Divergence theorem and closed surfaces

  • #1
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Main Question or Discussion Point

Hi,

I have a question about identifying closed and open surfaces.
Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed.
I mean for example a cylinder that is truncated, or tetrahedron in the first octant(x,y,z>=0)... I have seen some exercises that also add the "cover" to close the surface and some that don't.

How should I figure out what to do in a given exercise to help me understand when I should closed the surface or not.

Thank you!
 

Answers and Replies

  • #2
Orodruin
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A closed surface does not have a boundary. In the case of the cut cylinder, you need the end-caps to close the surface, otherwise you will have a one-dimensional boundary where you have cut it. Now, it may be that the end cap flux integrals are zero for particular vector fields, especially vector fields which is orthogonal to the end cap normal vectors.
 
  • #3
102
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A closed surface does not have a boundary. In the case of the cut cylinder, you need the end-caps to close the surface, otherwise you will have a one-dimensional boundary where you have cut it. Now, it may be that the end cap flux integrals are zero for particular vector fields, especially vector fields which is orthogonal to the end cap normal vectors.
Thank you for the answer.

Is it possible to notice if the author who writes the question wants you to add the cap or not? sometimes I see exercises that say "the solid hemisphere.." or something that says "the hemisphere with the disc in the xy-plane..." or "upper half of the hemisphere (some equation) that lies above the unit disc...", and similar questions.

Thank you.
 

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