Coordinates cartesian coordinate system

Equation (18) is valid when the polarizability of the dielectric is proportional to the electrostatic field strength [4]. The operator V in the Cartesian coordinate system has the form V = dldx,dldy,dldz). 

Figure 2-90. Cartesian coordinate system and Cartesian coordinates of methane.

Some of the common manipulations that are performed with vectors include the scalar product, vector product and scalar triple product, which we will illustrate using vectors ri, T2 and r3 that are defined in a rectangular Cartesian coordinate system ... 

Mathematical derivations presented in the following sections are, occasionally, given in the context of one- or two-dimensional Cartesian coordinate systems. These derivations can, however, be readily generalized and the adopted style is to make the explanations as simple as possible. 

Therefore the area coordinates defined by Equation (2.20) in a global Cartesian coordinate system are expressed as... 

In a fixed two-dimensional Cartesian coordinate system, the continuity equation for a free boundary is expressed as... 

Governing equations in two-dimensional Cartesian coordinate systems... 

MODELLING OF STEADY-STATE VISCOMETRIC FLOW -WORKING EQUATIONS OF THE CONTINUOUS PENALTY SCHEME IN CARTESIAN COORDINATE SYSTEMS... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

In a Cartesian coordinate system the differential of arc length of a line is defined... 

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

Cartesian coordinates system for locating points in space based on three coordinates, which are usually given the symbols x, y, z or i, j, k CBS (complete basis set) an ah initio method CC (coupled cluster) a correlated ah initio method CFF (consistent force field) a class of molecular mechanics force fields CFMM (continuous fast multipole method) a method for fast DFT calculations on large molecules... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

We now adopt a Cartesian coordinate system and write Eq. (4-54) for each of the components of the magnetization. Figure 4-5 shows the coordinate system. The... 

Figure 2-8. Equations of motion in a Cartesian coordinate system.

While the foregoing discussion of stress and strain is based on a Cartesian coordinate system, any orthogonal coordinate system may be used. 

KEY a. Infrared photobems are arranged in a Cartesian coordinate system on 7.6-cm centers... 

If a vector it is a function of a single scalar quantity s, the curve traced as a function of s by its terminus, with respect to a fixed origin, can be represented as shown in Fig. 8. Within the interval As the vector AR = R2 - Ri is in the direction of the secant to the curve, which approaches the tangent in the limit as As - 0. Tins argument corresponds to that presented in Section 2.3 and illustrated in Fig. 4 of that section. In terms of unit vectors in a Cartesian coordinate system... 

It is in general possible to find a Cartesian coordinate system attached to the molecule such that the tensor / takes a diagonal form. In terms of these so-called principal axes Equation (10) is simplified in that all cross terms are eliminated. Namely,... 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

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