Integral sign

The CCF is sensitive to variations in all the signal parameters carrier frequencies, phases, pulse forms, durations and amplitudes. Both signal amplitudes A, and J in formula (1) can factor outside the integral sign and do not define its value. Hence we let the CCF as N= MIA or in normalized form N = NIN . ... 

Since we assume that all higher derivatives are essentially zero, dp/dq = constant. Therefore, it may be taken out of the integral sign as follows ... 

Dijferentiation Here the application of the general rules for differentiating under the integral sign may be useful. 

Since this is a continuous function of (X, it may he differentiated under the integral sign... 

For physical absorption, values of the mass-transfer coefficients may not vary greatly, so a mean value could be adequate and coiild be taken outside the integral sign, but for reactive absorption the variation usually is too great. 

The expressions (9) and (10) can of course be used for both uniform and non-uniform fields. In the case of a uniform field, the X2 can be taken outside the integral sign the value of (10) will then reduce to (8) multiplied by the volume under consideration. 

If the temperatures of different parts of the working substance alter gradually during the process, the sign of summation must obviously be replaced by the integral sign, or ... 

This is true for finite changes, AZ, or for infinitesimal changes, dZ, where we replace the summation in equation (A 1.11) with an integral sign and a circle to represent an integration over a closed path. That is... 

The d orbitals are centrosymmetric and are of g symmetry. The light operator, being dipolar, is of u symmetry. The symmetry of the whole function under the integral sign in (4.7) - that is, for the product d r d- is g x m x g, namely u. The integral over all volumes of a m function vanishes identically. Since Q in (4.7) then vanishes, so does the intensity /. In short, d-d transitions are disallowed. 

The variable y in the expression under the integral sign is an auxiliary variable the value of the integral depends only on the limits of integration (i.e., on the value of u). The numerical values of the error function vary from zero for m = 0 to an upper limit of unity for m —(this value is practically attained already for u 2). Plots of functions erf(n) and erfc(n) are shown in Fig. 11.2. 

For design purposes it is convenient to write equations 11.97 and 11.98 in terms of transfer units (HTU) where the value of integral is the number of transfer units, and the group in front of the integral sign, which has units of length, is the height of a transfer unit. 

Since the reaction occurs under isothermal conditions, the rate constant may be taken outside the integral sign. Integration and simplification then gives... 

Since the reaction takes place isothermally at 500 °C, the reaction rate constant may be moved outside the integral sign. At this temperature it is equal to 0.124 sec- i. Thus... 

This equation is the basic relation for the mean residence time in a plug flow reactor with arbitrary reaction kinetics. Note that this expression differs from that for the space time (equation 8.2.9) by the inclusion of the term (1 + SAfA) and that this term appears inside the integral sign. The two quantities become identical only when 5a is zero (i.e., the fluid density is constant). The differences between the two characteristic times may be quite substantial, as we will see in Illustration 8.5. Of the two quantities, the reactor... 

In Equations 4.1 and 4.2, the numbers before the integral signs occur due to the indistinguishability of electrons and electron pairs, respectively. The single-particle density p(x) is defined as the diagonal element of the single-particle density matrix Pi(xi xi), viz.,... 

For a given permutation Py, there will be two permutations contributing from P that have rest of the orbitals in the same order, P ij with a positive sign and P - with a negative sign. Thus the expression under the integral sign in Equation 7.25 can be written as... 

Since the y values under the integral sign are the values at t and not at the time T the crustal segment formed, a correction for decay over t — T brings us back to the formation time... 

This equation holds true for any arbitrary volume, and so must be true for the volume element dV. We can therefore drop the integration sign and term dV to obtain... 

What is true for an arbitrary volume Q is true for the volume element dK, hence we can drop the integration signs and AV on both sides of the equation... 

In contrast to the design equations for batch and plug-flow reactors, eqns. (5) and (62), the design equation for the continuous stirred tank reactor does not contain an integral sign. Figure 14 shows [ A]o/r plotted... 

This is the general equation showing the time required to achieve a conversion for either isothermal or nonisothermal operation. The volume of reacting fluid and the reaction rate remain under the integral sign, for in general they both change as reaction proceeds. 

This operation requires differentiating under an integral sign. From the theorems of calculus, if... 

Here the first equality follows from the fact that f e C. The technical continuity condition on f and its first and second partial derivatives allows us to exchange the derivative and the integral sign (disguised as a complex scalar product). See, for example, [Bart, Theorem 31.7]. The third equality follows from the Hermitian symmetry of It follows that is an element... 

Differentiating under the integral sign with respect to T we find... 

The left sides of Equations 3 and 4 depending only on will be denoted as u( ). The expressions in brackets under the integral sign which are known for every and M will be called kernels and denoted as K(, M). In this way Equations 3 and 4 reduce to a single formula ... 

Along with the harmonic approximation, the Franck-Condon approximation is usually also used in calculating the probability of electron tunneling. In accordance with the latter approximation, the exchange matrix element V(q) in eqn. (20) is factorized outside the integral sign... 

---