Читать книгу RF/Microwave Engineering and Applications in Energy Systems - Abdullah Eroglu - Страница 35

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 1.15 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)