Arbitrary Orthogonal Coordinates

The curvilinear coordinates x1, x2, x3 are defined as functions of the rectangular Cartesian coordinates x,y, z  

In this case the third invariant of the metric tensor is given by 

In what follows, we present the basic differential operators in the orthogonal curvilinear coordinates x1, x2, x3. The corresponding unit vectors are denoted by ii, i2, and i3. The gradient of a scalar P is 

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The convective diffusion equation in arbitrary orthogonal coordinates has the form... 

Differentiation of bi-orthogonality conditions (2.186)-(2.189) with respect to an arbitrary soft coordinates cf yields the relations... 

FIGURE 10.1 Schematic potential energy surface V(R), showing change in potential energy along a notional reaction coordinate and in an arbitrary orthogonal direction. 

An exact determination of interaction energy for spherical and anisotropic particle systems and arbitrary electrolyte composition seems prohibitive due to the nonlinearity of the governing PB equation and the lack of appropriate orthogonal coordinate systems, except for the case of the two-sphere configuration. However, particle and protein adsorption occurs in rather concentrated electrolytes when the electrostatic interactions become short ranged in comparison with particle dimensions. This enables one to apply, for calculating particle interactions, the approximate... 

In addition to cell coordinates and directions, crystal planes are very important for the determination and analysis of structure. We begin with the cell s coordinate system, with axes x, y, and z. Recall that the axes are not necessarily orthogonal and that a, b, and c are the lattice parameters. Look at Figure 1.24. The equation of an arbitrary plane with intercepts A, B, and C, relative to the lattice parameters is given by... 

However, designating coordinate axes as x, y, z and so on, is arbitrary therefore, the various averages must be invariant with respect to relabeling axes. Thus, it follows that (af-) = j for all i and j. From the orthogonality of the transformation (5.41) we have... 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

Since one can attain the diagonal form Ga from an appropriate real orthogonal transformation R G R , one can then take the square roots of its principal values to attain a diagonal matrix ga (with signs arbitrary ), and return the latter to the original coordinate system using the backward transformation R-1 gd R to yield a matrix g. 

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

The composition vector in the (n — l)-dimensional space defined by the straight line reaction paths The composition vector g at time t The composition vector g at time t = 0 An arbitrary vector The jth element of the vector y A vector in the orthogonal system of coordinate The transpose of the vector y A small interval of time An interval of time... 

In a set of j projections with different projection vectors pi/, an Y-dimensional cross peak Q is projected orthogonally to the locations Qf Here,/is an arbitrary numeration of the set of j projections / = 1,. .., / In the 2D coordinate system of projection/ the projected cross peak has the position vector Ql = [v... 

Consider a point in a liquid x (x, y, z) at some moment in time, and in the vicinity of this point three mutually orthogonal area elements perpendicular to the vectors i, k. Through t(i), t(j), and t(k), we designate the stresses at these elements. Each of these stresses may be dissected with respect to the coordinate axes (Fig. 4.3). A projection of the stress t on the area element of arbitrary orientation with normal n can then be written as ... 

Changing the coordinate system thus changes a matrix by pre- and post-multiplication of a unitary matrix and its inverse, a procedure called a similarity transformation. Since the U matrix describes a rotation of the coordinate system in an arbitrary direction, one person s U may be another person s U . There is thus no significance whether the transformation is written as U AU or UAU and for an orthogonal transformation matrix (U = U ), the transformation may also be written as IPAU or UAU . 

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