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

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 1.15 as

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)

      Example 1.2 Gradient

      If function ϕ = x²y + 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?

      Solution

      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).

      1.3.3 Divergence

      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.

      Example 1.3 Divergence

      In cylindrical coordinate system, it is given that

      Calculate .

      Solution

      1.3.4 Curl

      The curl of a vector is used to identify how much the vector v curls