Vector. Radius

The point z can also be located by establishing polar coordinates in the complex plane where r is the radius vector and 0 is the phase angle. Draw suitable polar coordinates for the Argand plane. What is r for the point 7 = 3 + 4i7 What is 0 in degrees and radians ... 

A functional is a function of a function. Electron probability density p is a function p(r) of a point in space located by radius vector r measured from an origin (possibly an atomic mi dens), and the energy E of an electron distribution is a function of its probability density. E /(p). Therefore E is a functional of r denoted E [pfr). ... 

Figure 10.6 Two-dimensional representation of i and i (broken lines) and their resultant ifotai (solid line) for scattering by a molecule situated at the origin and illuminated by unpolarized light along the x axis. The intensity in any direction is proportional to the length of the radius vector at that angle. (Reprinted from Ref, 2, p. 168.)...

To find the equilibrium form of a crystal, the following Wullf construction [20] can be used, which will be explained here, for simplicity, in two dimensions. Set the centre of the crystal at the origin of a polar coordinate system r,6. The radius r is assumed proportional to the surface tension 7( ), where 6 defines the angle between the coordinate system of the crystal lattice and the normal direction of a point at the surface. The anisotropy here is given through the angular dependence. A cubic crystal, for example, shows in a two-dimensional cut a clover-leaf shape for 7( ). Now draw everywhere on this graph the normals to the radius vector r = The... 

In polar coordinates (Figure 1-34), r f(0), where r is the radius vector and 0 is the polar angle, and... 

The trajectories are logarithmic spirals (Fig. 6-4). For a > 0, they wind on the singular point (i.e., the rotation of the radius vector is clockwise) for a < 0, they unwind (i.e., the rotation of the radius vector is counterclockwise). 

The second equation merely shows that the radius vector rotates with a constant angular velocity. As to (6-14) it is integrated by the standard procedure which gives ... 

Finally, in the first approximation the amplitude (i.e., the radius vector) p0 of the singular point gives the radius of the periodic solution (which is a circle in the first approximation). 

Here r is the radius vector from the origin to a point R in the crystal, t is the electron-pair-bond function in the region near R, Pfc is the momentum vector corresponding to the three quantum numbers k (the density of states being calculated in the usual way), h is Planck s constant, and G is the normalizing factor. 

Here dli is the displacement along the radius vector, drawn from the point q, but dl2 is the element of the arc of the radius Lgp. Then we have ... 

That is, functions U q) and dU q)ldn on the surface S approach their values at the observation point, respectively. Also from Fig. 1.9 it follows that the normal n and radius vector r on this surface are opposite to each other, and therefore for points on this surface we have... 

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with infinitely large radius. In deriving this equation we assume that the potential U p) is a harmonic function, and the Green s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an infinity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to find a Green s function in the volume V that is equal to zero at each point of the boundary surface ... 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have... 

In order to study an arbitrary motion of a particle on the earth s surface we consider a more general case, when corresponding coordinate planes in both frames are not parallel to each other and they have different origins. Fig. 2.3c. The moving frame P rotates about the z-axis of the frame P and the distance between origins remains the same. We introduce three radius vectors r, ro, and ri, characterizing a position of a point p in both frames, as well as the origin Oi. It is obvious that... 

We have derived Equation (2.164), which shows how the field varies with the reduced latitude p on the surface of the spheroid. The reduced latitude is the angle between the radius vector and the equatorial plane. Fig. 2.7c. Also, it is useful to study the function y — y q>), where tp is the geographical latitude. This angle is formed by the normal to the ellipsoid at the given point p and the equatorial plane. Fig. 2.7b. First, we find expressions for coordinates v, y of the meridian ellipse. Its equation is... 

By definition, the angle between the z-axis and the radius vector R is equal to 6. At the same time, the gravitational field can be represented as a sum of the attraction force and centrifugal one, and it is almost opposite to the radius vector. Therefore, the radial component of the field is... 

Before we make use of Equation (2.265), let us transform the boundary condition (2.264) in the following way. With an accuracy of small quantities, which have the same order as the square of the geoid heights, a differentiation along the normal can be replaced by differentiation along the radius vector, and correspondingly the condition (2.264) becomes... 

By definition, the center of mass of the pendulum is characterized by the radius vector... 

Legendre s functions of the first and second kind with real argument Legendre s functions with imaginary argument radius of sphere radius vectors... 

Since the linear velocity vector v, is perpendicular to the radius vector r the magnitude Z, of the angular momentum is... 

The differential capacity is given by the slope of the tangent to the curve of the dependence of the electrode charge on the potential, while the integral capacity at a certain point on this dependence is given by the slope of the radius vector of this point drawn from the point Ep = pzc. 

As another example illustrating an explicit switch to normal coordinates, we consider a three-dimensional monoatomic simple lattice. In such a system, masses of all particles are the same and the positions of their stable equilibria are at the lattice sites which are given by radius vectors n (called lattice vectors). Instead of an unsystematic particle numbering (i = 1,2,..., N), it is now convenient to distinguish them by the lattice sites they belong to and to designate them by the index n. The... 

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