Gauss’ theorem surface

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 ... 

Let V be a region in space bounded by a closed surface S (of Lyapunov-type [24, 50]), and f (x) be a vector field acting on this region. A Lyapunov-type surface is one that is smooth. The divergence (Gauss) theorem establishes that the total flux of the vector field across the closed surface must be equal to the volume integral of the divergence of the vector (see Theorem 10.1.1). 

The first boundary condition at the surface is provided by integrating the Poisson Eqs. (la) and (lb) over the volume of a flat box, which includes the surface, with the large sides parallel to the surface and a vanishingly thin width. After using the Gauss theorem, one obtains ... 

The surface charge density is related to the gradient of potential by the Gauss theorem, namely... 

Equation 27J is not in a form that can be conveniently used by electrochemists. We would like to express the surface coverage in terms of the charge or the potential, rather than the field. The field is related to the charge through the Gauss theorem, namely... 

Now integrate over all space. By Gauss theorem, / d r V (f x Ti) may be transformed into a surface integral enveloping the fields the surface integration can be extended to infinitely remote boundaries where the fields ultimately vanish. This term therefore drops out, leaving... 

Building on this analysis, we may quantify the net hydrophobicity r] of a hydrogen bond by taking into account the surface flux of the dehydronic field generated by the hydrogen bond. This field is given by = volume of test hydrophobe). Thus, in accord with Gauss theorem we obtain... 

With the help of Gauss theorem the surface integral can be converted into a volume integral, giving... 

The convective term can then be formulated as a surface integral, and converted to a volume integral by use of Gauss theorem (App. A) ... 

The potential work term, denotes the rate at which work is done on each of the individual species c in the fluid per unit volume by the individual species body forces, gc, due to the diffusion of the various components in external fields such as an applied electro-magnetic potential. The term can be formulated as a surface integral, and as before converted to a volume integral by use of Gauss theorem (App. A) ... 

The convective and diffusive transport terms in (3.55) are rewritten as the sum of a volume and an interface surface integral using Gauss theorem (see app A). For each bulk phase we get ... 

The convective and diffusive terms can be written as a sum of a volume and a surface integral using the Gauss theorem ... 

Likewise, one imagines that the surface integral in (9.54) can be rewritten in terms of a volume integral using a generalization of the conventional Gauss theorem ... 

The change in the flux with position in the system may be related to the time derivative of the local concentration by applying Gauss theorem. Consider a system with volume V and surface area A. Suppose that substance i is flowing out of this volume. Then the rate of substance i leaving in moles per second can be found by integrating the flux /, over the surface area A, so that... 

---