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

(1.22)

Unit vector representation for a cylindrical coordinate system is shown in Figure 1.5.

A vector representation for a Cartesian coordinate system is given by (1.23).

(1.23)

The magnitude of vector is found from

(1.24)

The vector can be illustrated in the cylindrical coordinate system, as shown in Figure 1.6.

In Figure 1.6, the range of r , ϕ , and z are given as

The vector operations for dot and cross products for vectors and are given by

(1.25)

(1.26)

Figure 1.6 Representation of vector in a cylindrical coordinate system.

The base vector properties can be obtained as

(1.27)

Unit vector representation for a spherical coordinate system is shown in Figure 1.7.

A vector representation for a Cartesian coordinate system is given by (1.28).

(1.28)

The magnitude of vector is found from

(1.29)

The vector can be illustrated in the spherical coordinate system, as shown in Figure 1.8.

In Figure 1.8, the range of R , θ , and ϕ are given as

Figure 1.7 Representation of vector in a spherical coordinate system.

Figure 1.8 Representation of vector in a spherical coordinate system.

The vector operations for dot and cross products for vectors and are given by

(1.30)

(1.31)

The base vector properties are then equal to

(1.32)

In vector analysis, line, surface, and volume integrals are expressed using differential lengths, areas, and volumes.

Differential length represents infinitesimal change in any direction of the axis in the coordinate system and is represented by . In a Cartesian coordinate system, the differential length is given by

(1.33)

Its magnitude is defined by

(1.34)

The infinitesimal change for the surface area is defined by the differential area. The unit vector of a differential area points normal to the surface. The differential areas in a Cartesian coordinate system for the surface areas shown in Figure 1.9are defined as