Cartesian coordinates three-dimensional

Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vaiy in all three coordinate directions, the concept of deformation previously introduced must be generahzed. The rate of deformation tensor Dy has nine components. In Cartesian coordinates. 

The Stream Function Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function can be used to plot the streamlines of the flow and find the velocity. For two-dimensional Bow the velocity components can be calculated in Cartesian coordinates 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... 

Hie trivial geometric operation is known as the identity. If it is applied to an arbitrary vector in, say, three-dimensional vector f unchanged. In Cartesian coordinates in matrix form as... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

In this form the equation is rather cumbersome and not easily solved, so it is customary to express it in spherical polar coordinates r, 6, and, (p, where r is the distance from the nucleus and 6 and (p are angular coordinates, rather than in the Cartesian coordinates x, y, and z. The relationship of the polar coordinates to the Cartesian coordinates is shown in Figure 3.5. In this form V = e2/r, and the equation is easier to solve particularly because it can be expressed as the product R(r)Q(9)(dimensional functions R, the radial function, and 0 and , the angular functions. Corresponding to these three functions there are three quantum numbers, designated n, /, and m. 

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... 

The most important new feature of the Lorentz transformation, absent from the Galilean scheme, is this interdependence of space and time dimensions. At velocities approaching c it is no longer possible to consider the cartesian coordinates of three-dimensional space as being independent of time and the three-dimensional line element da = Jx2 + y2 + z2 is no longer invariant within the new relativity. Suppose a point source located at the origin emits a light wave at time t = 0. The equation of the wave front is that of a sphere, radius r, such that... 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

The three-dimensional transport equation for inert pollutant dispersion results from timesmoothing the equation of continuity of the emitted substance. In Cartesian coordinates the distribution of a pullutant is given by the partial differential equation of second order for the concentration C(x, y, z, t) 111 ... 

The geometric interpretation of a vector as a position in space requires reference to a coordinate system. Any point in three-dimensional space corresponds to a triplet of real numbers, the x, y and z coordinates of the point with respect to a set of Cartesian axes. Conversely, any vector consisting of three real numbers specifies a point in space. The position in space associated with a particular vector will change if a different coordinate system is selected. 

The three-dimensional diffusion equation in Cartesian coordinates is... 

The first step to making the theory more closely mimic the experiment is to consider not just one structure for a given chemical formula, but all possible structures. That is, we fully characterize the potential energy surface (PES) for a given chemical formula (this requires invocation of the Born-Oppenheimer approximation, as discussed in more detail in Chapters 4 and 15). The PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements the PES has 3N — 6 coordinate dimensions, where N is the number of atoms >3. This dimensionality derives from the three-dimensional nature of Cartesian space. Thus each structure, which is a point on the PES, can be defined by a vector X where... 

Returning to the issue of convergence, as noted above the structure of each snapshot in a simulation can be described in the space of the PCA eigenvectors, there being a coefficient for each vector that is a coordinate value just as an x coordinate in three-dimensional Cartesian space is the coefficient of the i Cartesian basis vector (1,0,0). If a simulation has converged, die distribution of coefficient values sampled for each PCA eigenvector should be normal,... 

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

To introduce the notation and concepts to be used below, let us first briefly recall some elementary aspects of the Euclidean geometry of a triangle of points V, V2, V3 in ordinary three-dimensional physical space. Each point Vi can be represented by a column vector vt (denoted with a single underbar) whose entries are the coordinates in a chosen Cartesian axis system at the origin of coordinates ... 

Vectors. A physical quantity that has both magnitude and direction can be represented by a directed line segment or vector in three-dimensional space. Let A be a vector quantity. (We use boldface type for vectors.) We set up a Cartesian coordinate system xyz and denote vectors of unit length along the x, y, and z axes by i, j, and k, respectively. If Ax, Ayy and Az are the projections of A on the x, y, and z axes, then A is given by... 

Let us consider a vector in ordinary three-dimensional space. We can specify the length and direction of this vector in the following way. We arrange to have one end of the vector lie at the origin of a Cartesian coordinate system. The other end is then at a point which may be specified by its three Cartesian coordinates, x, y, z. In fact, these three coordinates completely specify the vector itself provided it is understood that one end of the vector is at the origin of the coordinate system. We can then write these three coordinates as a column matrix, in this case one with three rows, x y z, and say that the matrix represents the vector in question. 

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