Figure 1.8 Representation of vector
The vector operations for dot and cross products for vectors
(1.30)
(1.31)
The base vector properties are then equal to
(1.32)
1.2.3 Differential Length (dl ), Differential Area (ds), and Differential Volume (dv )
In vector analysis, line, surface, and volume integrals are expressed using differential lengths, areas, and volumes.
1.2.3.1 dl, ds, and dv in a Cartesian Coordinate System
Differential length represents infinitesimal change in any direction of the axis in the coordinate system and is represented 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 1.9 are defined as
(1.35)
The infinitesimal change for the volume is defined by
(1.36)
The illustration of the changes in length, area, and volume for a Cartesian coordinate system are given in Figure 1.9.
1.2.3.2 dl, ds, and dv in a Cylindrical Coordinate System
In a cylindrical coordinate system, the differential length is given by
(1.37)
Its magnitude is defined by
(1.38)
Figure 1.9 Illustration of differential length, area, and volume in a Cartesian coordinate system.
The differential areas in a cylindrical coordinate system for the surface areas shown in Figure 1.10 are defined as
(1.39)
The infinitesimal change for the volume is defined by
(1.40)
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
