Curl operator

V X is the curl operator). Only the kinetic energy of the electrons are considered within the Bom-Oppenheimer approximation, and the generalized momentum becomes... 

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

These relationships show that the Fourier components are perpendicular to the propagating vector k, and consequently the field components can be combined in the complex plane as Ek + iHk, and together define the so-called tranversality condition. Equivalent arguments applied to the curl operator produce the remaining Maxwell equations in k space ... 

The Ampere law is next used with a covariant definition of the curl operator V flx = Vx+i A x (351)... 

Acting upon curl operator of f 1 given by equation (3), one derives the following expression [13a] (see also Ref. [29g])... 

Equations (16) and (19) are formally identical. It is suggested that a satisfies equation (17) for the two-state problem and a the similar one for the generalized two-state problem. Applying now the curl operator to both sides of the first equations in equations (16) and (19) and substituting a = a in the last one, one readily obtains that, in the former case, this results in the identity 0 = 0 because the right hand side of equation (11) vanishes for the two-state problem. In the latter case, one finds that the substitution a = a and curl operation, applied to the first equation in equation (19), do not commute, viz., replacing a by a eliminates the right hand side of the first equation in equation (19) and a further application of the curl to the resultant equation yields... 

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)... 

In many problems, it is useful to separate Maxwell s equations (8.8) - (8.11), i.e., to write the equations for the electric and magnetic fields separately. To do this, we will apply the curl operator Vx to the left- and right-hand sides of (8.9) and substitute the result into (8.8) ... 

Applying a curl operator to both sides of (12.80) and dividing by iuj. we derive the expressions for the magnetic field components ... 

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] ... 

Introducing these relations into the above expression for the Curl operator gives ... 

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