Integration theorem equation

From (A.48), (A.49), and (A. 104) s grad v = s k. Integrating this equation over the region and using the divergence theorem, it is found that... 

Hydrodynamic Equations.—Before deriving the hydro-dynamic equations, some integral theorems that are useful in the solution of the Boltzmann equation will be proved. Consider a function of velocity, G(Vx), which may also be a function of position and time let... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

Integrating this equation over a face S of the prism, bounded by a rectangular contour L, and using Stokes s theorem, we find... 

As shown in equation (22.10), the real part of the impedance tends toward a finite value as frequency tends toward infinity. The transfer function Z x) — Zr,oo tends toward zero with increasing frequency. As Z(x) is analytic, Cauchy s integral theorem, given in Appendix A as Theorem (A.2), can be written as... 

Equation (69) may be further simplified if we recall the integration theorem of Laplace transformation as generalized to fractional calculus, [31], namely,... 

FEMLAB has many options for boundary conditions. The finite element method is based on integrating the equations by parts and applying the divergence theorem. Thus, the allowable boundary conditions are really determined by the equations. The basic conditions of fluid mechanics are that either the velocity or forces must be specified. In a three-dimensional problem, you would need to specify three velocities, for example, at each boundary point, or some combination of velocities and forces. The finite element method provides the following boundary conditions (Finlayson, 1992) ... 

To obtain a general integral representation for solutions of the creeping-flow equations, it is necessary first to derive a general integral theorem reminiscent of the Green s theorem from vector calculus. 

Another consequence of the integral theorem (8-111) is that we can calculate inertial and non-Newtonian corrections to the force on a body directly from the creeping-flow solution. Let us begin by considering inertial corrections for a Newtonian fluid. In particular, let us recall that the creeping-flow equations are an approximation to the full Navier-Stokes equations we obtained by taking the limit Re -> 0. Thus, if we start with the ftdl equations of motion for a steady flow in the form... 

The first three relationships [Eqs (10.45)-(10.47)], derived by Hoare and Ruijgrok (1970), are a direct result of the integration theorem for Laplace transforms and the steepest descent approximation. Equation (10.48) comes directly from the derivative of Eq. (10.47). 

Since there are N ideal gas molecules, Euler s integral theorem for homogeneous thermodynamic state functions reveals that the chemical potential of a pure material is equivalent to the Gibbs free energy G T, p, N) on a per molecule basis (see equation 29-30(7) ... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

If one equates (29-27) and (29-28) and lets A approach unity and y y, then Euler s integral theorem provides the prescription to calculate any homogeneous function in terms of its extensive variables and the associated slopes ... 

Intercepts and Common Tangents to Agmixing ts. Composition in Binary Mixtures. Euler s integral theorem and the Gibbs-Duhem equation provide the tools to obtain expressions for Agmixing and (9 Agmixing/9y2)r,/) in binary mixtures. This information allows one to evaluate the tangent at any mixture composition via the point-slope formula. For example, if i i = and p,2 = M2 when the mole fraction of component 2 is y, then equations (29-73) and (29-76) yield ... 

Chemical Stability. Condition (29-123c) requires that (gmixture)22 > 0. Euler s integral theorem for a multicomponent mixture yields (i.e., see equation 29-69) ... 

The approximation method of choice is the LCAO-MO method, which is an acronym that stands for linear combinations of atomic orbitals to make molecular orbitals. Unlike VBT, however, the linear combinations used to construct MOs derive from the AOs on two or more different nuclei, whereas the linear combinations used to make hybrid orbitals in VBT involved only the valence orbitals on the central atom. Using the variation theorem, the energy of a particle can be determined from the integral in Equation (I0.I2), which is also written in its Dirac (or bra-ket) notation. 

---