Integration theorems over volumes

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 three most us ul equations for determining the density at the gas-liquid surface are the tirst YBG equation (4.34), the integral equation over the direct correlation function (4.S2), and the potential distribution theorem (4.69). At a planar surface with K- 0 these can be re-written in simpler forms. If the direction normal to the surface (the heig ) is that of the z-axis, then the only non-zero gradient is d/dz and we have cylindrical symmetry about this axis. The volume element can then be more usefully expressed in cylindrical or spherical polar coordinates. 

The surface integral in Eq. (3.32b) can be transformed into a volume integral over the enclosed domain by using Gauss s integral theorem, Eq. (3.7). Equating with Eq. (3.32a) leads to the electrostatic equilibrium condition, which is known as one of Maxwell s equations in integral form ... 

The Reynolds transport theorem is a general expression that provides the mathematical transformation from a system to a control volume. It is a mathematical expression that generally holds for continuous and integrable functions. We seek to examine how a function fix, y, z, t), defined in space over x, y, z and in time t, and integrated over a volume, V, can vary over time. Specifically, we wish to examine... 

Equation (3.9) is the Reynolds transport theorem. It displays how the operation of a time derivative over an integral whose limits of integration depend on time can be distributed over the integral and the limits of integration, i.e. the surface, S. The result may appear to be an abstract mathematical operation, but we shall use it to obtain our control volume relations. 

The reason why the relationship in Equation 7.41 is called the differential virial theorem is because if we take the dot product of both sides with vector r, multiply both sides by pir), and then integrate over the entire volume, it gives... 

The divergence of the flux vector is therefore the net rate of accumulation of the quantity which is transported in and out of the volume element dK This can be integrated over an arbitrary volume Cl limited by the surface I to give the divergence theorem of Gauss... 

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... 

All conservation equations in continuum mechanics can be derived from the general transport theorem. Define a variable F(t) as a volume integral over an arbitrary volume v(t) in an r-space... 

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 ... 

Space averages of e also may be defined by integrating over a fixed volume V of the chamber. Use of the divergence theorem in equation (12) shows that... 

We then integrate over all space. In the absence of the sample we can also set Bo = Ho and Vo = So- The third term then becomes (S — V) dSo = —AnV dSo, and the integration may be restricted to the volume of the sample, since V vanishes elsewhere. The fourth term reads Ho (dB — dH) = 4tt(Ho dAd) here, the integration may be restricted to the sample volume, since elsewhere A4 = 0. The fifth and sixth terms vanish automatically. When the integration is carried out each of the last four terms vanishes as well, on account of a vector theorem derived by Stratton. This leaves... 

We apply Gauss theorem, Table 1.3.1, line (k) to the first integral, converting it into an integral over the volume of the system, to obtain... 

An important property of the energy flow Ft out of domain V), containing sources j , is that its integral over a time period, Ftdt, is always non-negative independently of the type of the extraneous sources. To prove this we introduce a domain Oft Pi formed by the ball without domain (Figure 8-1). We can apply now Poynting s theorem (8.95) to the volume Oh Vi, taking into account that extraneous currents j = 0 in Or Vi ... 

Note that, due to radiation conditions, the surface integral over the sphere On goes to zero if the radius R tends to infinity, and the volume integral is equal to G (r r" w). As a result we arrive at the following important theorem. 

Integration of eqn (8.201) over an atomic volume for which the integral of V p(r) vanishes yields, term for term, the atomic virial theorem for a time-dep>endent system (eqn (8.193)) or for a stationary state (eqn (6.23)). Thus, eqn (8.201) is, in terms of its derivation and its integrated form, a local expression of the virial theorem. The atomic virial theorem provides the basis for the definition of the average energy of an atom, as discussed in Chapter 6. 

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