The shape of infinitesimal objects

[LucasGB]

I managed to show that the flux through an infinitesimally small cube equals the divergence of the vector field at that point. I also managed to show that the circulation around an infinitesimally small square equals the component of the curl perpendicular to that square at that point.

Should these demonstrations be considered special cases, since they deal with cubes and squares instead of general shapes, or can I consider them to be general proofs? Does the shape of the infinitesimal objects matter in that kind of proof?

 

[mnb96]

You might want to taka a look at the definitions of divergence and curl in general curvilinear coordinates.
Your demonstrations with flux and circulation around cubes and squares are then special cases for the Euclidean frame of reference, but hopefully someone else in this forum can confirm this.

 

[torquil]

Does the shape of the infinitesimal objects matter in that kind of proof?[/I]

In rigorous math, you have to show that it is shape-independent, even if your are taking the limit for infinitely small objects. I would guess that this is related to your assumptions on the continuity/regularity of the vectors fields.

Torquil

 

[wofsy]

I managed to show that the flux through an infinitesimally small cube equals the divergence of the vector field at that point. I also managed to show that the circulation around an infinitesimally small square equals the component of the curl perpendicular to that square at that point.

Should these demonstrations be considered special cases, since they deal with cubes and squares instead of general shapes, or can I consider them to be general proofs? Does the shape of the infinitesimal objects matter in that kind of proof?

the shape is arbitrary. Cubes are used because they are geometrically easy.

 

[LucasGB]

Well, torquil and wofsy are disagreeing. :) If I understand correctly, torquil thinks this does not constitute a rigorous proof, while wofsy does. What do you guys think of each other's arguments?

 

[lugita15]

I think that if you have already proved the theorem for an infinitesimal cube, then you have proved the theorem in general. This is because a very small (simply connected, smooth, etc.) region in $$\Re^{3}$$ can be approximated well by a very small cube. So an infinitesimal 3-dimensional region will be perfectly approximated by an infinitesimal cube. Ditto for the curl theorem and the infinitesimal square.

 

[LucasGB]

I think that if you have already proved the theorem for an infinitesimal cube, then you have proved the theorem in general. This is because a very small (simply connected, smooth, etc.) region in $$\Re^{3}$$ can be approximated well by a very small cube. So an infinitesimal 3-dimensional region will be perfectly approximated by an infinitesimal cube. Ditto for the curl theorem and the infinitesimal square.

I sure hope so. That would save me a lot of trouble.