Cartesian coordinate system plane

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 transformations of a cartesian coordinate system (x, y) in the plane of the circle can be used to generate a representation of the group. The... 

Plane of symmetry. If a plane can be placed in space such that for every atom of the molecule not in the plane there is an identical atom (which is to say, the same atomic number and isotope) on the other side of the plane at equal distance from it (i.e., a mirror image ), the molecule is said to possess a plane of symmetry. The Greek letter o is often used to represent both the plane of symmetry and the operation of mirror reflection that it performs. An example of a molecule possessing a plane of symmetry is methylcyclobutane, as illustrated in Figure B.l. Note that a planar molecule always has at least one ct, since tire plane of tire molecule satisfies the above symmetry criterion in a trivial way (the set of reflected atoms is the empty set). Note also that if we choose a Cartesian coordinate system in such a way tliat two of the Cartesian axes lie in the symmetry plane, say x and y, then for every atom found at position (x,y,z) where z there must be an identical atom at position (x,y,—z). 

The incident plane wave has only field components perpendicular to the direction of propagation. In contrast, the evanescent field has components along all directions X, y, and z of a Cartesian coordinate system attached to the IRE, as shown in Fig. 2. The direction of the incident field vector can be selected by use of a polarizer. The symbols II and denote electric field vectors parallel and perpendicular to the... 

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 ... 

As an example, let us work out a representation of the group C2v, which group consists of the operations , C2, Cartesian coordinate system, and let av be the xz plane and <7 be the yz plane. The matrices representing the transformations effected on a general point can easily be seen to be as follows ... 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

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)...

Consider the flame in a plane channel, and choose the cartesian coordinate system (x,y) moving with flame at a constant velocity equal to... 

In many cases we deal with rotational symmetric structures. Assuming that the axis of symmetry is identical to the y axis of an orthogonal cartesian coordinate system, then it is convenient to put one radius of curvature in the plane of the xy coordinate. This radius is given by... 

In the general case, when s-polarized light is converted into p-polarized light and/or vice versa, the standard SE approach is not adequate, because the off-diagonal elements of the reflection matrix r in the Jones matrix formalism are nonzero [114]. Generalized SE must be applied, for instance, to wurtzite-structure ZnO thin films, for which the c-axis is not parallel to the sample normal, i.e., (1120) ZnO thin films on (1102) sapphire [43,71]. Choosing a Cartesian coordinate system relative to the incident (Aj) and reflected plane waves ( > ), as shown in Fig. 3.4, the change of polarization upon reflection can be described by [117,120]... 

Figure 11. Unit orthogonal base vectors a , a2, and a3 for local Cartesian coordinate system associated with chain interior atom / . Atoms / — 1,/ ,/ + 1 lie in ah a2 plane and a2 bisects bond angle 8. Also shown are bond vectors W1 and r 1.

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... 

Lin (145) has carried out an extensive theoretical investigation of the radiative and nonradiative mechanisms involving vibronic, spin-orbit, and vibronic-spin-orbit coupling in formaldehyde. Earlier, Yeung (254) calculated the SVL values of Tg, and Yeung and Moore (255) calculated the SVL values of x g. Lin used the left-hand Cartesian coordinate system in which planar formaldehyde lies in the x-z plane rather than in the y-z planes for the right-hand coordinate, which is accepted as the standard spectroscopic convention. Here, we adhere to the latter... 

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. 

Let us consider a steady flow for which a Cartesian coordinate system (x, y, z) can be established such that -h x is the principal flow direction and all flow properties are independent of z. In this two-dimensional (x, y) flow, it will further be assumed that except in a layer extending parallel to the principal flow direction, all flow properties vary so slowly that transport effects are negligibly small. For convenience, the viscous, diffusive, and heat-conducting layer will be placed in the vicinity of the plane y = 0 (which, for example, may represent a stationary flat plate, or may divide two parallel... 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

Consider a set A and a (possibly approximate) symmetry element R, where the associated symmetry operator R leaves at least one point of the convex hull C of set A invariant. We assume that a reference point c g C, a fixed point of R, and a local Cartesian coordinate system of origin c are specified, where the coordinate axes are oriented according to the usual conventions with respect to the symmetry operator R. For example, if R is a Cy rotation axis, then the z axis of the local Cartesian system is chosen to coincide with this Cj axis, whereas if R is a reflection plane, then the z axis may be chosen perpendicular to this plane. 

The Cartesian coordinate system is usually chosen such that the z axis and the yz plane are parallel to the c axis and the b axis of the crystal lattice, respectively. Then the elements of T are given by... 

If a molecule has certain symmetry properties, important predictions about the solutions of the electronic Schrodinger equation can be made without having to solve the equation itself. Consider the case of a planar molecule, i.e. of a molecule whose nuclei lie in a plane. This plane is a symmetry plane for the molecule, and it can be shown that any eigenfunction is either symmetric or antisymmetric with respect to this plane. If one chooses the plane of the nuclei as the (y, z) plane of a Cartesian coordinate system, this means that... 

Figure 1.3 Sketch of an infinitesimal tetrahedron whose three faces coincide with the x-y, x-z, and y-z planes of the original (unprimed) Cartesian coordinate system. The third slant face appears to be oriented such that the axis x is normal to the ar ea of the slant face. The two remaining axes, y and z lie in the plane of the slant face but are orthogonal to one another as well as to the x -axis. Hence, x, j/, and z define a Cartesian coordinate system rotated with respect to the original one.

For convenience, let us adopt a Cartesian coordinate system in which the flat plate is assumed to occupy the xz plane, with the initially stationary fluid occupying the upper half space, v > 0. We denote the magnitude of the plate velocity as U so that... 

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