محاضرة 3

introduction
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 studen
is advised to concentrate simply on learning the mathematical techniques and to wait fo
heir applications in subsequent chapters.

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.

LINE, SURFACE, AND VOLUME INTEGRALS
The familiar concept of integration will now be extended to cases when the integrand in-
volves a vector. By a line we mean the path along a curve in space. We shall use terms such
as line, curve, and contour interchangeably.

4 DEL OPERATOR
The del operator, written V, is the vector differential operator. In Cartesian coordinates,
(3.16)
dx
V =
-
a
* +
Ty
a
> +
Tz
a
<
This vector differential operator, otherwise known as the gradient operator, is not a vector
in itself, but when it operates on a scalar function, for example, a vector ensues. The oper-
ator is useful in denning
1. The gradient of a scalar V, written, as W
2. The divergence of a vector A, written as V • A
3. The curl of a vector A, written as V X A
4. The Laplacian of a scalar V, written as V V
Each of these will be denned in detail in the subsequent sections. Before we do that, it
is appropriate to obtain expressions for the del operator V in cylindrical and spherical
coordinates