Gauss’s theorem

Gauss s theorem, in mathematics, says a change in a volume s density must be accompanied by flow through the boundary. Leak path analysis is a qualitative interpretation of Gauss s theorem. 

This equality is a special form of Gauss s theorem, and it will allow us to find several boundary conditions, which provide uniqueness of a solution of the forward problem. First, we make three comments ... 

At the same time it is worth to notice that in modern numerical methods of a solution of boundary value problems, based on replacement of differential equations by finite difference, these steps are performed simultaneously. In accordance with the theorem of uniqueness, the field inside the volume V is defined by a distribution of masses inside this volume and boundary conditions, and correspondingly it is natural to derive an equation establishing this link. With this purpose in mind we will again proceed from Gauss s theorem,... 

This relationship is called the second Green s formula and it represents Gauss s theorem when the vector X is given by Equation (1.98). In particular, letting ij/ — constant we obtain the first Green s formula ... 

It is now necessary to derive analogous relations for the divergence of a vector, viz. V - A. The calculation can be carried out in at least two ways. The direct analytic approach is long, but does not involve any methods other than those of vector algebra. Otherwise, it is necessary to develop the diver-geoce (Gauss s) theorem, after which the desired result is easily obtained (see Appendix VI). In either case it is given by... 

This integral relation is known as Gauss s theorem. The most familiar example is in electrostatics. 

If we let p tend to zero on both sides of equation (32.7) we find that we enu sum the hypergeoinctric series by Gauss s theorem (7.2) provided that /< < r + 1. We then have the result... 

Gauss s theorem, which states that the rate of change in [C] with respect to time (0 and at some depth (z) is equal to the negative spatial gradient of the mass flux (F),... 

By rearranging, we see that Gauss s theorem obeys the law of conservation of mass... 

Gauss s theorem, where K is the specific inductive capacity of... 

The survival probability of a pair before it is formed,p( °° r0, t0), is zero and the integral can be simplified by using Gauss s theorem (see Appendix A.3) to give... 

Because 8 is arbitrary and small and independent of Gauss s theorem for the volume integrals may be used and... 

Using Gauss s theorem to reduce the volume integral and applying the inner boundary condition of eqn. (353c) together with eqn. (352) with p = exp (+ G(R, t)... 

Equation (82) states that the slab is initially below the melting temperature. Gauss s theorem applied to the heat conduction equation over a region 2 with boundary B in the (x, V) plane bounded by the lines t = 0, f = t, x = a, and the curve x = X(t ) gives... 

For one-dimensional horizontal pipe flows without electrostatic effects, using Gauss s theorem, Eq. (11.5) yields... 

The minus sign is included since, with n pointing outward, V n > 0 when system points flow out of the volume element and therefore cause a decrease in the number of system points in the volume element. It is inconvenient to have a volume and a surface integral in the same equation, so we convert the surface integral to a volume integral using Gauss s theorem ... 

Important relationships can be developed by considering the effect of filling the space between the plates of a parallel-plate capacitor with a dielectric material, as shown in fig. 2.27. From Gauss s theorem the electric field E between and normal to two parallel plates carrying surface charge density a and separated by a vacuum is... 

Consider a small element of surface dA shaded in Fig, 13.VIII L, on the surface of a charged conducting sphere, and points M and M just outside and just inside the sphere. The force acting at M is (i) that / due to the small charged area dA plus (ii) that / due to the rest of the sphere. If M is in air (dielectric constant 1), the force at M is given by Gauss s theorem as Ank per cm., ... 

We introduce now a domain Vr, bounded by a sphere dVa of a radius / , with its center at the origin of some Cartesian coordinate system, x, y, z. Integrating both sides of equation (9.65) over the domain Vr, and applying Gauss s theorem, we find ... 

Thus, according to Gauss s theorem (13.100), we obtain the following integral... 

The surface integral in eqn (5.76) comes from the application of Gauss s theorem to the term involving V -(V,i/ 5 ). As before, all such surface integrals vanish except for r,- = fi because of the vanishing of on the boundaries at infinity. From this point on, the coordinate Tj and the volume element d-Tj will be set equal to r and dr, respectively, and Vj and to their corresponding unscripted quantities. 

Integration of the right-hand side of eqn (5.82) in the manner indicated in eqn (5.80) transforms it into an integral of the Laplacian of the charge density. A typical term in this integration can be transformed using Gauss s theorem to yield... 

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