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

Figure 1.14 Illustration of surface integral.

      1.2.5 Surface Integral

Consider the region given in Figure 1.14. The surface integral is then defined as the flux of the vector through surface S and expressed as

      (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)

      1.2.6 Volume Integral

      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.

1.3 Vector Operators and Theorems

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

      1.3.1 Del Operator

      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)

      1.3.2 Gradient

      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)