Volume integral

The two volume integrals change the dependence on x and introduce the number density of atoms n (number/cm ) such that the energy now varies as... 

In practice, only a limited number of atoms of the solid need to be considered because of the rapid falling-off of potential with distance, implied by Equation (1.8) moreover, for larger values of r, the summation process may be replaced by volume integration. With the ready availability of computers, however, it is now possible to sum up all the individual interactions over say 100 to 200 of the nearest pairs, and to confine the integration operation to the remainder. 

PBM (Photochemical Box Model) is a simple stationary single-cell model with a variable height lid designed to provide volume-integrated hour averages of ozone and otlier photochemical smog pollutants for an urban area for a single day of simulation. 

The foregoing are volume integrals evaluated over the entire volume of the rigid body and dw is an infinitesimal element of weight. If the body is of uniform density, then the center of gravity is also called the centroid. Centroids of common lines, areas, and volumes are shown in Tables 2-1, 2-2, and 2-3. For a composite body made up of elementary shapes with known centroids and known weights the center of gravity can be found from... 

A more universal fracture characteristic for use with ductile materials is the J integral . This is similar to CTOD but relates a volume integral to a surface integral and is independent of the path of the integral it can be classed as a material property. The J integral can also be used to predict critical stress levels for known crack lengths or vice versa. 

The volume integral will give a higher order term in k, so for now, we focus on the surface integral. The displacement due to the phonon is conveniently expanded in terms of the spherical waves e " =... 

The volume integral in Eq. (B.2) produces a quadratic term, which is roughly equal to (Vcj)) fy (Pr (d k). We then proceed in a completely identical fashion to our earlier estimate of g. Assuming the diplacements within the droplet are random, one gets for the integral P dik, where the factor of comes... 

As was pointed out earlier. Equation (1.6) allows us to find the attraction field everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the field g(p) can be greatly simplified. First, consider an elementary mass with density 6 q), located in the volume AV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By definition, these changes do not make a noticeable influence on the field because the observation point p is far away. In the limit, when... 

Taking into account Equations (1.101 and 1.103), the volume integral in the second Green s formula vanishes and we obtain... 

By definition, any plane 0 — constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of 0-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the 0-component, gg, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that — 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction field has only a radial component, g, and the field is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS — dSiR and the scalar component g is constant at points of the spherical surface, we have for the flux ... 

In the case when masses can be treated as elementary and it is possible to neglect the bar s mass, the volume integral disappears and we have... 

Thus, instead of a volume integral, the field is represented as a surface integral, which, of course, greatly simplifies calculations. If the function S(q) is constant, we have... 

Similar expressions can be written for horizontal components of the field of attraction. Thus, instead of the volume integral, we have derived an expression for the field that requires integration only over the surface. The formulas described in this section allow us, in many cases, to simplify the solution of the forward problem. 

Equation (20-70) is the unsteady-state component mass balance for fed-batch concentration at constant retentate volume. Integration yields the equations for concentration and yield in Table 20-19. 

In the case of the Porod-Kratky model, the polymer backbones have a constant curvature c0. Accounting for the polymer stiffness in generating the dense configuration of stream lines, the vector field used must have a homogeneous curvature field with a unique value cq in the entire simulation box T. In order to quantify the success in creating such a vector field, the deviation of the curvature from the ideal Porod-Kratky case, a volume integral has been used by Santos as a penalty function ... 

In the preceding F = fc(r, r), H = tc(r, vt)G = k(vt, v) and the normalization constant C is fixed by equating the volume integral of n to unity. For further tractability, Sano and Mozumder expand (r v) in a Taylor s series and retain the first two terms only. The validity of this procedure can be established a posteriori in a given situation. At first, the authors obtain equations for the time derivatives of the expectation values and the correlations of dynamical variables. Then, for convenience of closure and computer calculation, these are transformed into a set of six equations, which are solved numerically. The first of these computes lapse time through the relation... 

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 ... 

In summary, this zero-dead-volume integrated CL detection approach provides two major advantages, namely ... 

The volume integral of the last term, when transformed into a 3D surface integral by the theorem of Gauss, is required to vanish in the boundary surface [18]. The condition for stationary action thereby becomes... 

Chremmos, I.D., Uzunoglu, N.K., 2004, Transmission and radiation in a slab waveguide coupled to a whispering-gallery resonator volume-integral-equation analysis, J. Opt. Soc. Hm.H21(5) 839-846. 

Kottmann, J.P. and Martin, O.J.F., 2000, Accurate solution of the volume integral equation for high-permittivity scatterers,/EEE Trans. Antennas Propagat. 48(11) 1719-1726. 

Kucharski, A.A., 2000, Resonances in heterogeneous dielectric bodies with rotational symmetry-volume integral-equation formulation, IEEE Trans. Microwave Theory Tech. 48(5) 766-770. 

This is merely the volume integral of the concentration over the entire vessel. Therefore, the value of unity indicates that the total amount of solute in the system is constant. 

The main disadvantage of the full matrix analysis is that it requires volume integrals of all spectral peaks, including the diagonal. 

The present analysis clearly indicates that besides a complete peak volume matrix the full matrix analysis requires high accuracy of integrated peak volumes. Therefore, a good signal-to-noise ratio and good methods for peak volume integration are the main prerequisites for a successful full matrix analysis. 

Verify that z is normalized. The volume integral transformed into spherical coordinates is ... 

Using Green s theorem, it can be converted into a volume integral over fir, the tip side from the separation surface. Noticing that the sample wavefunction ip satisfies Schrodinger s equation, Eq. (3.2), in fir, and that the Green s function satisfies Eq. (3.8), we obtain immediately... 

Note that the temperature profile differs from the total heat profile because the heat per unit volume depends not only on the local temperature but also on the local density of the flame gas. In the preheat zone the profile of conducted heat coincides with the profile of total heat, whereas in the reaction zone the conducted heat gradually drops to zero. The difference between total heat and conducted heat represents the heat gained by chemical reaction. The volume integral of the total heat is approximately H, and the volume integral of the conducted heat is approximately H"... 

Black polymers, such as polyacetylene, absorb all visible light, but opaque polymers scatter the incident light. As shown by Lord Raleigh, the turbidity r is related to the scattered light per unit volume, integrated over all angles. 

---