Metric coefficient

In order to solve Laplace s equation we have to find also expressions for the metric coefficients. By definition, the elementary displacements along coordinate lines are... 

Recall that hi is a function of x and y, and that A i = /j2 = 1. Since hi is the only nonunity metric coefficient, and to simplify the nomenclature somewhat, we drop the subscript on hi,... 

For a fixed value of x, this is a third-order, nonlinear, ordinary differential equation. Recall that the metric coefficient h x,y) depends on both coordinates, but that the cone angle 9 is a fixed constant. 

The axial Poiseuille flow occurs when 6 = n/2, with x taking the role of z, and y taking the role of De/2 — r. The metric coefficient reduces to h = De/2 — y = r. The expected axisymmetric momentum equations for axial Poiseuille flow can be recovered by substituting f — ur and carrying out the independent variable transformation to exchange r for v. The chain rule for the independent variable transformation provides that... 

Verify that the proposed coordinate system is indeed orthogonal by showing that the off-diagonal metric coefficients vanish. 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

Electromagnetism can therefore be defined geometrically in curvilinear coordinates, and has vacuum properties such as scalar curvature, metric coefficient, affine connection, and Ricci tensor that manifest themselves fully on the 0(3) level ... 

If descriptor combinations are expressed as bit strings (often called fingerprints, as described in more detail later on), each test molecule is assigned a characteristic bit pattern, and pair-wise molecular similarity can be assessed by quantifying the overlap of bit strings using various similarity metrics (coefficients). Examples are shown in Table 1.4. 

C. Moeder, T. O Brien, R. Thompson, and G. Bicker, Determination of stochio-metric coefficient and apparent formation constant for a- and P-CD complexes of teipenes using reversed phase liquid chromatography, J. Chromatogr. 736 (1996), 1. 

Since the metric coefficients in Boyer-Lindquist coordinates are independent of t and (ft, both = c)t and if = are the Killing vectors of the metric, the timelike and axial, respectively, with Boyer-Lindquist components (1,0,0,0) and (0,0,0,1), respectively. 

The basic quantities describing an orthogonal coordinate system are the metric coefficients hi = i/gn, h2 = i/g, hs = 333, which satisfy the formula ... 

For the cylindrical coordinates the metric coefficients can be determined from (C.75) ... 

Lame coefficients, metric coefficients, or scale factors effective swept volume rate m /s)... 

The product terms such as dty duj (i A j) do not appear because of the orthogonality property of the chosen coordinate system. These metric coefficients can also be generated by means of the following formula [67] ... 

Fourier number based on square root of area critical value of Fourier number radiative parameter for point contact elastoplastic contact parameter gap conductance correlation equation metric coefficients, jacobian height of single and double cones Rockwell C hardness number Brinell hardness... 

Now using results given at the beginning of this section we will replace functions Um. oi) and Vm ct) by functions mKi mr) and m/i(mr), respectively. Taking into account that in the cylindrical system of coordinates the metric coefficient = r, we obtain expressions for reflection and screening coefficients, describing the field in the presence of one shell ... 

Due to channel curvature and changes in width, metric coefficients and rriy are introduced to correct for differences between the distances measured along any given surface and those measured along the corresponding axes. These are illustrated in Fig. 1. The values of nix and niy may vary from point to point, being functions of both x and y. Note that = 1 on the x axis and = 1 on the y axis. Due to the parallel nature of all horizontal coordinate surfaces, the value of niz is 1 everywhere. As Yotsukura 105) notes, this natural coordinate system is therefore based on the premise that the total velocity vector everywhere... 

Use of the natural coordinate system has introduced the metric coefficients nij. and rriy. Some uncertainty exists as to their full significance. However, they probably are not a big problem using the approach. Sayre and Yeh (83) report that they found values of rrij. ranging from 0.86 to 1.14 in an extremely sharp bend in the Missouri River, indicating that the value will likely be near unity if the X axis is placed in the central part of the channel. 

In an orthogonal coordinate system, let a/) denote the unit vectors in each of the three coordinate directions, and let denote distance along each of these axes. The coordinate system may he designated by the metric coefficients guygiz, gss, defined by... 

The new equations are equivalent, as expected. The distortion by the coordinate transformation (45) in E is compensated by the action of the metric coefficients. 

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