Partial derivative, meaning

The S5mibol stands for the set of amounts of all species, and subscript on a partial derivative means the amount of each species is constant—that is, the derivative is taken at constant composition of a closed system. Again we recognize partial derivatives as partial molar quantities and rewrite these relations as follows ... 

Partial Derivative The abbreviation z =f x, y) means that is a function of the two variables x and y. The derivative of z with respect to X, treating y as a constant, is called the partial derivative with respecd to x and is usually denoted as dz/dx or of x, y)/dx or simply/. Partial differentiation, hke full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations. 

Here e is a unit vector normal to the surface at the POI defined by e = pH xpv/ pH xpv, and subscripts of p denote the partial derivatives. Thus the mean, H, and Gaussian, K, curvatures are expressed as... 

As pictured in the top right-hand corner of Figure 4.16, diffusion takes places in two dimensions, denoted x and R, within the constrained diffusion layer, which can be expressed by means of the following set of partial derivative equation and initial and boundary conditions ... 

Summa summarum, Eq. (10.12) is certainly not the most efficient one for routine calculations of intrinsic bond energies. The reason is obvious—it lies with the partial derivatives (dski/dZk)p, which must be carried out at constant electron density p, meaning that this difficult calculation has to be made for each new s/ i, which is unpractical. 

This partial derivative is the velocity of the concentration front hi the bed. The constant pattern assumption presupposes that this velocity is constant, or in other words, is independent of the solution concentration. This means that all points on the breakthrough curve are traveling in the bed under the same velocity, and thus a constant shape of this curve is established (Wevers, 1959). According to the above equation, this could happen only if (Perry and Green, 1984)... 

Once the transformations have been made, a solution space has been defined. It is only within this framework that the concept of steepest ascents takes meaning. The proper direction in which to proceed can be determined by n + 1 cases if n is the number of controllable variables. In each of the n cases one variable is changed slightly from its value in the base case, while all other variables are held constant. This permits approximating the n partial derivatives of response with respect to each variable. The direction of steepest ascent is given by the vector which is the gradient of the response, R ... 

In order to evaluate P2 we need to consider how the governing equations for mass and energy balance themselves vary with changes in the variables. In the case of the present model this means evaluating various partial derivatives of (5.1) and (5.2) with respect to a and 0. Before proceeding, however, we should take a look at the elements of the Jacobian matrix evaluated for Hopf bifurcation conditions ... 

We used an analogous procedure (4) to derive the relations for activity coefficients assuming that the partial derivatives in Equation 4 can be expressed by means of a function f(mi,m2). Then for the activity coefficients of the first and second component in a three-component system, the following integrated relations are valid ... 

We will now discuss the steady-state solutions of Eq. 22-6. Remember that steady-state does not mean that all individual processes (diffusion, advection, reaction) are zero, but that their combined effect is such that at every location along the x-axis the concentration C remains constant. Thus, the left-hand side of Eq. 22-6 is zero. Since at steady-state time no longer matters, we can simplify C(x,t) to C(x) and replace the partial derivatives on the right-hand side of Eq. 22-6 by ordinary ones ... 

Numerical problems arising from the use of the Lorentzian for fitting spectra have also been reported [69]. These are related to the non-dif-ferentiable profile at the points where the exponential wings are attached to the Lorentzian core. Partial derivatives with respect to the line shape parameters are usually needed in least mean squares fitting procedures. 

Here the 0 of the limit means sufficiently small for the limit to exist which is to be understood more precisely in the context of the experiment. A corresponding z-meter in the multivariable case would have n xrdials, each of which is tweaked in turn (holding the remaining n — 1 dials fixed) to determine the successive partial derivatives Zi,i = 1,2,..., n. It is noteworthy that the multivariate dz carries sufficient information to evaluate each of its possible monovariate dxi derivatives zj, confirming its status as a more powerful type of mathematical object. 

Note that this identity clearly shows that (dz/dx)y (dz/dx)e, i.e., that the variable held constant matters in these derivatives (Strictly speaking, a lazy notation such as dz/dx has no meaning whatsoever ) Although the inconvenient notation of partial derivatives makes it somewhat tedious to keep the inactive (constant) background variables in mind, it is important from a physical and pedagogical standpoint that this be done as carefully as possible. (The tedium of this notation is avoided in the geometrical thermodynamics to be presented in Part III.)... 

However, it may happen that a non-linear mechanism cannot a priori be excluded. Therefore we now consider the elaboration of Rct and X for the CECDC mechanism treated in Sect. 4.2.2. It is tedious, but not difficult, to derive from eqn. (123) expressions for the partial derivatives F, O and R in terms of and the mean concentrations c and Cr. It can also be verified that these expressions reduce to simpler ones in the limiting cases for which eqn. (124) holds. The next step is to substitute c 0 and Cr by appropriate functions of c , Cr, and reversible case, this involves procedures similar to those mentioned in Sect. 4.3.1 and one may wonder whether the impedance parameters are of more diagnostic value than the d.c. current itself. 

The partial derivatives in eqns. (153) will be functions of the potential-dependent rate constants, kj, etc. and the mean concentrations c 0,c z, and cR. So, for application to experimental data, the results of the present section have to be combined with those of Sects. 5.1.1 or 5.1.2. The interested reader can find extensive theoretical predictions of the a.c. polarogram in ref. 131. 

A (a) The partial derivative of z with respect x is found by treating y as a constant. In order to make it clear that several variables are present, we sometimes use the notation ( )>, which means... 

In the concentration range regarding the ED processes, the effective diffusion coefficient (Z>B) can be predicted via the Gordon relationship (Reid et al, 1987), which accounts for the partial derivative of the natural logarithm of the mean molal activity coefficient (y+) with respect to molality (m) and solvent relative viscosity (rjr) ... 

The integral of a total differential, however, is the difference between two values of a state function and, therefore, cannot depend on the path of the integral. This holds even if this difference is evaluated by means of partial derivatives. 

This example is used in Frame 5 to illustrate the meaning and interpretation of partial derivatives. A is a function depending on two variables A = f(x,y) and this has implications for the meaning and interpretation of the differential coefficients corresponding to the various slopes which are represented as partial derivatives (3A/dx)y and (3A/3y)x rather than as ordinary derivatives d r/dx etc. as were discussed earlier in this Frame and for which only one variable is involved. 

The aroma compound will diffuse in as well as out of the cube because of its perpendicular side surface areas. Due to the greater decrease in the aroma near the soap s external surface, the flux out of the side of the cube closer to the surface is greater than the flux into the side of the cube that lies deeper in the soap. The difference between the aroma diffusing in and out will be positive which means one can consider the cube as an aroma source. As a consequence of the flux out of the cube, the concentration in the cube decreases with time. The concentration is also a function of time, c = c(x, y, z, t) and its decrease with time, i.e. the partial derivative —dc/dl in the cubic volume AV = Ax Ay Az, represents the net flux out of the cube and designated div/ the divergence of the flux. 

The Elementary Partial Derivatives.—We can set up a number of familiar partial derivatives and thermodynamic formulas, from the information which we already have. We have five variables, of which any two are independent, the rest dependent. We can then set up the partial derivative of any dependent variable with respect to any independent variable, keeping the other independent variable constant. A notation is necessary showing in each case what are the two independent variables. This is a need not ordinarily appreciated in mathematical treatments of partial differentiation, for there the independent variables are usually determined in advance and described in words, so that there is no ambiguity about them. Thus, a notation, peculiar to thermodynamics, has been adopted. In any partial derivative, it is obvious that the quantity being differentiated is one of the dependent variables, and the quantity with respect to which it is differentiated is one of the independent variables. It is only necessary to specify the other independent variable, the one which is held constant in the differentiation, and the convention is to indicate this by a subscript. Thus (dS/dT)P, which is ordinarily read as the partial of S with respect to T at constant P, is the derivative of S in which pressure and temperature are independent variables. This derivative would mean an entirely different thing from the derivative of S with respect to T at constant V, for instance. 

There are a number of partial derivatives which have elementary meanings. Thus, consider the thermal expansion. This is the fractional increase of volume per unit rise of temperature, at constant pressure ... 

This is a system of ordinary differential equations. Its solution is subject to the same boundary conditions (eqs 7-12) as the solution of eqs 4 and 5. However, the variables involved are no longer time-dependent, which means that the partial derivatives dq/dr and dT/dr can be replaced by the ordinary derivatives dc,/dr and dT/dr. 

The partial derivatives in this equation have definite physical meanings and are measurable quantities. For liquids they are related to two commonly tabulated properties ... 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

---