RF/Microwave Engineering and Applications in Energy Systems. Abdullah Eroglu

      The illustration of the changes in length, area, and volume for a cylindrical coordinate system is given in Figure 1.10.

      1.2.3.3 dl, ds, and dv in a Spherical Coordinate System

      In a spherical coordinate system, the differential length is given by

      (1.41)

      Its magnitude is defined by

      (1.42)

Figure 1.10 Illustration of differential length, area, and volume in a cylindrical coordinate system.

The differential areas in a spherical coordinate system for the surface areas shown in Figure 1.11 are defined as

      (1.43)

      The infinitesimal change for the volume is defined by

      (1.44)

      The illustration of the changes in length, area, and volume for a spherical coordinate system is given in Figure 1.11.

      1.2.4 Line Integral

A line integral is defined as the integral of the tangential components of a vector along the curve, such as the one shown in Figure 1.12. A line integral is then expressed as

Figure 1.11 Illustration of differential length, area, and volume in a spherical coordinate system.

(1.45)

      One of the practical examples for the line integral application is finding work that is required to move a charge from point a to point b, similar to the one shown in Figure 1.12. This can be expressed in terms of the line integral of the electric field, .

Figure 1.12 Vector

along the curve.

Example 1.1 Line Integral

Find the line integral of along the path given in Figure 1.13.

      Solution

      We need to find

Since the geometry is circular, it is easier to work with a cylindrical coordinate system, hence we transform vector to cylindrical coordinates as

Figure 1.13 Path for line integral Example 1.1.

      In a cylindrical coordinate system, the differential length along the path is defined as

      where r = 2 along the path. So