Yes So it doesn't prove my formula with vertices ??
The aim of this formula is to not use distance but just vertices to gain time !

Why not?
For tetrahedron: A*d/3 == Det(v1-v0,v2-v0,v3-v0)/6, where 'A' is the area of some triangle and 'd' is the distance from the opposing vertex to the plane of the triangle. In the Det formula v1,v2,v3 define the same triangle, while v0 is the opposing vertex.

[quote name='joohooo' timestamp='1354205457' post='5005341']
Yes So it doesn't prove my formula with vertices ??
The aim of this formula is to not use distance but just vertices to gain time !

Why not?
For tetrahedron: A*d/3 == Det(v1-v0,v2-v0,v3-v0)/6, where 'A' is the area of some triangle and 'd' is the distance from the opposing vertex to the plane of the triangle. In the Det formula v1,v2,v3 define the same triangle, while v0 is the opposing vertex.
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One more time thanks !

But ... I have one more question?? (yes again)
I agree with you for the Det but in the proof we have already use the origin for x0 ! Or in my formula the origin will be your V0 in the det formula as the opposing vertex for each triangle in the mesh. It's not a wrong thing to use the origin for the X0 in F and for the opposing vertex. In my mind , it's look like a ruse... no ?

I'm not sure whether I got you right, but why is it a bad thing to use the origin? You can choose x0 to be whatever you want, and as long as it's constant this formula will work.

I'll try to rephrase. In the vector field F you can choose x0 as any point in R^3. After some manipulations we can interpret the result as the sum of volumes of multiple tetrahedrons that are comprised of the faces of the mesh and have one vertex in common, that is x0.
Geometrically this makes a lot of sense for sphere-like (or more generally convex) meshes, just choose x0 to be somewhere within the object and the formula should come trivially to you without the divergence theorem. For arbitrary oriented manifolds (or when x0 is not within the enclosed volume) this is not so clear, but the divergence theorem takes care of that.

Yep,
I understand your last topic and that's why I decided to use this theorem (it's a very interesting thing).
But my worry is completely in the proof because we use the origin in the vector field (make a lot of sense to simplify the formula) AND I use the origin for one of my vertice (opposing vertex) in the Det formula to obtain the dot product of a cross product of three points (after simplification)...

Or in the formula we obtain A*d/3 where d == <x,n> because (<x,n>-<x0,n>)/3==<x,n>/3. To simplify the det formula I use x (opposing vertex)=Origin=xo. And it's here my problem....

This is not a problem since this is exactly the same point.
Let's recap it by steps:
1. You start with the volume integral.
2. Afterwards, you choose a vector field whose divergence is 1. This vector field is defined up to a constant x0.
3. As any x0 is good, you choose one (for instance the origin) and obtain the flux integral by the divergence theorem.
4. Once finished computing the flux integral, you choose to interpret the formula as the sum of volumes of tetrahedrons that share the vertex x0.
5. You rewrite the formula from step 4, with an equivalent formulation that uses Det. Again, this is just a change in interpretation.

In steps 2-5 x0 is fixated and chosen exactly once. Hence there's no problem.