Plane tangent

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

The "split" reflexions of the type ho.o (and hh.o) can be associated with the graphene sheets in the tangent planes perpendicular to the beam direction along... 

A line which intersects a circle at only one point is a tangent to the circle at that point. Every tangent is perpendicular to the radius drawn to the point of intersection. Spheres may have tangent lines or tangent planes. 

Now the equation of the tangent plane at the point 1 can, as we know from solid geometry, be obtained by writing the subscript 1 after each quantity in equation (1) and then putting ... 

From (9)—(12) we deduce at once that the tangent planes at the points 1 and 2 are coincident. Hence the theorem ... 

If two different states can exist permanently in contact, the points representing these states on the thermodynamic model have a common tangent plane. 

As the tangent plane rolls on the primitive surface, it may happen that the two branches of the connodal curve traced out by its motion ultimately coincide. The point of ultimate coincidence is called a plait point, and the corresponding homogeneous state, the critical state. 

The investigation above is due initially to Gibbs (Scient. Papers, I., 43—46 100—134), although in many parts we have followed the exposition of P. Saurel Joum. Phys. diem., 1902, 6, 474—491). It is chiefly noteworthy on account of the ease with which it permits of the deduction, from purely thermodynamic considerations, of all the principal properties of the critical point, many of which were rediscovered by van der Waals on the basis of molecular hypotheses. A different treatment is given by Duhem (Traite de Mecanique chimique, II., 129—191), who makes use of the thermodynamic potential. Although this has been introduced in equation (11) a the condition for equilibrium, we could have deduced the second part of that equation directly from the properties of the tangent plane, as was done by Gibbs (cf. 53). 

The tangent plane approximation is valid the curvature of the Ewald sphere is negligible at small scattering angles. 

WAXS and MAXS. Fiber symmetry means that, even in WAXS and MAXS, the scattering pattern is completely described by a slice in reciprocal space that contains the fiber axis. Nevertheless, for 20 > 9° the tangent plane approximation is no longer valid and the detector plane is mapped on a spherical surface in reciprocal space. 

If, in the detector plane, the effective slit is wider than the region of the pattern in which significant intensity is observed, the approximation of an infinite slit is valid. Let the slit be infinitively long in ft -direction but very narrow in 53-direction then in the tangent plane approximation the recorded scattering curve... 

For USAXS and SAXS data the tangent-plane approximation is valid and the relation between scattering angle and the units of reciprocal space are given by Eq. (2.7). If the scattering pattern is properly aligned with the vertical direction identical to a fiber axis or the polymer chain direction, then sy = 53. In similar manner the. vx-axis of the detector is related to the actual orientation of the sample with respect to the beam. 

Fig. 4.2. Newton-Raphson iteration for solving two nonlinear equations containing the unknown variables x and y. Planes are drawn tangent to the residual functions R and R2 at an initial estimate (r, > (o)) to the value of the root. The improved guess (v(l y(l)) is the point at which the tangent planes intersect each other and the plane R = 0.

To improve an initial guess (x ° y ), we reach above this point and project tangent planes from the surfaces of Ri and R2. The improved guess is the point... 

The corrections Ax and Ay to x and y are those that will project (o R = 0 along the tangent planes, according to,... 

Figure 5.48 Geometry discussed in Section 6.3 for tubule formation based on chiral elastic properties. Here, r is tubule radius, l is tubule length, n is molecular director, m is projection of n into local tangent plane (normalized to unit magnitude), <(> is angle in tangent plane between m and curvature direction (equator running around cylinder), and N is local normal vector. Adapted from Ref. 132 with permission of the author. Copyright 1996 by the American Physical Society.

The second-order necessary conditions require this matrix to be positive-semidefinite on the tangent plane to the active constraints at (0,0), as defined in expression (8.32b). Here, this tangent plane is the set... 

A more realistic statement concerning the orientation of a molecule at the surface is that the transition moment establishes an angle to the normal 9 but is random with respect to its projected orientation in the tangent plane. Figure 8.8 shows the associated coordinate system. The absorption by such a... 

The term spinode was proposed by van der Waals, who formulated it in analogy with the shape of a thorn ( spine ) given by the intersection of the tangent plane to the fr = f P) function for gaseous phases at the critical point (see figure 1 in van der Waals, 1890 see also Cahn, 1968, for an extended discussion of the etymology of the term). 

The covariance of the partition coefficients can be estimated by the correlation between the tangent planes of the response surfaces in a given mixture composition. This is explained in the next part of this paragraph. 

The correlation of the slopes of the tangent planes of the response surfaces of P, in P2 in the investigated mixture composition M is represented by r. 

In other words, the robustness of this ratio is a function of the robustness of the individual partition coefficients P, and Pj and of the parallelism of the tangent planes of the response surfaces in mixture composition M. 

If the tangent planes in M are more or less parallel then P, and P, are approximately equally affected by a variation in the mixture composition and the correlation of the response surfaces in M is high and the variance in the selectivity is small. If the tangent planes in M have opposite slopes then P, and Pj are affected in an opposite way by a variation in the mixture composition and the correlation of the response surfaces in Mwill be low and the variance in the selectivity high. The calculation of the correlation r (i.e. the parallelism of the response surfaces) is outlined below. 

The tangent plane in M for a response surface fitted with In P can also be described by using the adjusted mixture model (i.e. the restrictions of the mixture models are under consideration 0< x< 1 x,+ X2+ 3= 1) (equation (3) becomes equation (21a)) ... 

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