Vector calculus

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

By analogy with ordinary differential calculus, the ratio du/df is defined as the limit of the ratio as the interval t becomes progressively smaller. 

The derivative of a vector is the vector sum of the derivatives of its components. The usual rules for differentiation apply  

Suppose that f(x, y, z) is a scalar field, and we wish to investigate how / changes between the points r and r + dr. Here 

The first vector on the right-hand side is called the gradient of /, and it is written grad / in this text. 

An alternative notation involves the use of the so-called gradient operator V (pronounced del ), 

In spherical polar coordinates, the corresponding expression for grad / is 

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

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A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

It is conventional to write the multipole expansion as a series in 1/r, and so we need to find alternative expressions for terms higher than the first. From elementary vector calculus we have... 

In vector calculus, the flux 4> of an arbitrary vector field A through a surface S is given by the surface integral... 

Consider the bimolecular reaction of A and B. The concentration of B is depleted near the still-unreacted A by virtue of the very rapid reaction. This creates a concentration gradient. We shall assume that the reaction occurs at a critical distance tab- At distances r tab. [B] = 0. Beyond this distance, at r > rAB, [B] = [B]°, the bulk concentration of B at r = °°. We shall examine a simplified, two-dimensional derivation the solution in three dimensions must incorporate the mutual diffusion of A and B, requiring vector calculus, and is not presented here. 

The proof of this proposition follows fairly easily from the definition of matrix exponentiation and standard techniques of vector calculus. See any linear algebra textbook, such as [La, Chapter 9]. 

With this relationship in mind, vector calculus requires that the divergence of the vorticity field is exactly zero ... 

In the jargon of vector calculus, the vorticity field is said to be soleniodal. A flow for which the vorticity is exactly zero, u = 0, is, by definition, called irwtational. 

The procedure based on the multiplication rule as an initial postulate can also be generalized to the derivative of complex functions of a complex variable. Vector calculus would benefit from this approach, since the V operator also obeys Leibniz rule. In both of these cases we would have to generalize the basic Rule 1 and make it consistent with the corresponding case. 

The authors are hopeful that the book will serve as a practical manual for the researcher, both theorist as well as experimentalist, in atomic, molecular or chemical physics. With this in mind, we have paid special attention to the comprehensive information necessary to follow all calculations through to their conclusions. We also hope that the material in the appendices will take the place of a short handbook with formulae on vector calculus, spherical functions, Wigner D-functions, Clebsch-Gordan coefficient tables, etc. We fully realize that one may find in various pa-... 

Refer to Figure 1. As shown, dA has the unit vector h on its surface. The component of dA on the x-y plane is dxdy. As shown in the figure, the unit vector normal to dxdy is, which is in the direction of the z axis. In vector calculus, the component of dA can be obtained through the scalar product of h and 3. This product is designated by n 3. In other words. 

The mathematical properties of Xk ensure that the ordinary mathematical operations of vector calculus can be performed on the local instantaneous variables which are discontinuous [112, 58]. 

In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

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