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

(1.46)

The normal vector is normal on S at any point. The outward flux of the vector shown for a closed surface is then defined by

(1.47)

The volume integral for scalar quantities such as charge densities over the given volume is defined as

(1.48)

The volume integral given in (1.48)is a triple integral and can be solved in the corresponding coordinate system when geometry of the problem is given.

In this section, vector differential operators del, gradient, and curl are discussed. In addition, divergence and Stokes' theorem are presented.

The del operator, ∇, is used as a differential operator and can be used to find the gradient of a scalar, divergence of a vector, curl of a vector, or Laplacian of a scalar in electromagnetics. These operations can be expressed as

∇F for finding the gradient of a scalar

for finding the divergence of a vector

for finding the curl of a vector

∇2F for finding the Laplacian of a scalar

The del operator, in three different coordinate systems, can be written in the following forms

(1.49)

(1.50)

(1.51)

The gradient is used to identify the maximum change in direction and magnitude for a scalar field. Consider a scalar field ϕ ( x , y , z ). The gradient of ϕ ( x , y , z ) is defined as

(1.52)

The change in ϕ from can be found from

(1.53)

(1.54)

If we assume and s is the arc along a curve, then

(1.55)

is the unit vector tangent to the curve. Then, we can find the rate of change in ϕ for the distances with respect to vector shown in Figure 1.15as

Figure 1.15 Illustration of gradient.

(1.56)

in a cylindrical coordinate system as

(1.57)

and in a spherical coordinate system as

(1.58)

If function ϕ = x 2 y + yz is given at point (1,2−1). (a) Find its rate of change for a distance in the direction (b) What is the greatest possible rate of change with distance and in which direction does it occur at the same point?

1 Then, at point (1,2,−1)So, its rate of change is found fromwhere

2 The greatest possible rate of change at (1,2,−1) is found fromwhere at (1,2,−1).

The divergence of a vector field at a point is a measure of the net outward flux of the same vector per unit volume. The divergence of vector is defined as

(1.59)

The divergence can be represented in a cylindrical coordinate system as

(1.60)

In a spherical coordinate system, the divergence is given as

(1.61)

The divergence of a vector field gives a scalar result. In addition, the divergence of a scalar is not a valid operation.

In cylindrical coordinate system, it is given that