Cartesian coordinate Rectangular

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 ... 

The chain rule of differentiation applied to (A.8) provides, in rectangular Cartesian coordinates,... 

The set of unit vectors of dimension n defines an n-dimensional rectangular (or Cartesian) coordinate space 5 . Such a coordinate space S" can be thought of as being constructed from n base vectors of unit length which originate from a common point and which are mutually perpendicular. Hence, a coordinate space is a vector space which is used as a reference frame for representing other vector spaces. It is not uncommon that the dimension of a coordinate space (i.e. the number of mutually perpendicular base vectors of unit length) exceeds the dimension of the vector space that is embedded in it. In that case the latter is said to be a subspace of the former. For example, the basis of 5 is ... 

Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis,... 

The phase space (r space) of the system is the Euclidean space spanned by the 2n rectangular Cartesian coordinates qL and pt. Every possible mechanical state of the system is represented by exactly one point in phase space (and conversely each point in phase space represents exactly one mechanical state). 

Equation A.7 represents conservation of mass for a general flow in rectangular Cartesian coordinates. 

Consider the fluid s x-component of motion in a rectangular Cartesian coordinate system. By following the flow, the rate of change of a fluid element s momentum is given by the substantive derivative of the momentum. By Newton s second law of motion, this can be equated to the net force acting on the element. For an element of fluid having volume Sx ySz, the equation of motion can be written for the x-component as follows ... 

Equation A.22 is the Navier-Stokes equation for the x-component of motion in rectangular Cartesian coordinates. The corresponding equations for they and z components are obvious. 

Figure 7.2 Transforming the Cartesian reference frame a) cylindrical cross section of the screw and barrel with flow out of the surface of the page, and b) the unwound rectangular channel with a stationary barrel and the Cartesian coordinate frame positioned on the screw, is the velocity of the screw core in the z direction and it is negative...

The tensor quantities given in this chapter are all second rank, and are sometimes referred to as matrices, according to common usage, so that the two terms, tensor and matrix, are used interchangeably. In many cases, the components (or coefficients) of second-rank tensors are represented by 3 x 3 matrices. Symbols for tensors (matriees) are printed in bold italic type, while symbols for the components are printed in italic type. In general, the base tensors are those for a rectangular Cartesian coordinate system. 

This equation was solved numerically with a fourth-order Runge-Kutta scheme. It was usually more convenient to recast (12.25) as a differential equation in the rectangular Cartesian coordinates sometimes, however, the advantage was tipped in favor of the polar coordinates. The results shown in Fig. 12.4 were obtained with a mixture of the two approaches. 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

We want to formulate the mass balance for a rectangular test volume (V = AxAyAz) whose edges are parallel to the axes of a Cartesian coordinate system (Fig. 18.4). For the one-dimensional case, the flux is assumed to be parallel to the x-axis and independent of y and z, yet variable along the x-axis. It is expressed by Fx(x) where the subscript x indicates the axis along which the flux occurs and the parenthetical x refers to the location on the x-axis (e.g., x = 0). If C is the average concentration in the test volume, then the change of total mass, VC, per unit time is ... 

The stress tensor describes the forces transmitted to an element of material through its contacts with adjacent elements (78). Traction is the force per unit area acting outwardly on the material adjacent to a material plane, and transmitted through its contact with material across the plane. If the components of traction are known for any set of three planes passing through a point, the traction across any plane through the point can be calculated. The stress at a material point is determined by an assembly erf nine components of traction, three for each plane. If the orientations of the three planes are chosen to be normal to the coordinate directions of a rectangular Cartesian coordinate system, the Cartesian components of the stress are obtained ... 

Most rheological data on polymer liquids of known structure has been obtained in simple shearing deformations. The velocity field for homogeneous simple shear in rectangular Cartesian coordinates may be expressed ... 

Fig. 2.24 Translation and deformation of an initially rectangular fluid element in cartesian coordinates.

A model for wet scrubbing in a cross-flow is illustrated in Fig. 7.21. Consider a rectangular scrubbing domain of length L, height H, and width of unity in Cartesian coordinates. Assume that the gas-solid suspension flow is moving horizontally, and that the solid particles are spherical and of uniform size. The particle concentration across any plane perpendicular to the flow is assumed to be uniform. The water droplets fall vertically and are uniformly distributed in the flow system. 

Let Xi and Ab be the coordinates in a rectangular Cartesian coordinate system with X1 along the ordinate axis and X2 along the abscissa axis. One of the axes in the new coordinate system is now chosen to be collinear with the abscissa axis in the rectangular coordinate system, and we let the coordinate along this axis be the first term in the expression for X2 in Eq. (D.22), namely 2(ay — ay). The situation is sketched in Fig. D.1.1. The other axis with a coordinate proportional to the other distance ay — a 1 forms an angle 4> with the first. This angle is determined from the requirements that the projections of this coordinate on the X coordinate axis is a (a 2 — ay) and on the X2 coordinate axis is aym ay — ay )/.sy. If we let the proportionality constant of X2 — ay be / , then we have... 

Figure 7.3 Sampling principles in 2D k space (a) Cylindrical coordinates. The angle of the field-gradient direction with respect to the x axis is given by 6= arctan Gy / Gx, (b) Cartesian coordinates. For rectangular gradient pulse shapes ky = -g Gy t1 and kx = -g Gx t2. Such sampling schemes are applicable to a slice which can be selected when the rf pulse is applied selectively in the presence of a gradient Gz. The areas of k space accessible by the pulse sequences shown are shaded in gray. TX transmitter signal ...

We present first a brief discussion of the stress tensor and the concepts of its mean and deviatoric parts. A rectangular Cartesian coordinate system (x, i 1,2,3) is used throughout. The stress tensor referred to this coordinate system is denoted... 

Parameters Appearing on the Stability Criterion (Bounded Analysis) for Rectangular Bubble Columns Cartesian Coordinates... 

A molecule is the smallest fundamental group of atoms of a chemical compoimd that can take part in a chemiccd reaction. The atoms of the molecule are organized in a 3D structure the molecular matrix M is a rectangular matrix Ax3 whose rows represent the molecule atoms and the columns the atom Cartesian coordinates x, y, z) with respect to any rectangular coordinate system with axes X, Y, Z. The cartesian coordinates of a molecule usually correspond to some optimized molecular geometry obtained by the methods of -> computational chemistry. The molecular geometry can also be obtained from crystallographic coordinates or from 2D-3D automatic converters. 

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