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
Example 1.1 Line Integral
Find the line integral of
Solution
We need to find
Since the geometry is circular, it is easier to work with a cylindrical coordinate system, hence we transform vector
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
