Nabla operator

Instead of the dot product the vector product of the nabla operator can also be formed to produce a function called curl or rot,... 

We shall assume that our system is spherically symmetric so with the nabla operator in spherical coordinates, the diffusion equation may be written... 

V - nabla operator, Vt]a = (Vtjai> Vt uA), H - a field conjugate to the parameter tp. In Eq. 5 the dependence of order parameters on coordinates is used implicitly. 

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

The relativistic correction of the mass variation with velocity depends essentially on the fourth power of the nabla operator [68b]. In fact one can write the involved integral as ... 

Here the first term with V the Nabla operator is the kinetic energy, the second is the potential energy due to the nuclear charge, and the last term is the total electrostatic interaction energy over all pairs of electrons. 

The Nabla operator (V) is in fact two-dimensional in this equation. The pressure gradient dp/dz is a constant, i.e., place independent. Obviously, the assumption of constant density is very reasonable even for gases, since we are dealing with low-pressure-drop reactors. 

Here it is the vector of the horizontal velocity, U the vertically integrated horizontal velocity, w is the vertical velocity, r is the sea-level elevation, V/, the horizontal nabla operator, q a source term of water flux, T the temperature, S the salinity, p the pressure, and p is the density. Moreover,/is the inertial frequency,/= 2 1 sin 0, where 1 Zn (1 I 1/365.2425)724 h is the earth s angular velocity and 0 is the latitude. Turbulent viscosity is indicated by the term D, . Wind forcing enters the scheme as a vertical boundary condition. The equations are solved usually in spherical coordinates, but are written here for simplicity in Cartesian form. 

In (1.25), the terms inside the brackets can be reformulated by use of vector and tensor notations. By comparing the terms inside the brackets with the mathematical definitions of the nabla or del operator, the vector product between this nabla operator and the mass flux vector we recognize that these... 

The resulting expression for the nabla operator (C.86) are then employed to deduce the transformation formulas for the gradient, divergence, and curl operators in any orthogonal curvilinear coordinate system [11] ... 

Finally, the self-part of the dipolar Ewald energy, U, cannot be derived by applying the Nabla operator as for the other terms, because the corresponding self-term for charged systems is already a constant. Specifically, one realizes from Eq. (6.17b) that the Coulomb self-energy can be rewritten as... 

To be able to formulate the theory one has to treat first of all the problem of the unbounded operator Er. If E is homogeneous (which is fulfilled in a good approximation within a laser beam) we can apply, following Mott and Jones119 and Kittel120 (see also ref. 116), the nabla-operator V to a Bloch function... 

V Vector nabla operator (1) Over symbol denoting deriva-... 

In the first integral, we have the same situation as earlier in this chapter. In the second integral, we write the nabla operator in Cartesian coordinates, obtain a scalar product of two gradients, and then get three integrals equal to one another (they contain x, y, z), and it is sufficient to calculate one of them by spherical coordinates using the formula H.2 in Ajjjendix H available at booksite.elsevier.com/978-0-444-59436-5, p. e9l. 

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