Cartesian theorem

The elements of these vectors can be evaluated using an off-diagonal fomt of the Hellmann-Feynmann theorem, which in Cartesian coordinates, Xa, is... 

The Euler characteristic is usually calculated in two ways by using the Gauss-Bonnet theorem [7,85,207,210,222] [Eq. (8)] or by combining the Cartesian and Gauss theorems [Eqs. (122) and (123)], which is also called the Euler formula [23,76,224]. 

To end this section, note that the Cartesian formula (120) is suitable to calculate the local Gaussian curvature while an exact value of the Euler characteristic is obtained from the Gauss-Bonnet theorem. 

In order to attain maximum clarity, we will carry out below a complete proof of the anti-dynamo theorem for the case of plane motion with vx = 0 in Cartesian coordinates (cf. [6]). Then we will discuss the general case more briefly. 

In the case of a scalar field, the irreducible matrix D is a unit matrix, and drops out of. I1. For rotation through an angle S9t about the Cartesian axis ek, the rotational submatrix of the Lorentz matrix is given by Xkx = ()Hkekl]x], where el]k is the totally antisymmetric Levi-Civita tensor. For the one-electron Schrodinger field f, Noether s theorem defines three conserved components of a spatial axial vector,... 

The conservation equations for continuous flow of species K will be derived by using the idea of a control volume r t) enclosed by its control surface o t) and lying wholly within a region occupied by the continuum here t denotes the time. In this appendix only, the notation of Cartesian tensors will be used. Let i = 1, 2, 3) denote the Cartesian coordinates of a point in space. In Cartesian tensor notation, the divergence theorem for any scalar function belonging to the Kth continuum a (x, t), becomes... 

Technically, the nuclear coordinates and orbital parameters, which consist of the Cartesian coordinates of the orbital center and its exponent, are all variational parameters and have to be optimized until the viral theorem is satisfied as a necessary condition (at least the total kinetic energy is equal and opposite to the total energy). For a sufficient condition, it is required that the obtained electronic and geometrical structure must make sense chemically and experimentally. 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

We introduce now a domain Vr, bounded by a sphere dVa of a radius / , with its center at the origin of some Cartesian coordinate system, x, y, z. Integrating both sides of equation (9.65) over the domain Vr, and applying Gauss s theorem, we find ... 

Here 8, is Kronecker s delta i, j= 1,2,3 correspond to Cartesian (the crystalline) axes, 8(t) is the Dirac delta-function. The random variable H (f) must also obey Isserlis s theorem [21]. By introducing [9] the dipole vector... 

The classical phase space is formally defined in terms of generalized coordinates and momenta because it is in terms of these variables that Liouville s theorem holds. However, in Cartesian coordinates as used in the present section it is usually stiU true that pi = mci under the particular system conditions specified considering the kinetic theory of dilute gases, hence phase space can therefore be defined in terms of the coordinate and velocity variables in this particular case. Nevertheless, in the general case, for example in the presence of a magnetic field, the relation between pi and Cj is more complicated and the classical formulation is required [83]. 

Here the space variables r and s in the Cartesian coordinate frame of the projection have been replaced by the Cartesian coordinates x and y in the laboratory frame (cf. Fig. 5.4.1), and

rotation angle between both frames. This equation is another formulation of the projection cross-section theorem (cf. eqn (5.4.12)), which states that the Fourier transform p(k, cp) of a projection P(r, (p) is defined on a line p kcos(p, k sin p) at an angle

[Pg.203]

Since the transformation properties of spherical harmonics are well known, the spherical-tensor notation has some advantages, particularly in the derivation of general theorems however, the reality of cartesian tensors also has its attractions, especially for small values of /. Normally, the moments of a particular three-dimensional molecule are most conveniently given in an x,y,z frame. 

This is identical to (2-8), though now expressed in Cartesian component notation.6 Time derivatives at fixed spatial position are often called the Eulerian time derivatives, whereas those taken at a fixed material point are known as Lagrangian. Although we have derived a simple relationship relating the convected or material derivative to the ordinary partial derivative at a fixed point, this cannot be applied directly to (2-6) without derivation of a general relationship, known as the Reynolds transport theorem. 

Since many of the operators that appear in the exact Hamiltonian or in the effective Hamiltonian involve products of angular momenta, some elementary angular momentum properties are summarized in the next section. Matrix elements of angular momentum products are frequently difficult to calculate. A tremendous simplification is obtained by working with spherical tensor operator components and, in this way, making use of the Wigner-Eckart Theorem (Section 3.4.5). A more elementary but cumbersome treatment, based on Cartesian operator components, is presented in Section 2.3. 

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