Total curvature

Dogleg Total curvature in the wellbore consisting of a change of inclination and/or direction between two points. 

Total curvature Implies three-dimensional curvature. 

From analytical geometry follows that the total curvature of the plumb line is... 

The total curvature energy of a spherical vesicle is given by 4tt(2/cc + k). As all experimental data on phospholipids indicate that kc is not small, one is inclined to conclude that the vesicles are thermodynamically unstable the reduction of the number of vesicles, e.g. by vesicle fusion or by Ostwald ripening, will reduce the overall curvature energy. However, such lines of thought overlook the possibility that k is sufficiently negative to allow the overall curvature free energy of vesicles to remain small. 

We can interpret the quantity 2k — b(k — 2) as the curvature of the faces of gonality b Euler formula is the condition that the total curvature is a constant, equal to 4k, for fc-valent plane graphs. This curvature has an interpretation and applications in Computational Group Theory, see [Par06] and [LySc77, Chapter 9]. 

The r-gons are now regular r-gons and the angle at its vertices is LyL - Consider a point A where q r-gons meet The curvature of A is the difference between 2m and the sum of the angles of the r-gons, i.e. 2jt — The total curvature of the... 

In the master formula for the analysis of the pseudo-Jahn-Teller effect, the total curvature of the adiabatic potential surface, K, is partitioned in a so-called non vibronic part K0 and the vibronic one Kv, namely K = K0 + Kv, with... 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

Table 2b. The total curvature and its non-vibronic (K0) and vibronic parts (Kv)...

Figure 7.14 A regular Mobius strip with its single boundary curve. The absolute local curvature of the double cover is constant and the total curvature is zero.

The second solution is to reduce the size of the experimental domain. Usually the smaller the domain, the less the total curvature, and therefore there exists a simpler satisfactory model within the reduced domain. Third- or higher-order designs are rarely used (except for mixture models) and for the quadratic model to be sufficient the extent of the experimental region must normally be restricted. 

Exploring the molecular mechanisms of flexoelectricity is a central task of the liquid crystal approach in membrane biophysics. 43,15-21 pjjg flexoelectric coefficient can be represented as an integral over the curvature derivative of the distribution of the normal component of polarization P z, c+) across the membrane (c+ = ci - - C2 is the total curvature). Both direct and converse flexocoefficients can be expressed in this manner and these can be shown to be equal ... 

The shape of essentially flat membranes stretched across the periodic boundary conditions of a simulation box can be described by specifying their vertical displacement h(r) above some horizontal reference plane, say of size L x L. In this so-called Monge parametrization, the bending contribution due to the total curvature term (ignoring for now on the spontaneous curvature Kq) is given by ... 

When at rest, the surface of the fluid is vertical and corresponds to the (y, z) plane. Consider now the case when the line is displaced by a small distance u y) = Ug cos qy along the a -axis. The surface of the fluid is now distorted, with a local displacement C(y, z). At ground level, we have ( (y,0) = u y). Since we neglect gravity, the pressure inside the fluid is the atmospheric pressure. In accordance with Laplace s law, this implies a net total curvature of zero for the interface, which we write as... 

In terms of the two radii of curvature the total curvature, /, and Gaussian curvature, K, are defined by... 

We have now derived the Laplace equation for the case of a spherical liquid droplet. In a more general form, the Laplace equation relates the total curvature at a point of the surface to the difference in pressure on both sides of the surface at that point. 

The surface Laplacian of the total curvature expressed in terms of R(s) and il/(s) reads... 

Finally, suppose that the patterned composite film structure on the left in Figure 3.19 is capped with a uniform thin passivation layer, as shown on the right in the figure. If both surface layers are thin compared to the substrate, then their contributions to substrate curvature are additive, as was demonstrated in Section 2.4.7. In such a case, the total curvature is the sum of the contribution from the patterned film as discussed in this section and that due to the uniform film as discussed in Chapter 2. 

Even though the mechanisms that cause the rotation in these systems are different, there is a physical model that describes the formation of helical structures with perversions in both cases despite the difference in order of scale and it is known as the Calugareanu theorem [106]. This model takes into account the fact that in a filament with intrinsic curvature the total curvature must remain constant. As the tension is removed from the filament, and the helix starts to form, the formation of two helices... 

The circular approximation fails at vapor pressures close to saturation. For P/Po O, the total curvature and thus the Laplace pressure goes to zero. In addition, the circumference of the meniscus might become large. As a result, the contribution of the capillary pressure term in the total force might become insignificant and the direct action of the surface tension dominates. For two equal spheres in contact, this leads to an adhesion [584] of... 

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