Jacobian matrix, transformation

Obviously the described transformation depends on the existence of an inverse for the Jacobian matrix (i.e. det/must always be non-zero). 

DERIV. Calculates the inverse of the Jacobian matrix used in isoparametric transformations. 

Note, these many "coefficients" are the elements which make up the Jacobian matrix used whenever one wishes to transform a function from one coordinate representation to another. One very familiar result should be in transforming the volume element dxdydz to... 

The Jacobian matrix specifying the transformation from (9, ) to (C, k) is immediately derived from equation (7.107) and is... 

Although the r/E-fit and the p-Kr r, (-rM) fit are not equivalent (the former determines three more variables), it could be shown [55] that the molecular structures determined by the r/e-fit and the r -fit are strictly identical, including the covariance matrix. This is due to the specific form of the Jacobian matrix X of the coupled least-squares problem r/ , which permits a decomposition by a non-singular transformation into a smaller least-squares problem rM plus a subsequent direct calculation of the constant rovib contributions Eg. The r -part of the problem alone determines the molecular structure which must then be used (including the covariance matrix of the structural parameters) for the calculation of the contributions Eg. When rotational constants of new isotopomers are to be predicted from the structure determined, the r/E-method performs much better than the r -method due to the presence of the additional rovib parameters . ... 

It is thus possible to eliminate the angular motion from the problem and to write down an effective body-fixed Hamiltonian within any (J, M, k) rotational manifold, that depends only on the internal coordinates. The transformation from the laboratory-fixed to the translation-free frame is linear and so its jacobian is simply a constant that can be ignored and since the transformation from the t to the zi is essentially a constant orthogonal one, it has a unit jacobian. The transformation from the translation-free nuclear coordinates to the body-fixed ones is, however, non-linear and has a jacobian J -1 where J is the matrix constructed from the nuclear first derivative terms. 

This form of the general Jacobian element allows for the straightforward solution of the SSOZ equation for molecules of arbitrary symmetry. However, in the numerical solution using Gillan s methods, most of the computation time is involved in calculating the elements of the Jacobian matrix, rather than in the calculation of its inverse or in the calculation of transforms. Indeed, as the forward and backward Fourier transforms can be carried out using a fast Fourier transform routine, the time-limiting step is the double summation over / and j in Eq. (4.3.36). With this restriction in mind, it is... 

The Jacobian matrix of this transformation at the point (0, 0) has the form... 

When the function F(x) has a critical point at x = 0, the transformation defined above, (2.28), does not determine a one-to-one change of variables nearby the point x = 0, because the first row of the Jacobian matrix vanishes and, consequently, so does its determinant. As a result, such a function cannot be represented as F(x ) = x/ in the vicinity of its critical point. In a case, however, when the critical point x = 0 is structurally stable (a precise criterion will be provided in further part of this section), the function F may be reduced in the vicinity of this point to a simpler form. 

Jacobian matrix memory kernel Laplace exponent infinitesimal generator Laplace transform Laplacian matrix monodromy matrix... 

The pull-back of a differential 1-form f under a phase space mapping is defined as the action of f after transformation by the Jacobian matrix of the mapping. It is written so... 

After this transformation, the reduced Jacobian matrix will then take the form (Figure 5.4-3), which is more convenient for the computation of 8... 

With the transformation carried out for the last row of the Jacobian matrix O, such transformation also changes the last element of the vector g and that element is replaced by... 

J" (introduced in the main text) is the Jacobian matrix of partial derivatives for transforming from generalized coordinates to mass-weighted coordinates. These substitutions result in... 

The Jacobian matrix for transforming from mass-weighted Cartesian to generalized coordinates can be narrowed in a similar manner. A derivative dx jd(f is taken with respect to flexible generalized coordinate (f with all other generalized coordinates held constant, so no stiff coordinates change along the direction 9x/9g. Thus the Jacobian matrix J" may be divided as... 

Let us just mention that sometimes we are not interested in transforming positions (position vectors, triplets of coordinates), but small (differential) changes in positions r as we saw in Eq. (12). In this case, we are not concerned with translations between the coordinate systems, only misalignments are of interest. Instead of using the rotation matrix, we may use the Jacobian of transformation, getting... 

Co-ordinate transformation matrix Displacement vector Jacobian Matrix... 

In this section, we first introduce the standard form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term. 

Although they did not obtain a closed-form analytic expression for the three-dimensional case, they dealt with a trasformed one-matrix for the single Slater determinant constructed from plane waves, and rewrote the energy in terms of this transformed matrix. The conditions on the transformation were not imposed through the Jacobian but rather through the equations ... 

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 = det is the determinant of the 3N x 3N transformation matrix 8R /8g , which gives the Jacobian for the transformation from generalized to Cartesian coordinates. This follows from the fact that the right-hand side (RHS) of Eq. (2.16) for g p is a matrix product of this transformation matrix with its transpose, and that the determinant of a matrix product is a product of determinants. By similar reasoning, we find that... 

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. 

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