Cartesian coordinate system defined

Equation (4.35) can now expanded in the Cartesian coordinate system defined in figure 4.1. We replace RpRi cos0ip by J]XPIXII where the X, represent X, Y, Z. Hence equation (4.35) becomes j... 

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 ... 

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

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... 

Tuckennan et al. [38] showed how to systematically derive time-reversible, areapreserving MD algorithms from the Liouville formulation of classical mechanics. Here, we briefly introduce the Liouville approach to the MTS method. The Liouville operator for a system of N degrees of freedom in Cartesian coordinates is defined as... 

Any three-dimensional orthogonal coordinate system may be specified in terms of the three coordinates q, q2 and q3. Because of the orthogonality of the coordinate surfaces, it is possible to set up, at any point, an orthogonal set of three unit vectors ex, e2, e3, in the directions of increasing qx, q2, q3, respectively. It is important to select the qt such that the unit vectors define a right-handed system of axes. The set of three unit vectors defines a Cartesian coordinate system that coincides with the curvilinear system in... 

In the Cartesian coordinate system the deformation rate tensor, D, is defined as ... 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

The relationships between the matrices representing the reflection in different coordinate systems are expressible in terms of the matrix S that defines the relationships between the coordinate systems themselves. Suppose x, y) and x, if ) are two pairs of normalized vectors oriented along the axes of two Cartesian coordinate systems related by a hnear transformation ... 

Consider first a Cartesian coordinate system with unit basis vectors ei, e2, e3. We define the effect of a positive rotation about the z axis through an angle 0, corresponding to a symmetry operation G, say, as a counterclockwise rotation since... 

Figure 2.1 The Cartesian coordinate system used to represent the points (3.3) and (-3,-3) in the plane defined in terms of coordinates referenced to the origin (0,0)...

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

We now choose a space-fixed Cartesian coordinate system XYZ relative to which our standard basis functions are defined. Another coordinate system X Y Z, initially coinciding with the XYZ system, has been rotated by the operator R. This operator may represent any rotation. We shall need a general rotation which we specify as follows first rotate around Z through ip, then around Y through 0, and finally... 

In this theory, equilibrium flow is obtained using thin shear layer (TSL) approximation of the governing Navier- Stokes equation. However, to investigate the stability of the fluid dynamical system the disturbance equations are obtained from the full time dependent Navier- Stokes equations, with the equilibrium condition defined by the steady laminar flow. We obtain these in Cartesian coordinate system given by. 

A molecule-fixed Cartesian coordinate system is oriented such that its origin coincides with the center of mass and the axes coincide with the main inertial axes of the molecule. A rotation axis can thus be identified by a Cartesian axis, for example C (z). The main axis is usually defined as the 2 axis. Planes are identified by the axes they contain a(xy) or tr.y. 

In the case of the hexahalogenoniobate ions the negative charges represented by the halide ions lie on the axes of a Cartesian coordinate system [3]. The ligand coordinates may then be defined in the... 

In many applications of continuum mechanics the center of mass of objects has special significance. It appears advantageous to use an analogous concept in fuzzy set theory, taking the value of membership function at each point x of a fuzzy set A in the role of mass density and the integral defined in Eq. (100) as the total mass of the fuzzy set A. This is easily accomplished if the underlying set X can be interpreted as a Euclidean space with a well-defined Cartesian coordinate system. In this case, the center of mass c( ) of fuzzy set A is defined as... 

---