for open surfaces (Stokes) that can be used.
  • #1
NicolasM
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Is there some way we can apply divergence (Gauss') theorem for an open surface, with boundaries? Like a paraboloid that ends at some point, but isn't closed with a plane on the top.

I found this at Wikipedia:
It can not directly be used to calculate the flux through surfaces with boundaries

but I couldn't find some further explanation, like under what conditions it can be applied, or in what way we can "indirectly" use it.
 
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  • #2
Well you can use divergence theorem to calculate flux through your paraboloid WITH plane attached. Then if you can calculate flux through the plane you can substract it from what you got; what is left is obviously flux through paraboloid.
 
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  • #3
IMHO, it makes no sense to speak of the divergence theorem for a non-closed surface because of the way the theorem is derived. I don't know if you already know this (or maybe I'm the one who lacks knowledge) but in the derivation of this theorem one makes reference to the addition of the flux in all infinitesimally small cubes that make up a surface and the only flux that "survives" is the one from the outward facing-sides on the most exterior cubes of your closed surface.
 
  • #4
No, but there is a curl theorem for open surfaces (stokes)
 

What is the divergence theorem for a non-closed surface?

The divergence theorem for a non-closed surface is a mathematical principle that relates the flow of a vector field through a surface to the divergence of that field within the volume enclosed by the surface. It is a generalization of the traditional divergence theorem, which only applies to closed surfaces.

How is the divergence theorem for a non-closed surface different from the traditional divergence theorem?

The traditional divergence theorem only applies to closed surfaces, while the divergence theorem for a non-closed surface can be applied to any surface, regardless of whether it is closed or not. This makes it a more versatile and powerful tool in mathematical modeling and analysis.

What is the significance of the divergence theorem for a non-closed surface in physics?

In physics, the divergence theorem for a non-closed surface is used to relate the flow of a vector field, such as fluid velocity or electric current, through a surface to the sources or sinks of that field within the enclosed volume. This allows for a better understanding and prediction of physical phenomena.

Can the divergence theorem for a non-closed surface be applied in three-dimensional space?

Yes, the divergence theorem for a non-closed surface can be applied in three-dimensional space, as well as in higher dimensions. It is a generalization of the traditional theorem, which only applies in three dimensions.

Are there any limitations to the application of the divergence theorem for a non-closed surface?

The divergence theorem for a non-closed surface may not be applicable in situations where the vector field is not continuous or where the surface is not smooth and well-defined. In these cases, other mathematical tools and techniques may be needed to analyze the system.

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