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

Calculate .

The curl of a vector is used to identify how much the vector v curls around a reference point. The curl of a vector is expressed as

(1.62)

Curl can be given in a cylindrical or spherical coordinate system as

(1.63)

and

(1.64)

It is important to note that curl operation on any vector results in another vector. Curl cannot operate on a scalar quantity. In addition, the following two properties follow for curl operation.

(1.65)

(1.66)

Equation (1.65)implies that the curl of a vector does not diverge and the curl of a gradient of a scalar does not exist as expected.

The divergence theorems states that the volume integral of the divergence of a vector is equal to the surface integral of the same vector enclosing that volume. It is mathematically given by

(1.67)

While this theorem can be applied for an electric flux density, it is valid for any vector. When there are adjoining incremental small volumes, an arbitrary shape is considered to form a larger volume which is enclosed by surface S . Then, flux leaving incremental volume enters the adjacent incremental volume, as shown in Figure 1.16[1]. Hence, the net flux contribution for a surface integral due to interior surfaces is zero. The contributions that are nonzero occur only for the surfaces enclosing the outer surface of the volume.

Figure 1.16 Illustration of divergence theorem.

Verify the divergence theorem for vector

(1.68)

and the geometry of the pyramid given in Figure 1.17. Surfaces are defined by 1 – aob, 2 – aoc, 3 – boc, 4 – acb.

Let's calculate the right‐hand side of Eq. (1.67).

(1.69)

In (1.69), surface areas are defined by

(1.70a)

(1.70b)

(1.70c)

To be able to find the differential surface area vector for surface 4, we need to determine the normal vector for that surface. The normal vector can be found using the equation

Figure 1.17 Geometry of Example 1.4.

(1.71)

In (1.71), f is the equation that defines the surface. The equation that defines the surface is given as

(1.72)

Constants A , B , and C are obtained from the geometry at the intercept points which are (a,0,0), (0,b,0), and (0,0,c). When intercept point a is on the axis with y = 0, and z = 0 are substituted into (1.72), we obtain

(1.73)

Similarly, from intercept point (0,b,0)

(1.74)

and from intercept point (0,0,c)

(1.75)

Then, function f is defined by

(1.76)

Hence, from (1.71)and (1.76)

(1.77)

We can then calculate the surface area as

(1.78)

where

(1.79)