Divergence, of vector

The mean H curvature is given by the divergence of the unit vector [30] normal to the surface at r,... 

The divergence of a vector field u, written div u, is given in Cartesian coordinates by... 

Alternatively, one may attempt to estimate the integral over the derivative of the displacement field that entered in the expression for the coupling constant g= pc Jy cPr du/2. Since da is the divergence of a vector, the integral is reduced to that over a surface within the droplet s boundary ... 

It is now necessary to derive analogous relations for the divergence of a vector, viz. V - A. The calculation can be carried out in at least two ways. The direct analytic approach is long, but does not involve any methods other than those of vector algebra. Otherwise, it is necessary to develop the diver-geoce (Gauss s) theorem, after which the desired result is easily obtained (see Appendix VI). In either case it is given by... 

The scalar product of the vector operator V and a vector A yields a scalar quantity, the divergence of A. Thus,... 

The divergence of a vector v with components vt,v2. .., v is the scalar number noted either dive or, with the nabla notation V t, defined as ... 

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... 

On the other hand, it is well known that there is a relationship between Lyapunov exponents and the divergence of the vector field deduced from the differential equations describing a dynamical system. This relation provides a test on the numerical values obtained from the simulation algorithm. This relationship is, according to the definition of Lyapunov exponents ... 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

The second term in Eq. 2.33 requires taking the divergence of a tensor. This operation, V pVV which produces a vector, is expanded in several coordinate systems in Section A.ll. In noncartesian coordinate systems, since the unit-vector derivatives do not all vanish, the divergence of a tensor produces some unexpected terms. 

With this relationship in mind, vector calculus requires that the divergence of the vorticity field is exactly zero ... 

The relative volumetric expansion is seen to be the sum of the normal strain rates, which is the divergence of the vector velocity field. The sum of the normal strain rates is also an invariant of the strain-rate tensor, Eq. 2.95. Therefore, as might be anticipated, the relative volumetric dilatation and V V are invariant to the orientation of the coordinate system. 

In general, as seen in Section 2.8.4, the vector of surface forces (per unit volume) on a differential element can be represented as the divergence of the tensor stress field... 

The last term [in square brackets] is in fact the divergence of the velocity vector. Thus, through straightforward algebraic manipulation,... 

In cylindrical coordinates write a general expression for the the divergence of a vector V = uez + ver -(- we. Take care with inclusion and evaluation of unit vectors where appropriate. 

Carry out all the operations to evaluate the divergence of the stress tensor, V T. Be careful to consider that some unit-vector derivatives do not vanish. Check the results with those provided in the Appendix. 

Recognizing the terms in the parenthesis on the right-hand side as the divergence of the mass-flux vector and dV = rdrdOdz, it can be seen that the procedure has recovered the Gauss divergence theorem (Eq. 2.29). That is,... 

The right-hand side of the expression above can be recognized as the divergence of the heat-flux vector,... 

The first term on the right-hand side of the identity above involves the divergence of the stress tensor, which also appears in the vector form of the momentum (Navier-Stokes) equations, Eq. 3.53. The momentum equation can be easily rearranged as... 

In this expression one term vanishes because V V = 0 for an incompressible flow and Vw = 0 because the divergence of the curl of a vector vanishes (vorticity is the curl of the velocity vector). For the same reason the last term on the right-hand side of the vorticity equation also vanishes. As a result the vorticity-transport equation is further reduced to... 

The divergence of a second-order tensor produces a vector. 

The equations in this section retain some compact notation, including the substantial derivative operator D/Dt, the divergence of the velocity vector V-V, and the Laplacian operator V2. The expansion of these operations into the various coordinate systems may be found in Appendix A. 

The stress state is represented as a symmetric tensor T, whose components may be expanded into various coordinate systems. The specific-coordinate-system expansions of the divergence of the velocity vector V V may be found in Section A.10. 

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