Gauss’ integral theorem

The integral which extends over the area of the region has been converted to the volume integral of the divergence of q according to Gauss integral theorem. 

The Euler characteristic, %, of a closed surface is related to the local Gaussian curvature K r) via the Gauss-Bonnet theorem [Eq. (8)]. A number of different schemes have been proposed to calculate the local curvatures and the integral in Eq. (8). 

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

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)... 

It is possible, and very useful, to write the surface integral in terms of a volume integral via the use of the Gauss divergence theorem, which states that... 

Applying the Gauss divergence theorem (Section A. 15), the net force on the control volume can be represented in terms of a volume integral as... 

The surface integral can be converted to a volume integral using the Gauss divergence theorem, Eq. 2.29,... 

By the Gauss divergence theorem, the surface integral can be rewritten as a volume integral, yielding... 

Explain why the control-volume volume dV — r2 sin 6d6drd

[Pg.138]

Assume that there is a volume V that is enclosed by a surface A whose orientation is described by a normal outward-pointing unit vector n. The Gauss divergence theorem, which is used to relate surface integrals to volume integrals, is stated as... 

The Gauss-Bonnet theorem, which relates integrals of Gaussian curvature (1/(/ii) in three dimensions) over a surface to integrals of mean curvature (1/iii + I/R2 in three dimensions) over boundaries of the surface, is particularly simple in two dimensions. In two dimensions, the (N — 6)-rule is equivalent to the Gauss-Bonnet theorem. 

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 ... 

The right-hand side may be converted to a surface integral using Gauss s theorem, Eq. (5.14), and we find... 

Nonequilibrium thermodynamics often uses the Gauss-Ostrogradsky theorem, which states that the flux of a vector through a surface a is equal to the volume integral of the divergence of the vector v for the space of volume Fbounded by that surface... 

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... 

Figure 1.8 Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (flj) and the geodesic curvature along the arcs AB, BC, CD and DA.

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... 

The force may be equivalently expressed using Gauss s theorem as an integration of the force density — V r (r) over the basin of the atom,... 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

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