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chapter 03

# chapter 03 - Chapter 3 VECTOR CALCULUS No man really...

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Chapter 3 VECTOR CALCULUS No man really becomes a fool until he stops asking questions. —CHARLES P. STEINMETZ 3.1 INTRODUCTION Chapter 1 is mainly on vector addition, subtraction, and multiplication in Cartesian coordi- nates, and Chapter 2 extends all these to other coordinate systems. This chapter deals with vector calculus—integration and differentiation of vectors. The concepts introduced in this chapter provide a convenient language for expressing certain fundamental ideas in electromagnetics or mathematics in general. A student may feel uneasy about these concepts at first—not seeing "what good" they are. Such a student is advised to concentrate simply on learning the mathematical techniques and to wait for their applications in subsequent chapters. J.2 DIFFERENTIAL LENGTH, AREA, AND VOLUME Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, cylindrical, and spherical coordinate systems. A. Cartesian Coordinates From Figure 3.1, we notice that (1) Differential displacement is given by d\ = dx a x + dy a y + dz a z (3.1) 53

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54 Vector Calculus •A- Figure 3.1 Differential elements in the right-handed Cartesian coordinate system. (2) Differential normal area is given by dS = dy dz dxdz dzdy a* a v a, and illustrated in Figure 3.2. (3) Differential volume is given by dv = dx dy dz (3.2) (3.3) dy a^ (a) dz < (b) ia z dy (c) Figure 3.2 Differential normal areas in Cartesian coordinates: (a) dS = dy dz a^, (b) dS = dxdz a y , (c) dS = dx dy a,
3.2 DIFFERENTIAL LENGTH, AREA, AND VOLUME 55 These differential elements are very important as they will be referred to again and again throughout the book. The student is encouraged not to memorize them, however, but to learn to derive them from Figure 3.1. Notice from eqs. (3.1) to (3.3) that d\ and dS are vectors whereas dv is a scalar. Observe from Figure 3.1 that if we move from point P to Q (or Q to P), for example, d\ = dy a y because we are moving in the y-direction and if we move from Q to S (or S to Q), d\ = dy a y + dz a z because we have to move dy along y, dz along z, and dx = 0 (no movement along x). Similarly, to move from D to Q would mean that dl = dxa x + dya y + dz a z . The way dS is denned is important. The differential surface (or area) element dS may generally be defined as dS = dSa n (3.4) where dS is the area of the surface element and a n is a unit vector normal to the surface dS (and directed away from the volume if dS is part of the surface describing a volume). If we consider surface ABCD in Figure 3.1, for example, dS = dydza x whereas for surface PQRS, dS = -dy dz a x because a n = -a x is normal to PQRS. What we have to remember at all times about differential elements is d\ and how to get dS and dv from it. Once d\ is remembered, dS and dv can easily be found. For example, dS along a x can be obtained from d\ in eq. (3.1) by multiplying the components of d\ along a^, and a z ; that is, dy dz a x . Similarly, dS along a z is the product of the components of d\ along a x and a y ; that is dx dy a z . Also, dv can be obtained from d\ as the product of the three components of dl; that is, dx dy dz.

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chapter 03 - Chapter 3 VECTOR CALCULUS No man really...

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