Jacobian of the transformation

An interesting property of these integrators is that the Jacobian of the transformation from the state at time 0 to the time At is... 

SOME TENSOR CALCULUS RELATIONS 265 the Jacobian of the transformation is written as... 

From these relations between the initial and final velocities of each of the particles, it follows that the jacobian of the transformation is unify, so that... 

Now, if the transformed variable is normalized to Zw by the Jacobian of the transformation, we obtain... 

In view of the above equations, the Jacobian of the transformation yields... 

In Fig. 5, we present plots for the transformation function /(r), for the Jacobian of the transformation J(f (r) r), and for the derivative of the transformation function, df r)fdr corresponding to the example at hand. Selected values for the transformed orbitals are presented in Table 1, where we have also listed the corresponding values of the Hartree-Fock-Raffenetti orbitals, as well as the difference between untransformed and transformed orbitals for the sake of completeness. 

Moreover, since we assume that the set t,(r) is orthonormal, it follows that Skikji P) = Sk,kj ) = kfkj- In order to establish the connection with Cioslowski s work, let us define the matrix element of the Jacobian of the transformation as ... 

Note that, in this system of coordinates, the correction force includes a term involving the Jacobian of the transformation from generalized to Cartesian coordinates, which arises from the In contribution to In / in Eq. (2.157) for V. ... 

Standard deviations in unit-cell parameters may be calculated analytically by error propagation. In these programs, however, the Jacobian of the transformation from Sj,. .., s6 to unit-cell parameters and volume is evaluated numerically and used to transform the variance-covariance matrix of Si,. .., s6 into the variances of the cell parameters and volume from which standard deviations are calculated. If suitable standard deviations are not obtained for certain of the unit cell parameters, it is easy to program the computer to measure additional reflections which strongly correlate with the desired parameters, and repeat the final calculations with this additional data. 

With values for kd, f and <1 (0) we could convert the abscissa from T space to L space. After having determined the range in L space required to adequately cover the size distribution, we chose equally spaced values of and calculated the corresponding values. The amplitudes G(r, K) for this new set of ri were determined by interpolation from the original continuous distribution curve. By taking into account both the Jacobian of the transformation and the form factors calculated from theoretical principles (8), we could convert the ordinate by setting... 

Theorem 2 (Monotonicity). LetJ] = f(0) be a smooth transformation of 9, such, that exists, and let J be the Jacobian of the transformation Then... 

Since the equations of motion constitute a set of coupled ordinary differential equations, the solution in (59) is a unique function of the initial conditions. Thus, (59) can be viewed as an invertible coordinate transformation from a set of initial system coordinates xq at time, fo, to the coordinates Xt at time, t. As a result, using (58) and (59), the Jacobian of the transformation will tell us how the phase space volume element dxo transforms under (59). We can... 

The determinant on the right-hand side is the Jacobian of transformation from the error vector to the coordinate vector. More explicitly, the Jacobian of the transformation is... 

The Jacobian of the transformation acquires a particular realization for initial and final densities expanded about the same single center. In such a case, we have ... 

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