Gausss Theorem

The other way to calculate the volume inside the triangulated surface is to use the Ostrogradski-Gauss theorem. It relates the surface integral from a vector field j to a volume integral from its divergence ... 

On the other hand, the total integral from the Gaussian curvature can be expressed by using the Gauss theorem ... 

The Euler characteristic is usually calculated in two ways by using the Gauss-Bonnet theorem [7,85,207,210,222] [Eq. (8)] or by combining the Cartesian and Gauss theorems [Eqs. (122) and (123)], which is also called the Euler formula [23,76,224]. 

The simplification introduced with the use of Gauss theorem is most valuable for the physical picture it conveys, that of a valence electron cloud in the field of a nucleus partially screened by its core electrons. Using it in Eq. (3.24), we get... 

Finally, this valence region energy can be expressed, with the help of Gauss theorem, by an equivalent, considerably simplified approximation that features only the nuclear-electronic potential energy of the valence electrons in the field of an expanded nucleus partially screened by its core electrons. 

The key is in the treatment of core-other core and core-other nucleus interactions. Simple approximations were presented in that matter to get Eq. (4.35). Assuming Gauss theorem— the potential is just as though all the core electronic charge... 

Applying Gauss theorem, leading to the Politzer-Parr core-valence separation in atoms [61]... 

Here r is the distance from the atom centre. In this book the units are chosen so that Qi is measured in electron charges ( = valence units) and q is set equal to 1.0. According to Gauss theorem, if the electron density of the atom is spherically symmetric, fj- mono gives a fully correct description of the field generated by the atom in the region outside the atom itself, i.e. in the region where the electron density of the atom has fallen to zero. 

In Q 18.6 (below) you are asked to derive the three-dimensional version of Gauss theorem (Eq. 18-12). From there it is straightforward to show that, provided that diffusivity D is the same in all directions, the three-dimensional form of Fick s Second Law (Eq 18-14) has the form ... 

Expand Fick s law and Gauss theorem (Eq. 18-12) to three dimensions and derive Fick s second law for the general situation that the diffusivities Dx, Dy, and Dz are not equal (anisotropic diffusion) and vary in space. Show that the result can then be reduced to Eq. (1) of Box 18.3 provided that D is isotropic (Dx = Dy=Dz) and spatially constant. 

Transport Processes and Gauss Theorem One-Dimensional Diffusion/Advection/Reaction Equation Box 22.1 One-Dimensional Diffusion/Advection/Reaction Equation at Steady-State... 

In Chapter 18 we derived the Gauss theorem (Eq. 18-12), which allows us to relate a general mass flux, Fx, to the temporal change of the local concentration, dC/dt (see Fig. 18.4). Applying this law to the total flux,... 

Using Gauss theorem (Eq. 18-12) for the concentration per unit bulk volume, CW yields ... 

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