Jacobian determinant

The issue is whether it is possible that (dRf/dX dx = 0 for some nonzero intensive differential dR h which could then be taken as one of the / independent intensive differentials (S 10.2-5) to describe the system, with corresponding conjugate extensive differentials dX i. The usual condition of independence of these differentials is the nonvanishing of the Jacobian determinant ... 

Finally, note that in Eq. (2.12) we have included no Maslov phase factor related to the square root of the Jacobian determinant D,... 

Jacobian determinant, 394 Jager s equation for molecular diameter and viscosity, 95 Jamin effect, 137... 

To evaluate f ( 2) using eqn (E3.8) we need the values of the Jacobian determinant /(/i ) for all s, 6, direct computation of this determinant would involve products of integrals of the form... 

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

The Jacobian determinant of transformation between the material and fixed coordinate systems, is defined by ... 

It would not be practical or even desirable, however, to carry out a classical calculated in the above framework. The practical difficulty would be related to finding the roots of (117), the usual multi-dimensional root-search problem, and the result would be undesirable because zeros in the Jacobian determinant cause singularities, classical rainbows , in the classical probability distribution in (116), To remedy both of these features one averages the classical expression over a quantum number increment about n2 and over some increment about 2 ... 

A new periodic orbit with period T + ST, in the vicinity of the above T-periodic solution exists if the Jacobian determinant of the left-hand-side of Equation (58), computed at the initial conditions Xio, is different... 

The general procedure to be followed is to put each of these equal to zero, and so solve for an approximation for the fii, to first order in e, and then proceed by successive approximation, using the complete expressions for the fii, to obtain expressions for the fii in powers of e, to correspond to a periodic solution of the full perturbed motion. This could be done, taking r = 0, and so obtaining a solution of period T, provided that the Jacobian determinant,... 

This is an example for Euler s chain relation. Euler s chain relationship can be derived more formally with Jacobian determinants, as shown in Example 1.23. Rewriting Eq. (1.19) without arguments results in... 

A powerful method for the conversion of differentials uses Jacobian determinants, introduced in Sect. 1.14.6. We anticipate the conversion starting from Eq. (1.22) in terms of Jacobian determinants ... 

Using Jacobian determinants, as explained in more detail in Sect. 1.14.6, for the energy U in natural variables S, E, n, the law of Schwarz can be expressed as [9]... 

In three variables, there are two additional analogous equations. Similar relations are valid also for other thermodynamic functions. The relation Eq. (1.28) is claimed to be the formulation of the first law of thermodynamics in the form of Jacobian determinants [9],... 

The Jacobian determinant is the determinant of the Jacobian matrix. Both are sometimes simply addressed as Jacobian. If we have a set of functions... 

The Jacobian determinant can be handled like an ordinary fraction ... 

The properties of Jacobian determinants allow a formalism to convert differentials occurring in thermodynamics [21]. 

In the first step, develop the partial derivative into a Jacobian determinant. Then lookup in Table 1.4 both numerator and denominator and insert the abbreviation. In the final step, from the table footnotes again expand the abbreviation into a Jacobian determinant. 

Here we have converted the variable set for the Gibbs free energy into its natural variables by multiplying the numerator and denominator by 9(r, p) with subsequent rearranging. Expanding the Jacobian determinant of Eq. (1.86), but keeping the Jacobian notation except to the final equality results in... 

When the partial derivatives are directly inserted in the expansion of the Jacobian determinant, it may be more difficult to verify which terms will cancel. The calculation without the use of the Shaw tables is more lengthy. The experienced user may not need the Shaw tables. However, these tables give valuable hints on how to substitute properly to arrive at the desired conversion. 

However, observe that the corresponding pressure changes from dp(S, V,. ..)/dV into dV(S, p, -)/dp. Thus, keep in mind that the entropy is still the dependent variable. As an exercise, we change the variables using Jacobian determinants. For simplicity, we are using two independent variables in the list of variables ... 

However, we will treat the matter more generally and we recalculate the differential for temperature and volume as the independent variables. We use functional determinants, also addressed as Jacobian determinants. In terms of functional determinants, we can rewrite... 

---