Differentiation With Respect To Time

This equation can be put in a more useful form by differentiating with respect to time and rearranging... 

Differentiating with respect to time, the stress rate relation becomes... 

To obtain the heat flow Qj, equation (2.48) is differentiated with respect to time... 

To check that the equation for s really does obey the law of conservation of energy, we differentiate with respect to time as follows ... 

Here i, j, k and ii, ji, ki are unit vectors in the inertial and rotating frames of reference, respectively. Performing a differentiation with respect to time we obtain for the velocity of the point p... 

Performing a differentiation with respect to time twice, we obtain the acceleration in the inertial system... 

In Eqs. (5.1) and (5.2), m is the reduced mass of the colliding system, V is the interaction potential at ion-molecule separation r, 6 is the angle between the direction of r and the center-of-mass velocity, and the dot indicates differentiation with respect to time. Integration of (5.1) just gives the angular momentum L, which is conserved in the collision. Substitution in (5.2) gives... 

Now in order to apply the Boltzmann Superposition Principle (Equation 4.60) we need to express this as a strain rate. Differentiating with respect to time gives us... 

This equation can be differentiated with respect to time to obtain the equations... 

The three-step model was developed as a consequence of the extreme complexity of a PBC system. This author had a wish to describe the PBC-process as simple as possible and to define the main objectives of a PBC system. The main objectives of a PBC system are indicated by the efficiencies of each unit operation, that is, the conversion efficiency, the combustion efficiency, and the boiler efficiency. The advantage of the three-step model, as with any steady-state system theory, is that it presents a clear overview of the major objectives and relationships between main process flows of a PBC system. The disadvantage of a system theory is the low resolution, that is, the physical quantity of interest cannot be differentiated with respect to time and space. A partial differential theory of each subsystem is required to obtain higher resolution. However, a steady-state approach is often good enough. 

Note that the right-hand side of eqn. (A.17) is the same as that of eqn. (A.15) since the differential with respect to time of Caqui zero. The variables in eqn. (A.17) may now be Laplace transformed. 

Note that A(z )does not change with time. Furthermore, differentiation with respect to time and integration over space can be interchanged.)... 

Differentiation with respect to time t and using Eq. 24-63 yields ... 

Symbols printed in boldface represent vectors the hat above a vector signifies a dimensionless unit vector, for example R = R / J r = r(t) describes the position of the charge q differentiation with respect to time is indicated by dots so that r signifies acceleration and t = t — R/c is the retarded time. Accordingly, the power radiated due to accelerated charge per unit solid angle, Q, is given by... 

When eq. (3.34) is, in analogy to eq. (3.13), differentiated with respect to time, the distribution function is differentiated in the integrand. In this way, the equation of continuity5 as derived in Zimm s paper, takes over the role which eq. (3.10) plays in the derivation of eq. (3.14). The final result is, however, the same, viz. eq. (3.15). 

The dot notation delned in eqns 2.1 and 2.2 is used to represent differentiation with respect to time. 

For a particle moving without a force, both potential and kinetic energies are constant. Generally, they will vary during the motion, but in such a way that the total energy given by eqn 2.8 remains constant. To show this, we differentiate with respect to time ... 

Another very important result concerning non-circular orbits is that the angular momentum remains constant during the motion. Qualitatively, this comes from the fact that the velocity is larger when the distance to the centre is smaller. We can show it mathematically by starting with the definition in eqn 3.40 and differentiating with respect to time, and then using Newton s Second Law ... 

For elastic bodies, the shear stress is related to the shear strain by the shear modulus. For viscous fluids, the shear stress is related to the shear strain rate by the viscosity. We note that for laminar viscous flow in a Margules viscometer (Figure 10.7), radial fluid displacement is zero (gr = 0). Thus, differentiating with respect to time ... 

Exponential Profiles These have the form c(f) = ycxp( fit). Differentiating with respect to time, one obtains... 

Gamma Profiles These profiles follow the form c (f) = yt-0 exp (—fit), which is reported in the literature as the gamma-function model [244], This model was used to fit pharmacokinetic data empirically [245,246]. Differentiating with respect to time, we obtain... 

Just as space can be divided into unequally spaced intervals, so might time also be unevenly divided. As with spatial intervals, there is the choice between discretising on an uneven time grid or using a transformation to a new time scale. Since, except for BDF methods, one usually differentiates with respect to time using only two time points (levels), transformation does not make sense here. 

This expression may be easily deduced from the corresponding relation for the desorption experiment (eq 7) by differentiation with respect to time. 

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