Discussion Overview
The discussion revolves around the application of the divergence theorem (Gauss' theorem) to non-closed surfaces, specifically exploring whether it can be used for an open surface with boundaries, such as a paraboloid that does not have a top plane. Participants seek to clarify the conditions under which the theorem might be applicable or if alternative approaches exist.
Discussion Character
Main Points Raised
- One participant questions the applicability of the divergence theorem to an open surface, seeking clarification on conditions for its use.
- Another participant suggests that the theorem can be applied to the paraboloid if a plane is added, allowing for the calculation of flux through the closed surface and then subtracting the flux through the plane to find the flux through the paraboloid.
- A third participant argues that discussing the divergence theorem for non-closed surfaces is nonsensical due to the theorem's derivation, which relies on the concept of closed surfaces and the flux contributions from outward-facing sides of infinitesimal cubes.
- One participant mentions the existence of Stokes' theorem as an alternative for open surfaces.
Areas of Agreement / Disagreement
Participants express differing views on the applicability of the divergence theorem to non-closed surfaces, with no consensus reached on whether it can be used directly or indirectly. The discussion remains unresolved regarding the validity of applying the theorem in this context.
Contextual Notes
Limitations include the lack of clarity on specific conditions under which the divergence theorem might be applied to non-closed surfaces and the assumptions underlying the derivation of the theorem itself.