Vector identities

For ready reference, we list a selection of useful vector identities. 

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The ordered set of measurements made on each sample is called a data vector. The group of data vectors, identically ordered, for all of the samples is called the data matrix. If the data matrix is arranged such that successive rows of the matrix correspond to the different samples, then the columns correspond to the variables as in Figure 1. Each variable, or aspect of the sample that is measured, defines an axis in space the samples thus possess a data stmcture when plotted as points in that / -dimensional vector space, where n is the number of variables. 

J is the Fourier transform of the electron current which gives rise to the magnetic field B. Using these equations with the following vector identity,... 

Not only are the lengths of the pattern and weights vectors identical, the individual entries in them share the same interpretation. 

Using the definitions above many vector identities can be derived and these are listed in many sources e.g. [10]. Some useful identities are... 

Using the vector identity (16), it is found that E is irrotational ... 

This result, however, is an identity of Minkowski spacetime itself, namely, 8 8 operating on a function of produces the same result as 8V8M operating on a function of . Equation (879) does not mean that Aa can take any value. We reach the important conclusion that the vector identity (872) of U(l) is a property of three-dimensional space itself and can always be interpreted as such. Therefore even on the U(l) level, Eq. (872) does not mean that % can take any value. Even on the U(l) level, therefore, potentials can be interpreted physically, as was the intent of Faraday and Maxwell. On the 0(3) level, potentials are always physical. 

We remarked at the beginning of this section that the equation of motion is the cornerstone of any discussion of fluid dynamics. When one considers the various coordinate systems in which it may be expressed, the vector identities that may transform it, or the approximations that may be used to simplify it, Equation (28a) takes on many forms, some of which are scarcely recognizable as the same relationship. The purpose of this section is to illustrate that —despite its complexity and variations —the equation of motion is really nothing more than a statement of Newton s second law of motionl... 

While the operation (V V) V is relatively straightforward in cartesian coordinates, it is a bit trickier in other coordinate systems. In general, the operation is defined by a vector identity as... 

Expand each of these expressions directly in cylindrical coordinates, and compare with the results in the first problem using the vector identity. Explain your preference for the alternate ways of writing the operator. 

The result, of course, must be the same as that using the vector identity (Eqs. 2.55 and 2.56). Take care to explain the role of the nonvanishing unit vectors. 

A further vector identity defines the Laplacian of a vector field as... 

The second term on the left-hand side can be expanded by vector identity, and then simplified ... 

In this case the pressure is eliminated altogether, since by vector identity, the curl of the gradient of a scalar field vanishes. From the definition of vorticity, Eq. 2.103, a simple diffusion equation emerges for the vorticity... 

The second term can be expanded further, using a vector identity, to give... 

Using the general vector identity, VxVxE = V(V E)-V2E, and that V E = 0 in a space free of charge, the following wave equation results ... 

Characterization of the vector batch used in non-clinical safety studies is also performed in compliance with GLPs. Care should be taken to ensure that adequate and appropriate processes are used to produce the batch(es) required, and the production process should be similar to that proposed for production of the vector for human clinical trials, or at least performed in such a way that adequate documentation of the production methods is available for comparative purposes. The GLP regulations state that the identity, purity, and composition of the vector batch (test article) used in a safety study must be known and documented. In addition, the stability of the vector preparation in the specific container used for the study must also be known prior to initiation of the study or acquired concomitantly with the study itself. These test article characterization experiments require that the researcher produce additional amounts of vector identical to the material used in... 

By using the vector identities relating Eulerian and Lagrangian frames together with the equation of continuity, one can convert Eq. (10) to an equivalent form ... 

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