Divergence operator

Gradient operator Laplace operator Dot product Cross product Divergence operator Curl operator Vector transposition Complex conjugate... 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

Following this new Copemican revolution, far from being horrified by the remoteness of the stars and by the tremendous divergence operating across the whole Universe, astronomers set abont drawing up an overview of the new paradigm and determining its structural and chemical evolution. 

With no variations across the channel, the divergence operator contributes only an axial derivative,... 

To begin with we concentrate on equation (2a), which is purely static. We define the Green s function g(x,x ) for the divergence operator ... 

Begin with the Poisson equation but keep the e matrix inside the divergence operation V (eV0) = -4jrpext (see Fig. L3.24). The net electric-charge density pext at a given point depends on the magnitude of potential as in Debye-Huckel theory. As before in relation (L3.175),... 

During a diffusion process, e.g. the migration of an additive from a plastic into the atmosphere, a change in the concentration of the diffusing substance takes place at every location throughout the plastic. The mass flux caused by diffusion is represented by a vector quantity whereas the concentration c and its derivative of time t is a scalar quantity and is connected by the flux with help of the divergence operator. The following example serves to emphasize this relationship. 

V- Divergence operator V X Curl operator t Vector transposition t Complex conjugate The Variational Principle states that an approximate wave function has an energy which is above or equal to the exact energy. The equality holds only if the wave function is exact. The proof is as follows. Assume that we know the exact solutions to the Schrodinger equation. H. 1 = 0.1.2 oo CB.l)... 

Note that the fourth equation in (8.47) can be derived by applying the divergence operator V- to the left- and right-hand sides of the first equation in (8.47),... 

The forth equation in (8.62) can be obtained by applying the divergence operator V to both sides of the first equation and allowing for the continuity equation for the extraneous currents and charges. In particularly, the Helmholtz equations (8.31) can be written as follows... 

Gauss theorem to the volume integral containing Laplacian and divergence operators ... 

We will apply the divergence operation to both sides of this equation. Since... 

Equation [1-5] is pertinent to a one-dimensional system, such as a long, narrow tube full of water, where significant variations in concentration may be assumed to occur only along the length of the tube. In a three-dimensional situation, the advection-dispersion-reaction equation can be represented most succinctly using vector notation, where V is the divergence operator ... 

The rate of deformation and the pressure are frame-indifferent (e.g., see [6], p 141 [54], p 400 [28, 31, 34]) so we can simply re-write the divergence operator and the stress terms into the rotating reference frame notation. The transformed momentum equation delds ... 

The Laplacian operator on the LHS of the pressure equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum equations. The RHS of the pressure equation consists of a sum of derivatives of the convective terms in the three components of the momentum equation. In all these terms, the outer derivative stems from the continuity equation while the inner derivative arises from the momentum equation. In a numerical approximation, it is essential that the consistency of these operators is maintained. The approximations of the terms in the Poisson equations must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint may lead to convergence problems as the continuity equation is not appropriately satisfied. 

In this relation, the operator 6/6xi outside the parentheses on the LHS is the divergence operator inherited from the continuity equation, while is... 

Applying the continuity divergence operator on the terms in this relation, and recognizing that the first divergence term in the resulting relation represents an extended estimate of the transient term in the continuity equation, yields ... 

The class boundaries are independent of time and spatial position so the time derivation and the divergence operator can be moved outside the integral ... 

Now the vector operators defining the derivatives of a vector are introduced. The divergence operator div simply gives the derivative of the vector in terms of its Cartesian components. Thus,... 

The divergence operator is a vector derivative operator that produces a scalar when applied to a vector function. 

We can choose this quantity to satisfy eq. 1.132. Performing the divergence operation on eq. 1.173, we have ... 

The product rule for the divergence operator is applied to both terms on the right-hand side of equation (9-27). In any coordinate system, the divergence of the product of a scalar and a vector is expanded as a product of the scalar and the divergence of the vector plus the scalar (i.e., dot) product of the vector and the gradient of the scalar. This vector identity was employed in equation (9-14). The pseudo-binary mass transfer equation for component i is... 

In this case the balance of energy may be written in the global format of Eq. (4), which may be localized by the standard argumentation yielding Eq. (5), where div( ) is the divergence operator. 

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