Notation vector

Setting the diabatic basis equal to the adiabatic basis at the degenerate point, Ro, the expansions can be written in vector notation as... 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as... 

Nuclear Magnetic Resonance. The iateraction of a nucleus with Bq is usually described usiag vector notation and models as ia Figure 2 where the bulk magnetization, Af, and the static field Bq are initially parallel to A radio frequency pulse is appHed ia the xy plane for a duration of t p.s,... 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

In order to facilitate writing, expressions such as (3-99) usually will be abbreviated by means of vector notation as follows... 

We can collect the three components in each of these equations into a single vector notation ... 

It is useful to adopt a double vector notation, so that X, Y ) represents the of coefficients In this notation... 

Here (j is the CG update parameter. In the above equations, e = e (tj) o electric field strength, = g (fj) o objective functional J with respect to the field strength (evaluated at a field strength of e t) and dk = d (t ) o search direction at the feth iteration. The time has been discretized into N time steps, such as that tj=jx )t, where j = 0,1,2, , N. Different CG methods correspond to different choices for the scalar (j. ... 

Vector notation is being used here because this is the easiest way to define the unit-cell. The reason for using both unit lattice vectors and translation vectors lies in the fact that we can now specify unit-cell parameters in terms of a, b, and c (which are the intercepts of the translation vectors on the lattice). These cell parameters are very useful since they specify the actual length eind size of the unit cell, usually in A., as we shall see. 

Equivalently, expectation values of three-dimensional dynamical quantities may be evaluated for each dimension and then combined, if appropriate, into vector notation. For example, the two Ehrenfest theorems in three dimensions are... 

Here, V is vector notation for the set of all component energies Vy, and A, j gives the coefficient of Vy in the ith run. The Ay, without subscript i, indicate the values of A in the target ensemble. The histograms collected in the runs are multidimensional in that they are tabulated as functions of the component energies as well as the order parameter . Similarly, the final result of the WHAM calculation is a multidimensional probability distribution in V J and . 

The synthesis of ideas described above was first achieved by Maxwell and is described by two equations that summarize the essential experimental observations pertaining to the reciprocally related electromagnetic effects1. In SI units and vector notation these equations, valid at every ordinary point... 

We are trying to discover what waves are generated inside and outside a crystal when it is illuminated by X-rays. The propagation of any electromagnetic waves in any medium is accurately described by Maxwell s equations. These are, in vector notation. 

Any hydrodynamic consideration of a drop moving in a liquid field starts with the Navier-Stokes equations of motion, as given in representative books on fluid mechanics (L2, Sll). Using vector notation to conserve space, these equations may be written (B3, B4)... 

A sketch of a pipe section is given in Figure E4.3.1, with the coordinate system and the vector notation. Typically, a length of 50 diameters from the inlet is required before a fully developed (long pipe) flow is assumed to be reached. The following can be assumed for this problem statement ... 

Spatial intensity distribution of the interference field depends on the phase difference between the beams Sy = ki - kj (in vector notation) and their mutual angles 9 (where i,j are the beam index) as ... 

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