Function vectors

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

The non-adiabatic coupling matrix t will be defined in a way similar to that in the Section V.A [see Eq. (51)], namely, as a product between a vector-function t(i) and a constant antisymmetric matrix g written in the form... 

In Section XIV.A.2, we intend to obtain the vector function x q, 0) for a given distribution of conical intersections. Thus, first we have to derive an expression for a conical intersection removed from the origin, namely, assumed to be located at some point, ( /),0jo), in the plane. 

If A, S and C are differentiable vector functions of scalar t and is a differentiable function of / then... 

If V is the volume bounded by a closed surface S and A is a vector function of position with continuous derivatives, then... 

Firstly, let us formulate an auxiliary statement concerning boundary values for the vector-functions having square integrable divergence (Baiocchi, Capelo, 1984 Temam, 1979). Consider a bounded domain H c i . Introduce the Hilbert space... 

In the two-dimensional theory of solids, the following theorem is also useful. For vector-functions M = introduce the space... 

The inner product in the space Q /j of all grid vector-functions given on the grid ijjp and vanishing on the boundary is defined by... 

Assuming that the distribution of masses inside the volume V is given, this vector function g p) depends only on the coordinates of the observation point p, and by definition it is a field. It is appropriate to treat the masses in the volume V as sources of the field g p). In other words, these masses generate the field at any point of the space, and this field may be supposed to exist whether a mass is present or absent at this point. When we place an elementary mass at some point p, it becomes subject to a force equal to... 

To derive the boundary conditions, we introduce some vector function X(p) ... 

As seen from the above equations, the (mxp) matrix (Ve1)1 is the Jacobean matrix, J, of the vector function e, and the (mxp) matrix(VfT)T is the Jacobean matrix, G, of the vector function f(x,k). The srth element of the Jacobean matrix J. is given by... 

In MATLAB, polynomials are stored exactly the same as vectors. Functions in MATLAB will interpret them properly if we follow the convention that a vector stores the coefficients of a polynomial in descending order—it begins with the highest order term and always ends with a constant even if it is zero. Some examples ... 

As pointed out in the foregoing, there are two specific peculiarities qualitatively distinguishing these systems from the classical ones. These peculiarities are intramolecular chemical inhomogeneity of polymer chains and the dependence of the composition of macromolecules X on their length l. Experimental data for several nonclassical systems indicate that at a fixed monomer mixture composition x° and temperature such dependence of X on l is of universal character for any concentration of initiator and chain transfer agent [63,72,76]. This function X(l), within the context of the theory proposed here, is obtainable from the solution of kinetic equations (Eq. 62), supplemented by thermodynamic equations (Eq. 63). For heavily swollen globules, when [Pg.178]

Next, we refer to the requirements to be fulfilled by the matrix D, namely, that it is diagonal and that it has the diagonal numbers that are of norm 1. In order for that to happen, the vector-function t(r) has to fulfill along a given (closed) path F the condition ... 

In formulating physical problems it is often necessary to associate with every point (x, y, z) of a region R of space some vector a(x, y, z). It is usual to call a(x,y,z) a vector function and to say that a vector field exists in II. If a scalar x, y, z) is defined at every point of R then a scalar field is said to exist in R. 

Clearly grad is a vector function whose (x, y, z) components are the first partial derivatives of . The gradient of a vector function is undefined. Consider an infinitesimal vector displacement such that... 

The operator V- may be applied to any vector function and the result is called the divergence, e.g. 

Since curl a is a vector function one can form its divergence... 

Likewise, since grad is a vector function it is allowed to take its curl,... 

Hence curl grad = 0 for all (j>. Again, conversely may be inferred that if 6 is a vector function with identically zero curl, then 6 must be a gradient of some scalar function. Vector functions with identically zero curl are said to be irrotational. 

For a vector function h(x), such as occurs in a series of nonlinear multivariable constraints... 

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