Derivatives notation

Hi) Change of Variable Held Constant Under the same change of variables (x, y) —> (x, 0), we can also obtain the partial derivative (dz/dx)e (with the new variable 9 held constant). Starting again from (1.10), we divide by dx at constant 6 on both sides (using proper partial derivative notation for the constrained ratios) to obtain... 

Cy is the heat capacity at constant volume. We have used the partial derivative notation (see Appendix A) for dUr/dTr, the rate of change of internal energy with temperature when volume is held constant. We cannot use a partial derivative for 8qr/dTr because, as we discussed, 8qr is the amount of heat transferred and not the change in something, which is required for a derivative. The added heat for a finite temperature range may be found by integrating Eq. (16) in the form 8cy = Cy dTy. ... 

Note that when a figure C depends on several independent variables x and t, its variations vis-a-vis one of these variables are given by its partial derivative 9C/9x with respect to the variable. When C depends on a single variable, the usual derivation is used dC/dx. The intrinsic difference between the two kinds of derivations is indicated by the symbolic use of 9 or d in the derivative notations. 

The justification of this reformulation is apparently related to the physical interpretation of the mathematical operators used here. In this case the total derivative notation is adopted for an operator determining the time derivation at a fixed point in space, and not for a time derivative along a fluid particle trajectory (see [182], p. 79 [183], p. 51 [82], p. 13). 

Since each derivative operator now acts on a function of a single coordinate, we use total, rather than partial, derivative notation. 

A direct and transparent derivation of the second virial coefficient follows from the canonical ensemble. To make the notation and argument simpler, we first assume pairwise additivity of the total potential with no angular contribution. The extension to angularly-mdependent non-pairwise additive potentials is straightforward. The total potential... 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

A formal derivation of the location of the zeros of Cg t) for a general adiabatic Hamiltonian can be given, following proofs of the adiabatic principle (e.g., [250-252]). The last source, [252] derives an evolution operator U, which is written there, with some slight notational change, in the form... 

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

The matrix elements of these derivatives are to be evaluated with R equal to its equilibrium value Rq. However, to keep the notation simple, we shall still write R in place of Rq in later text unless ambiguity may occur. 

The obvious way to form a similarity between the Wigner rotation matrix and the adiabatic-to-diabatic transformation mabix defined in Eqs. (28) is to consider the (unbreakable) multidegeneracy case that is based, just like Wigner rotation matrix, on a single axis of rotation. For this sake, we consider the particular set of T matrices as defined in Eq. (51) and derive the relevant adiabatic-to-diabatic transfonnation matrices. In what follows, the degree of similarity between the two types of matrices will be presented for three special cases, namely, the two-state case which in Wigner s notation is the case, j =, the tri-state case (i.e.,7 = 1) and the tetra-state case (i.e.,7 = ). 

In what follows, the subscript M will be omitted to simplify the notations. If the initial point is P po, qa) and we are interested in deriving the value of A(= Am) at a final point Q p, q) then one integral equation to be solved is... 

We have used a common notation from mechanics in Eq. (5-4) by denoting velocity, the first time derivative of a , x, and acceleration, the second time derivative, x. In a conservative system (one having no frictional loss), potential energy is dependent only on the location and the force on a particle = —f, hence, by differentiating Eq. (5-3),... 

In developing these ideas quantitatively, we shall derive expressions for the light scattered by a volume element in the scattering medium. The symbol i is used to represent this quantity its physical significance is also shown in Fig. 10.1. [Our problem with notation in this chapter is too many i s ] Before actually deriving this, let us examine the relationship between i and 1 or, more exactly, between I /Iq and IJIq. 

To simplify the notations we do not indicate the dependence of the solutions on the parameters s, 5. Our aim is first to prove the existence of solutions to (5.185)-(5.188) and next to justify the passage to limits as c, 5 —> 0. A priori estimates uniform with respect to s, 5 are needed to study the passage to the limits, and we shall derive all the necessary inequalities while the existence theorem is proved. 

Consider the Sobolev space IFf (Dc) of functions whose derivatives up to the second order in flc are integrable with the first power. Introduce the notation... 

Shorthand notations have been developed to avoid repetitive systematic names of unsaturated fatty acids. Eor example, linolenic or (7j -9,i7j -12-,i7j -15-octadecatrienoic acid can be represented by 18 3(9, 12, 15 ). The Greek letter A has been used to indicate presence and position of double bonds, eg a fatty acid, but it should never be used in a systematic name. An equally inappropriate but popular designation is derived by counting... 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

The derivation for the general case of the horizontal rod anode in half space with the notation in Table 24-1, line 9 [7] gives ... 

Carra and Forni (1974) derived the criteria that Carberry (1976) referred to in his book. These are equivalent to the original derivation of Aris and Amundson (1958). The notation is easier to understand and closer to the notation in this book. Eliminating some typographical errors, the criteria are ... 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

These formulae explain the scission products of the two alkaloids and the conversion of evodiamine into rutaecarpine, and were accepted by Asahina. A partial synthesis of rutaecarpine was effected by Asahina, Irie and Ohta, who prepared the o-nitrobenzoyl derivative of 3-)3-amino-ethylindole-2-carboxylic acid, and reduced this to the corresponding amine (partial formula I), which on warming with phosphorus oxychloride in carbon tetrachloride solution furnished rutaecarpine. This synthesis was completed in 1928 by the same authors by the preparation of 3-)S-amino-ethylindole-2-carboxylic acid by the action of alcoholic potassium hydroxide on 2-keto-2 3 4 5-tetrahydro-3-carboline. An equally simple synthesis was effected almost simultaneously by Asahina, Manske and Robinson, who condensed methyl anthranilate with 2-keto-2 3 4 5-tetrahydro-3-carboline (for notation, see p. 492) by the use of phosphorus trichloride (see partial formulae II). Ohta has also synthesised rutaecarpine by heating a mixture of 2-keto-2 3 4 5-tetrahydrocarboline with isatoic anhydride at 195° for 20 minutes. 

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

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