Vector differentiation

Given that a vector is a function of certain independent variables A = A(jcj, x2,. 3), then 

Assume that a differentiable scalar field depends on certain independent variables as S = S(xi, x2, X3). The gradient of the scalar field produces a vector, described below in different coordinate systems. 

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The vector differential operator del (or nahla) written as V is defined by... 

Discrete-time solution of the state vector differential equation... 

I have assumed that the reader has no prior knowledge of concepts specific to computational chemistry, but has a working understanding of introductory quantum mechanics and elementary mathematics, especially linear algebra, vector, differential and integral calculus. The following features specific to chemistry are used in the present book without further introduction. Adequate descriptions may be found in a number of quantum chemistry textbooks (J. P. Lowe, Quantum Chemistry, Academic Press, 1993 1. N. Levine, Quantum Chemistry, Prentice Hall, 1992 P. W. Atkins, Molecular Quantum Mechanics, Oxford University Press, 1983). 

Figure 1 The two diagrams show rotation of vectors equal in magnitude, on the left, and unequal vectors differentiated by absorption, on the right. The two situations lead to ORD and CD respectively.

The following program module extends the formula (5.21) to vector differential equations of the form (5.1), simply by considering , f and... 

The dynamic model can be written in the following vector differential equation form... 

The vector differential operator, V, is the most widely used vector and tensor differential operator for the balance equations. In Cartesian coordinates it is defined as... 

By vector differentiation we mean the operation of differentiating one vector with respect to a second vector. The result is a matrix in which each column is the derivative of the first vector with respect to... 

It may be worth while to review the different kinds of multiplicity involved in the symbols appearing in Eqs. (3-6) and (3-15). Equation (3-6) is merely a shorthand way of writing the material balance for each of the key components, each term being a row matrix having as many elements as there are independent reactions. The equation asserts that when these matrices are combined as indicated, each element in the resulting matrix will be zero. The elements in the first two terms are obtained by vector differential operation, but the elements are scalars. Equation (3-15), on the other hand, is a scalar equation, from the point of view of both vector analysis and matrix algebra, although some of its terms involve vector operations and matrix products. No account need be taken of the interrelation of the vectors and matrices in these equations, but the order of vector differential operators and their operands as well as of all matrix products must be observed. 

The surface integral in formula (F.16) can be written in a more compact form if we introduce the following vector differential operator ... 

It follows from Eq. (A 124) that the vector derivative operator changes the grade of the object it operates on by 1. For example, the vector derivative of the scalar X(xa) is a vector (because a X = 0 for any scalar X, so aX = a A X), and the vector derivative of the vector f(xa) is a scalar plus a bivector. The differentiation with respect to the vector variable xa greatly resembles the differentiation with respect to some scalar variable xa. For example, the vector differentiation is distributive,... 

This term can be reformulated by use of the vector differential operators for differentiation of products into the difference between two terms (e.g., [11], p. 

An effective approximation to Equation 1 is obtained by segmenting the water body of interest into n volume elements of volume Vj and representing the derivatives in Equation 1 by differences. Let V be the n X n diagonal matrix of volumes V, A, the n X n matrix of dispersive and advective transport terms SPy the n vector of source terms SPjy averaged over the volume Vjy- and P, the n vector of concentrations P which are the concentrations in the volumes. Then the finite difference equations can be expressed as a vector differential equation... 

The solution of Equations 47, 48, and 49 requires numerical techniques. For such nonlinear equations, it is usually wise to employ a simple numerical integration scheme which is easily understood and pay the price of increased computational time for execution rather than using a complex, efficient, numerical integration scheme where unstable behavior is a distinct possibility. A variety of simple methods are available for integrating a set of ordinary first order differential equations. In particular, the method of Huen, described in Ref. 65, is effective and stable. It is self-starting and consists of a predictor and a corrector step. Let y = f(t,y) be the vector differential equation and let h be the step size. 

Pebernard, S., R.D. Iggo. 2004. Determinants of interferon-stimulated gene induction by RNAi vectors. Differentiation 72, 103-111. 

With the notation v = du/dE equation (16) may be written as the vector differential equation... 

Generalizing to the multivariable case, a time-invariant, linear dynamic system may be defined by the vector differential equations ... 

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