Orthogonalized coordinate system

Now we intend to derive nonpenetration conditions for plates and shells with cracks. Let a domain Q, d B with the smooth boundary T coincide with a mid-surface of a shallow shell. Let L, be an unclosed curve in fl perhaps intersecting L (see Fig.1.2). We assume that F, is described by a smooth function X2 = i ixi). Denoting = fl T we obtain the description of the shell (or the plate) with the crack. This means that the crack surface is a cylindrical surface in R, i.e. it can be described as X2 = i ixi), —h < z < h, where xi,X2,z) is the orthogonal coordinate system, and 2h is the thickness of the shell. Let us choose the unit normal vector V = 1, 2) at F,, ... 

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

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... 

All PCA loading vectors are orthogonal to each other PCA is a rotation of the original orthogonal coordinate system resulting in a smaller number of axes. 

An often-overlooked issue is the inherent non-orthogonality of coordinate systems used to portray data points. Almost universally a Euclidean coordinate system is used. This assumes that the original variables are orthogonal, that is, are uncorrelated, when it is well known that this is generally not the case. Typically, principal component analysis (PCA) is performed to generate a putative orthogonal coordinate system each of whose axes correspond to directions of maximum variance in the transformed space. This, however, is not quite cor-... 

Ligand positions about the iron are designated X, Y, Z, —X, -Y, and -Z, where X, Y, and Z, define the directions of an idealized right-handed orthogonal coordinate system centered about the iron site. 

As a first step, the treatment in this chapter is limited to electromagnetic field theory in orthogonal coordinate systems. Subsequent steps would include more advanced tensor representations and a complete quantization of the extended field equations. 

Direction cosines, which can be used to define the direction of a vector in an orthogonal coordinate system, play an essential role in accomplishing coordinate transformations. As illustrated in Fig. A. 1, there is a vector V oriented in a (z,r,9) coordinate system. Because our concern here is only the direction of the vector, the physical dimensions are sufficiently small so that the curvature in the 9 coordinate is not seen (i.e., the coordinate system... 

Fig. A.l Three direction cosines are needed to orient a vector V in an orthogonal coordinate system.

Given a state at a point (e.g., stress or strain rate) that can be described by a symmetric tensor in some orthogonal coordinate system, it is always possible to represent that particular state in a rotated coordinate system for which the tensor has purely diagonal components. The axes for such a rotated coordinate system are called the principal axes, and the diagonal components are called the principal components. Finding the principal states and the principal directions is an eigenvalue problem. 

If a matrix contains only one row, it is called a row matrix or a row vector. The matrix B shown above is an example of a 1 X 3 row vector. Similarly, a matrix containing only one column is known as a column matrix or column vector. The matrix C shown above is a 6 X 1 column vector. One use of vectors is to represent the location of a point in an orthogonal coordinate system. For example, a particular point in a three-dimensional space can be represented by the 1X3 row vector... 

In Fig. 17a a total stationary orthogonal coordinate system is shown, and in Fig. 17b we see the corresponding distorted image as observed by the traveler, who is represented by the small circle (i, o) passing from left to right. The... 

Figure 17. Orthogonal coordinate system of the stationary world (a) appears distorted into that of (b) to the traveler who exists at the small circle (i, o) and is moving to the right at a speed of 0.6c. From the diagram we find that flat surfaces parallel to motion are not changed but those perpendicular to motion are transformed into hyperboloids, while the plane (i—i) through the observer is transformed into a cone. The back side can be seen on all objects that have passed this cone. The separation of advancing hyperboloids is increased, while that of those moving away is decreased.

Let us compare the apparent distortion of flat surfaces moving past an observer at increasing relativistic velocities. Figure 17b represents an orthogonal coordinate system moving from right to left at the speed of 0.6c past the stationary observer at B. As the velocity is increased and approaches that of... 

Here a new aspect arises. When the optical axis of the crystal points in the z-direction of an orthogonal coordinate system, the xy-plane is isotropic. Therefore the physical quantities are denoted according to their relation with respect to the optical axis ... 

A crystal structure usually is described by the unit cell dimensions, space group and coordinates of the atoms (or orientation and position of the molecules) in the asymmetric unit. This, in fact, is the order in which the information is obtained when a crystal structure is determined by X-ray or neutron diffraction experiments. However, an equivalent way to describe a structure is to place the center of a molecule at the origin of an orthogonal coordinate system and to specify its molecular surroundings. This alternative is especially powerful in crystals with one molecule per asymmetric unit because the orientations of the surrounding molecules are related to the central molecule by crystallographic symmetry. The coordination sphere or environment of the structure then is defined as those surrounding molecules which are in van der Waals contact, or nearly in contact, with the central molecule. 

As already discussed in Section 2.4, in 3-D any point (denoted by the vector r) can be described by its three coordinate projections x, y, z (in emits such as m, nm, A, or pm) using an orthogonal coordinate system with emit vectors eX/ ex, ex hence r = xex + yeY + zez- In noncrystallographic textbooks, the position vector r is usually given in a Cartesian (orthogonal) system. 

PROBLEM 7.10.8. There are six choices of orthogonalized coordinate systems ... 

In the configuration space of the nuclei another orthogonal coordinate system can be defined, consisting of (F — 1) coordinates Qv (v = 1 — 1)... 

One of the postulates of quantum mechanics is that the state of a particle, in this case an electron, is described fully by a wavefunction fi, from which all its observable properties can be determined. If an orthogonal coordinate system is being used, 4 will be a function of x, y and z, and denoted more fully by T(x, y, z). 

Consider an orthogonal coordinate system in which Z is one of the coordinates and the other two, x and y, are distances along surfaces of constant Z. Let u and v be velocity components in the x and y directions, respectively, in this new reference frame. When use is made of equation (71) it is found that a formal transformation of (88) into the new coordinate system yields... 

This transformation is made by replacing the original coordinate system Xj...jc of the experimental space defined by the experimental variables with a new orthogonal coordinate system in which the origin is located at the stationary... 

Considering a generalized orthogonal coordinate system, the orthogonal curvilinear coordinates are defined as qa- In this O-system the base vectors Gq, are defined as unit vectors along the coordinates. The position of the point P is given by the coordinates, or by the position vector r = r qa,t). 

The governing equations can be transformed directly from Cartesian coordinates into cylindrical coordinates without considering the vector notation. In this appendix the relationships between the Cartesian coordinates and the cylindrical coordinates are defined solely, but the method of coordinate transformation is generic and can thus be applied to any orthogonal coordinate system. 

---