Differential, coefficient Order

In order to simplify the mathematical treatment, London assumes that the distance between a and b in the molecule a—b remains unchanged on the approach of the atom c, and only the interaction with c is taken into consideration. Thus the value of Aab remains constant and the variation of the energy depends only on Ab i-e. on the distance between b and c. The value of increases with decrease of the distance between b and c until this distance is identical with the a—b distance. The energy of the system is then a maximum, which can be determined by taking the differential coefficient of E with respect to Abe and equating to zero, viz... 

We have already argued that discussions of bond strength should most naturally be given in terms of two-particle matrices. But before doing so it is worth looking briefly at earlier discussions in terms of the one-particle matrix. In Hiickel theory, where in (5) there are no terms g(ij), (6) shows that the energy involves only y hence a one-particle discussion is reasonable. The simplest example is the Coulson bond order, which is simply an off-diagonal element in the expansion of (9) in terms of atomic orbitals it is also the differential coefficient of (H) with respect to the appropriate resonance... 

The differential coefficient derived from any function of a variable may be either another function of the variable, or a constant. The new function may be differentiated again in order to obtain the second differential coefficient. We can obtain the third and higher derivatives in the same way. Thus, if y = a 3,... 

A function may thus change its sign by becoming zero or infinity, it is therefore necessary for the first differential coefficient of the function to assume either of these values in order that it may have a maximum or a minimum value. Consequently, in order to find all the values of x for which y possesses a maximum or a minimum value, the first differential coefficient must be equated to zero or infinity and the values of x which satisfy these conditions determined. 

Hence the rule In order to find a point of inflexion we must equate the second differential coefficient of the function to zero find the value of x which satisfies these conditions and test if the t second differential coefficient does really change sign by substituting in the second differential coefficient a value of a a little greater and one a little less than the critical value. If there is no change of sign we are not dealing with a point of inflexion... 

In order to expand any function by Maclaurin s theorem, the successive differential coefficients of u are to be computed and x then equated to zero. This fixes the values of the different constants. 

A differential equation is ordinary or partial, according as there is one or more than one independent variables present. Ordinary differential equations will be treated first. Equations like (2) and (3) above are said to be of the first order, because the highest derivative present is of the first order. For a similar reason (4) and (6) are of the second order, (5) of the third order. The order of a differential equation, therefore, is fixed by that of the highest differential coefficient it contains. The degree of a differential equation is the highest power of the highest order of of differential coefficient it contains. This equation is of the second order and first degree ... 

A linear differential equation of the first order involves only the first power of the dependent variable y and of its first differential coefficients. The general type, sometimes called Leibnitz equation, is... 

Similar to the calculus of functions, we can differentiate matrices by differentiating the elements of the matrix in each case. The usual designation for the derivative of a square matrix Y is dY/dx if the matrix elements are functions of the scalar x. The differential coefficient of a product is also similar to that of a product of scalar functions however, the order of the matrices in the product must be maintained. Therefore,... 

In order to compute the change of, say, the enthalpy it is desirable to use temperature and pressure as the independent variables and also to express the various partial differential coefficients in terms of experimental magnitudes. This may be illustrated as follows. 

However, although these equations have theoretical support, and equation (34) has fewer coefficients than have equation (25) or the tenth-order Chebyshev equation, its fit to the data for oxygen is not as good and there are systematic deviations which the polynomials do not exhibit. It therefore seems questionable whether derived thermodynamic quantities which depend upon the differential coefficients dp/dJ or d p/dr are certain to be more accurate when calculated from equation (34) than they are when calculated from equation (25) or the Chebyshev equation. 

Under consideration from the purely mathematical point of view, the surfaces with which we occupy ourselves were the object of research of several geometCTs we will profit from further results they obtained only 1 will say, as of now, that the radii R and R can be expressed using differential coefficients, which converts formula [2] into a differential equation of the second order by making = P> = r, = t,... 

If we take the axis of revolution as the x-axis, we will have, as one knows, p and q being respectively the differential coefficients of the first and the second order of y with respect to x. 

Let us take on this curve two points belonging one to a convex arc, the other to a concave arc, and placed in the same manner on these two arcs, i.e. at the same distances from the starts of these arcs. If, for brevity, we represent by y the term psin jx of our equation, the value of y will be the same, except for the sign, for the two points, so that the ordinates of those will be respectively r+y and r — y, which gives, according to a known formula, for the values of the two normals, (r+y) +p and (r - y) Vl + p, where p is, as always, the differential coefficient it should be noticed that, by the nature of the liquid shape, these normals are both positive ones one will have to remember moreover, for the understanding of the formulas which follow, that the quantity y, or Psin jX, is taken in itself, and consequently is primarily positive. As for the radius of curvature, it is clear that its value is, except for the sign, the same for the two points if thus q indicates the differential coefficient of the second order one will have, also according to a known expression, for the radius of curvature at the first... 

It is worthwhile, albeit tedious, to work out the condition that must satisfied in order for equation (A1.1.117) to hold true. Expanding the trial fiinction according to equation (A1.1.113). assuming that the basis frmctions and expansion coefficients are real and making use of the teclmiqiie of implicit differentiation, one finds... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

A reversed pitot tube, also known as a pitometer, has one pressure opening facing upstream and the other facing downstream. Coefficient C for this type is on the order of 0.85. This gives about a 40 percent increase in pressure differential as compared with standard pitot tubes and is an advantage at low velocities. There are commercially available very compact types of pitometers which require relatively small openings for their insertion into a duct. 

In general, consider a system whose output is x t), whose input is y t) and contains constant coefficients of values a, h, c,..., z. If the dynamics of the system produce a first-order differential equation, it would be represented as... 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

A water body is considered to be a one-diiuensional estuary when it is subjected to tidal reversals (i.e., reversals in direction of tlie water quality parameter are dominant). Since the describing (differential) equations for the distribution of eitlier reactive or conserv ative (nomciictive) pollutants are linear, second-order equations, tlie principle of superposition discussed previously also applies to estuaries. The principal additional parameter introduced in the describing equation is a tid il dispersion coefficient E. Methods for estimating this tidiil coefficient are provided by Thomaim and Mueller... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

As a result, a considerable amount of effort has been expended in designing various methods for providing difference approximations of differential equations. The simplest and, in a certain sense, natural method is connected with selecting a, suitable pattern and imposing on this pattern a difference equation with undetermined coefficients which may depend on nodal points and step. Requirements of solvability and approximation of a certain order cause some limitations on a proper choice of coefficients. However, those constraints are rather mild and we get an infinite set (for instance, a multi-parameter family) of schemes. There is some consensus of opinion that this is acceptable if we wish to get more and more properties of schemes such as homogeneity, conservatism, etc., leaving us with narrower classes of admissible schemes. 

By the n-layer composite scheme of period m (of order m) we generally mean a system of differential equatioins with operator coefficients... 

Since the measurements of conductance change are not directly related to the composition of the solution, as an alternative method numerical integration of the differential rate equations implied by the proposed mechanism was employed. The second order rate coefficients obtained by this method are... 

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