Vector derivative identities

In the following, S is assumed to be a continuous, differentiable scalar. The vectors V, A, 

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By subtracting the mechanical-energy contributions from the total energy equation, a thermal energy equation can be derived. It is this equation that proves to be most useful in the solution of chemically reacting flow problems. By a vector-tensor identity for symmetric tensors, the work-rate term in the previous sections can be expanded as... 

Anyone who has studied abstract algebra may recognize a way to think about (A8.5) that has attractive mathematical elegance. Think of Ai and as the Cartesian components of a vector Rj in a two-dimensional space . Then think of (A8.5) as embodying an operation T that transforms the vector Rj into the vector Rj +1. Finally think of the cyclic property of the system under study as requiring that six successive performances of the operation carry the vector to identity with the initial vector in other words, T6 = /, where / is the identity operation. Use these ideas to derive the results obtained by the use of finite-difference equations in the appendix. 

In CFA we can derive biplots for each of the three types of transformed contingency tables which we have discussed in Section 32.3 (i.e., by means of row-, column- and double-closure). These three transformations produce, respectively, the deviations (from expected values) of the row-closed profiles F, of the column-closed profiles G and of the double-closed data Z. It should be reminded that each of these transformations is associated with a different metric as defined by W and W. Because of this, the generalized singular vectors A and B will be different also. The usual latent vectors U, V and the matrix of singular values A, however, are identical in all three cases, as will be shown below. Note that the usual singular vectors U and V are extracted from the matrix. ... 

Using the definitions above many vector identities can be derived and these are listed in many sources e.g. [10]. Some useful identities are... 

Vectors whose components have continuous values correspond to the more traditional types of vectors found in the physical sciences. They are of identical form to the discrete-valued vectors (see Eq. 2.16) except that the components, vA(xk), are continuous valued. In chemoinformatics, however, the nature of the components is considerably different from those typically found in physics. For example, physiochemical properties, such as logP, solubility, melting point, molecular volume, Hammett ap parameters, and surface charge, as well as other descriptors derived explicitly for the purpose, such as BCUTs... 

Ty [see (11.27)—(11.30)]. As described previously, scalar products for S) and V) (i.e., involving properties CP, fiT, aP) are obtained by matrix inversion from those for T) and — P) (i.e., involving Cv, /3S, 1 v). The vector-algebraic procedure to be described will automatically express any desired derivative in terms of the six properties in Table 12.2, and these expressions may subsequently be reduced (if desired) to involve only three independent properties by identities previously introduced [cf. (11.39)-(11.42)], consistent with the /(/ + l)/2 rule. ... 

This/2x / matrix (matrix of vectors), or the conjugate m matrix, contains the higher-order response functions needed to fully incorporate higher-order (U ") derivatives into the thermodynamic geometry. Geometrical identities for higher-order response functions can then be obtained in analogy to Sections 12.1-12.5. 

Recognition of the diffusional metric geometry Mv allows one to apply powerful vector-geometrical techniques to the derivation of complex identities between measured transport properties, analogous to those discussed in Chapter 12. However, further discussion of the rich science and applications of near-equilibrium transport phenomena is beyond the scope of the present treatment [see, e.g., R. B. Bird, W. E. Stewart, and E. N. Lightfoot. Transport Phenomena, revised 2nd edn (Wiley, New York, 2007)]. 

Figure 6.8 a shows the phase determination using a second heavy-atom derivative F h is the structure factor for the second heavy atom. The radius of the smaller circle is IF Hpl, the amplitude of F Hp for the second heavy-atom derivative. For this derivative, Fp = F Hp — F H. Construction as before shows that the phase angles of F and Fj are possible phases for this reflection. In Fig. 6.8 b, the circles from Figs. 6.7b and 6.8a are superimposed, showing that Fp is identical to F. This common solution to the two vector equations is Fp, the desired structure factor. The phase of this reflection is therefore the angle labeled a in the figure, the only phase compatible with data from both derivatives. 

The diabatization within the time-dependent framework produced the expected potential matrix W presented in equation (56) but enforced the four vector curl equation which is given in equation (54). This set of equations contains not only derivatives with respect to the spatial coordinates but also with respect to time. In fact this non-Abelian curl equation is completely identical to YM curl equation which has its origin in field theory. 

These quantum mechanical results are identical to the ones derived before based on the vector model (equs. (9.12)), which were used to interpret Fig. 9.2. 

Equation 3.3 uses the fact that velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. Equation 3.3 is actually a vector equation. If the vectors are expressed in Cartesian coordinates (Section 1.3) it is identical to the three equations... 

This derivation uses an algebraic identity valid for Pauli matrices and any two vectors A, B,... 

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