Differential Vector Operations

The gradient of a scalar function (p = (p r) is defined as the three-dimensional vector of its Cartesian partial derivatives. 

Application of the Laplacian to a vector field A has to be understood as acting separately on each component. The result is then again a three-vector, of course. We note some important identities for the above vector operations. 

For any differentiable vector fields A, B and scalar functions (p, tp the following relations hold  

The proof of all these relations is straightforward and thus omitted here. Eq. (A.17) states that any rotational vector field has no sources, and Eq. (A.18) summarizes the fact that the curl of any gradient field is zero. 

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We present here the expressions for various differential vector operators that can be of great utihty for transforming the equations in different coordinate systems. Given a curvilinear coordinate system with basis vectors (/ i5i /t252 /tsSs),... 

Basically, the gradient applies to a scalar function in three variables, say /(x, y, z). The V operator in cartesian coordinates is a differential vector operator, like... 

In contradistinction with the previous discussion about reducing scalar state variables, the relationship with the next localized variable cannot be obtained by a differential vector operator, but by an ordinary derivation, becanse here the two variables U/ and U/ are both scalars. Notwithstanding, this is possible between the two last levels by a gradient operator ... 

Section 5.3. There, the operator T> s d/dx was used in the solution of ordinary differential equations. In Chapter 5 the vector operator del , represented by the symbol V, was introduced. It was shown that its algebraic form is dependent on the choice of curvilinear coordinates. 

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. 

We list here some of the useful vector operations in a differential setting that are generally encountered in chemical engineering ... 

Answer Begin by identifying the unit normal vector from the solid to the fluid across the surface at r = / n = Sr. Then (1) take the dot product of n with the total momentum flux tensor, (2) evaluate this vector-tensor operation at the fluid-solid interface, and (3) multiply the result by the differential surface element, dS = R sind d9 d, to generate a differential vector force. Hence,... 

The differential operator Vx consisting of the cross product of the differential vector V with another vector is called the rotational or curl operator, and the notation in terms of a function rot or curl is sometimes used. An example of such a link is the relationship between the electromagnetic induction B and the electromagnetic potential vector A, written as... 

The mathematical operator consisting of the dot product, or scalar product, of the differential vector V with another vector is called the divergence operator. For instance, the charge concentration p in a volume is linked to the electric displacement D through one of Maxwell s relationships ... 

It can be useful to recall that the vector operator is not properly defined by the expressions of its components, for they depend on the adopted space representation, Cartesian, spherical, etc., but by Equation 5.8 representing the area and by stating that Wis orthogonal to both U and V. Figure 5.3b shows the geometrical representation of the cross product. The ability of the cross product to express an area works also in the reciprocal way, that is, the differentiation with respect to an area is equal to the cross product of two differential vectors, supposedly oriented in different directions and distinguished by subscripts 1 and 2 ... 

This book is intended for first-year graduate and advanced undergraduate courses in qnantnm chemistry. This text provides students with an in-depth treatment of quantnm chemistry, and enables them to nnderstand the basic principles. The limited mathematics backgronnd of many chemistry stndents is taken into account, and reviews of necessary mathematics (snch as complex nnmbers, differential equations, operators, and vectors) are inclnded. Derivations are presented in fnll, step-by-step detail so that students at all levels can easily follow and nnderstand. A rich variety of homework problems (both qnantitative and conceptnal) is given for each chapter. 

The vector operators N2, N3. .. denote quadratic, cubic. .. operators in Vand its spatial derivatives, whereas the operators C and Bi represent matrix differential operators of the indicated order on V Computer algebra can be used to perform the ejqjansion. 

We use the mass-scaled (MS) body-fixed (BF) center of mass Jacobi coordinates of the reactant D-I-H2 to define the differential Hamiltonian operator. The two Jacobi vectors are R for the scattering coordinate, and r for the vibrational coordinate. The internal coordinates are q = (r, i , 7) where r = [rj is the MS bond length... 

Many physical laws assume a particularly simple form when written in terms of the vector differential operators. We define a vector operator V del which, in rectangular coordinates, is... 

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

Adesina has shown that it is superfluous to carry out the inversion required by Equation 5-255 at every iteration of the tri-diagonal matrix J. The vector y"is readily computed from simple operations between the tri-diagonal elements of the Jacobian matrix and the vector. The methodology can be employed for any reaction kinetics. The only requirement is that the rate expression be twice differentiable with respect to the conversion. The following reviews a second order reaction and determines the intermediate conversions for a series of CFSTRs. 

Finally, 3 " (j)[f (x )] is a short symbol expressing the m-th order partial derivative operators, acting first over the function f (x) and then, the resultant function, evaluated at the point x . The differential operators can be defined in the same manner as the terms present in equation (9), but using as second argument the nabla vector ... 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

Indeed, we have discussed the matrix elements involved in these formulas (see Eqs. (36), (52), and (56)) as well as the physical meaning of the Fourier coefficients pk p] t). However, the mathematical expressions are often rather involved and it is convenient, especially in specific applications, to introduce a diagram technique in order to represent the various terms of these general formulas.28 We first notice that in Eqs. (41) and (42), the momenta p,- essentially appear as parameters indeed, according to Eq. (52) only the wave-vectors are explicitly modified by the interactions. This is the reason why we shall only represent these wave numbers graphically it should, however, be kept in mind that the momenta are effectively affected by the interactions through the differential operators d/dp ... 

Now, three types of differential operations involving real-valued functions, vector and co-vectors fields are described. These operations are frequently used in the analysis and design of nonlinear control systems (for more details see [15, 26]). 

The differential operator divergence allows the passage from vectorial to scalar fields. For a vector [Pg.810]

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

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