Applying Gauss' theorem
and integrating over the surface of a sphere of radius r, one obtains for the energy-momentum distribution the following expression:
Therefore, as shown in Figure 2(b), the value of 3-cochain [rho] on i-th tet [V.sub.i] can be derived by 3-D Gauss' theorem:
And notice that, Gauss' theorem and Stokes' theorem can also be applied for complex structures, such as polygons or polyhedrons.
Then, by Gauss' theorem, the discrete divergence of this vector field [[nabla].sub.s] x D can be represented as a dual 2-cochain [sigma].
Using Gauss' theorem, we can rewrite this as a surface integral:
Again, we employ Gauss' theorem to express the total energy as a surface integral:
One more time, we employ Gauss' theorem to express the total energy as a surface integral:
Like our above illustration of the Poisson equation, a misunderstanding of Gauss' Theorem,
From this, we can conclude that although we may possess measurements [nabla] u and [[nabla].sup.2]u, we cannot determine the nature of the scalar field u simply from the Poisson equation or Gauss' Theorem.
So one can consider the hypothesis of continuity as a first approximation, and one can re-examine the Gauss' theorem