Origin of coordinate system

Figure 13. Cartesian [center-of-mass (CM)] contour diagrams for NH+ produced from reaction of N+ with H2. Numbers indicate relative product intensity corresponding to each contour. Direction of N+ reactant beam is 0° in center-of-mass system. For clarity, beam profiles have been displaced from their true positions (located by dots and 0°). Tip of velocity vector of center of mass with respect to laboratory system is located at origin of coordinate system (+). Scale for production velocities in center-of-mass system is shown at bottom left of each diagram (a) reactant N+ ions formed by impact of 160-eV electrons on N2 two components can be discerned, one approximately symmetric about the center of mass and the other ascribed to N+(IZ3), forward scattered with its maximum intensity near spectator stripping velocity (b) ground-state N+(3/>) reactant ions formed in a microwave discharge in N2. Only one feature is apparent—contours are nearly symmetric about center-of-mass velocity.12 ...

The electric moments are not invariant under a shift of the origin of the coordinate system. The first non-vanishing moment is however usually independent of the choice of the coordinates. The origin of coordinate system is thus often chosen as the center of gravity or the center of the charge distribution. 

Near the origin of coordinates system the function A tends to the vector potential of the magnetic dipole in a uniform medium, that is ... 

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

The polar coordinate system describes the location of a point (denoted as [r,0]) in a plane by specifying a distance r and an angle 0 from the origin of the system. There are several relationships between polar and rectangular coordinates, diagrammed in Figure 1-30. From the Pythagorean Theorem... 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

We place an ion a at the origin of our system of coordinates and we look for the probability nga(r) of finding an ion of species p at a distance r. Clearly, this is a very complicated problem because ngtt(r) will be influenced by the presence of the other ions in the system. We shall, however, make the simplifying assumption that this probability may be written as ... 

Fig. 5.3. Relations between the fluorescence intensity components resulting from the Curie symmetry principle. The fluorescent sample is placed at the origin of the system of coordinates.

All PCA loading vectors are orthogonal to each other PCA is a rotation of the original orthogonal coordinate system resulting in a smaller number of axes. 

An often-overlooked issue is the inherent non-orthogonality of coordinate systems used to portray data points. Almost universally a Euclidean coordinate system is used. This assumes that the original variables are orthogonal, that is, are uncorrelated, when it is well known that this is generally not the case. Typically, principal component analysis (PCA) is performed to generate a putative orthogonal coordinate system each of whose axes correspond to directions of maximum variance in the transformed space. This, however, is not quite cor-... 

To introduce the notation and concepts to be used below, let us first briefly recall some elementary aspects of the Euclidean geometry of a triangle of points V, V2, V3 in ordinary three-dimensional physical space. Each point Vi can be represented by a column vector vt (denoted with a single underbar) whose entries are the coordinates in a chosen Cartesian axis system at the origin of coordinates ... 

If the interaction between two objects does not depend explicitly on the lime coordinate, then the actions lhal take place do nol depend on when one starts to measure lime i.e.. the properties of the system are invariant with respect to a translation of the origin of coordinates along the lime axis. This symmetry is associated with conservation of energy. Use of a 4-dimen.sional coordinate system allows one to associate conservation of momentum and energy in a unified manner with the geometrical symmetry of space-time. 

The column vector x represents the original (undrawn) coordinate system to which the ellipse is referred, xr is the transpose of x, and A represents the ellipse. The components of A are easily shown to be given by ... 

It is not difficult to observe that, by using this system of dimensionless coordinates for each factor, the upper level corresponds to -i-l, the lower level is -1 and the fundamental level of each factor is 0. Consequently, the values of the coordinates of the experimental plan centre will be zero. Indeed, the centre of the experiments and the origin of the system of coordinates have the same position. In our current example, we can consider that the membrane remains unchanged during the experiments, i.e. the membrane porosity (e) and the zeolite concentration (Cj.) are not included in the process factors. 

In a diatomic molecule the electronic centre of gravity will be found on the line of centres which may be taken as the x axis. If the origin of the system of coordinates is located at the centre of gravity of the... 

In order to measure the commonness between the two configurations, we have applied the canonical correlation analysis (CCA) [1], The CCA applies linear transformations to the original two sets of coordinate systems to produce new coordinate systems such that the correlation coefficients between y th coordinates in both sets is the largest. CCA applied to the nMDS-generated 2D coordinate systems produced the first coordinates with the correlation coefficient 0.79 and the second with 0.32. These values may be understood as quantitative measures of the commonness between the two different types of... 

During the past few years it has been reported that the reversible electron electrode can be realized in solutions of solvated electrons in hexamethylphospo-triamide against a background of lithium 3-i65) sodium 21,165,166) the system is reversible in these solutions is evidenced by the fact that the polarization curve in linear coordinates passes trough the origin of coordinates without any kink When the potential is more positive than the equilibrium potential,... 

The cell that contains the origin of coordinates for the whole system is used as the reference cell. The two-center matrix element that involves only the ground state primitive basis functions < >0 is... 

Analysis of this type of expression shows that LC is just another one-electron function with w = 0 or 1, directed at a special angle, that depends on the coefficients a, b and c. An infinite number of such linear combinations is possible, each defining another one-electron eigenfunction directed at one of an infinite number of angles, measured with respect to the original laboratory coordinate system. The important conclusion is that each linear combination corresponds to a new choice of axes. Selection of the polar axis along any Z always leaves Px and Py undefined as separate entities. In particular, there is no hope ever to simulate the tetrahedral structure of methane in terms of a linear combination of carbon electron eigenfunctions. That requires four linear combinations, each with a different polar axis, which is physically impossible. 

Let the displacement of each nucleus be expressed in terms of rectangular coordinate systems with the origin of each system at the equilibrium position of each nucleus. Then the kinetic energy of an A-atom molecule would be expressed... 

Let us choose the cylindrical system of coordinates (r, 0, z) and the vertical magnetic dipole is placed at the origin of this system (Fig. 4.1). The moment of the magnetic dipole is oriented along the z-axis. We will look for a solution using only the z-component of the vector-potential, A. As follows from Maxwell s equations the vector potential must satisfy several conditions ... 

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