Volume and Area in physical equations

[ShayanJ]

In some parts of the physics,sometimes it happens that the volume of a region of space or the area of a surface enters into a formula.In such situations,most of the time,the author argues that "although I have derived this formula for such a shape,it is independent of the shape of the region/surface". For example the formula for Fermi energy.In textbooks a cube is considered for the proof but at the end,the volume is simply put there and every one seems to be fine with it.
Now I can imagine only two possibilities:
1-The author(or any other physicist)knows a proof which doesn't need any assumption about the shape.
2-There is a general theorem which says any formula proved for a particular shape,is true for all shapes!
I know the 2nd possibility seems weird,but well...Its not that unreasonable to be impossible!
Anyway,I want to know if the first possibility is correct, are those proofs published in books as the ones dependent on the shape?
And if the second possibility is correct,I'm just longing to know about that magical theorem!
Thanks

 

[tiny-tim]

Hi Shyan!

In some parts of the physics,sometimes it happens that the volume of a region of space or the area of a surface enters into a formula.In such situations,most of the time,the author argues that "although I have derived this formula for such a shape,it is independent of the shape of the region/surface".

It's basically a theorem that the ancient greeks used …

they knew they could approximate an area by dividing it into tiny rectangles and adding their areas, and taking the limit as the rectangles get tinier.

It doesn't matter how you divide it into rectangles (or other shapes, like fish! ), the limit is always the same.

If you want to know the temperature or the mass or something else of a region, you can divide it into tiny cubes, and the same theorem says that the limit will give you the correct answer.

Of course, a mathematician would say that you need the temperature, mass, etc to be a sufficiently continuous function: but a physicist can normally safely assume this to be true.

When it isn't, ie when there's a singularity, that's usually obvious from the physical description, and won't mislead anyone … eg if the electromagnetic field isn't sufficiently continuous, then we say that there's a charge, and we adapt the formula by adding a special term involving charge!

 

[jtbell]

(or other shapes, like fish! )

For example:

http://tavmjong.free.fr/INKSCAPE/MANUAL/images/QUICKSTART/TILEPAT/tilepat_fish.png

 

[tiny-tim]

For example:

http://tavmjong.free.fr/INKSCAPE/MANUAL/images/QUICKSTART/TILEPAT/tilepat_fish.png

happy new year, jt!

(and merry fishmas! )​

 

[ShayanJ]

Happy new year to both and thanks...
And jt, your fishial(!) elements are cool but I suggest don't use them in proving any formula!

Anyway, your answer seems reasonable about problems involving a kind of integration but its validity for calculating the Fermi energy of electrons in a region of space with arbitrary shape is far from trivial!
I mean, I just can't say: " if I can divide the space into little cubes and the formula for Fermi energy of electrons in each cube is like this,then the formula for a region of space of any shape is the same!"
One thing that may be making me doubt is the the example of an infinite well in QM.Just consider that the well is not rectangular and has a little curvature at the edges.As I remember, such a problem needs a reconsideration which is usually done by perturbation theory.