Jacobian volume element

The origin of this residual Hartree-Fock error at the large—I) limit is readily identified [40]. The Hartree-Fock wavefunction, constructed as a product of one-electron orbitals, lacks any explicit dependence on the angle 6 between the electron radii. Hence this angle enters only in the Jacobian volume element, which contains (sin ) therefore... 

Transformation to Unit Jacobian. The radial part of the Jacobian volume element is Jp = thus, the transformation... 

But what are we to make of the new dimension-scaled potential energy function, V r) = D — l)/2r, when D = Herrick and Still-inger [10] have observed that (U—l)/2r = S(x) in the one-dimensional limit, where x is the cartesian coordinate, —oo < x < oo, and S(x) is the Dirac delta function. This surprising result is a consequence of the dimension dependence of the Jacobian volume element, r dr, which leads to the identity [14]... 

The volume of a Y -space-volume-element does not change in the course of time if each of its points traces out a trajectory in Y space determined by the equations of motion. Equivalently, the Jacobian... 

The term J is the determinant of the Jacobian matrix upon changing from Cartesian to generalized coordinates. It measures the change in volume element between dxdp, and d polar coordinate J = r and therefore dxdy = r dr <17. The derivative of A is therefore the sum of two contributions the mechanical forces acting along (dU/<9 ), and the change of volume element. The term -1//3 d In J /<9 is effectively an entropic contribution. 

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 transformation of the variables and the region with respect to which the integration is made involves the determinant of the Jacobian, detJ = J. For instance, a volume element becomes... 

For coordinate transformations we generally have the following relation between the volume elements dq = J dQ, where J is the absolute value of the Jacobian J, which is given by the determinant... 

Another important property of Hamiltonian systems can be deduced by considering the change in the volume element of phase space dT as the ensemble evolves from an initial time t = 0 to a time t. Hamilton s equations of motion can be viewed as generating a transformation from an initial set of coordinates and momenta (qi(0),. . . , qN(0), Pi(0),. . . , Pn(0) to a set of coordinates and momenta (qi(t),. . . , qN(t)> Pi(t)> Pjv(t) via the evolution operator exp(iLt) in Eq. [9]. We can, therefore, use this transformation to calculate the Jacobian that results when the volume element dF for the phase space coordinates at time t is transformed into the volume element dTq for the phase space coordinates at t = 0. The volume element will transform according to... 

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 invariance rests on the property of the Jacobian determinant of a canonical transformation D = det = 1. Consequently, the volume element is... 

The Jacobian of the transformation from x,y, z to r,9,z is denoted by Jxyz/r9z in the following integration over a differential volume element ... 

The volume element in spherical polar coordinates can be determined from the Jacobian ... 

Although it is quite possible to discuss other transformations such as rotations,these would take us too far afield here. To proceed it is important to observe the effects of these transformations on various quantities that come into the evaluation of time correlation function in Eq. (12). For example, the transformation might have an effect on the volume element d F. This would be given by the Jacobian J of the transformation from F to F. In addition, the Hamiltonian, Liouvillian, and equilibrium distribution function might change under the various transformations. In Table 1 we summarize how these quantities are transformed. The primes on the headings of the columns indicate the values of the transformed quantities, whereas the unprimed quantities in the body of the table indicate the untransformed values. 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

To express the differential volume element in terms of new variables is most readily done using Jacobians. Given the coordinate transformation... 

The factor r sin(6>), which is called a Jacobian, is required to complete the element of volume in spherical polar coordinates. The form of this Jacobian can be deduced from the fact that an infinitesimal length in the r direction is dr, an infinitesimal arc length in the 9 direction is rdO, and an infinitesimal arc length in the cj) direction is rsin(0), dcj) if the angles are measured in radians. The element of volume is the product of these mutually perpendicular infinitesimal lengths. Spherical polar coordinates were depicted in Figure 17.3. The volume element is crudely depicted by finite increments in Figure 17.10. 

For other coordinate systems, a factor analogous to the factor r sin(0) must be used. This factor is called a Jacobian. For example, for cylindrical polar coordinates, where the coordinates are z, (p (the same angle as in spherical polar coordinates, and p (the projection of r into the xy plane), the Jacobian is the factor p, so that the element of volume is pdpdzdp. We use the symbol d r for the three-dimensional volume element in any coordinate system, so that dxdydz, r sin 6)d[Pg.1241]

In 3D we also need the two transformations used with the 2D isoparametric element. In the first place, the global derivatives of the formulation, dNi/dx, must be expresses in terms of local derivatives, dNi/d . Second, the integration of volume (or surface) needs to be performed in the appropriate coordinate system with the correct limits of integration. The global and local derivatives are related through a Jacobian transformation matrix as follows... 

Changing variables in three dimensions requires the appropriate Jacobian. Figure 9.5 depicts the element of volume in spherical polar coordinates. The length of the little box in the r direction is dr, the length in the 6 direction r dO, and the length in the

[Pg.127]

The continuum approach to the description of mechanical behavior starts with the assumption that a body at the macroscopic level may be regarded as composed of material that is continuously distributed. Such a body occupies a region of three-dimensional space. The region occupied by the body will of comse vary with time as the body deforms. The region occupied by the body in the reference configuration at the time t = 0 is denoted by Q, and a material point may be identified by the position vector x. The properties and the behavior of the body can be described in terms of fimctions of position x in the body and time t. The motion is orientation-preserving that is, the Jacobian J, defined by j = Aet dyi/dXj must be positive. Hence, every element of nonzero volume in Q is mapped to an element of nonzero volume in Qi (Figure 1.4). 

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