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As a note, it is assumed that the magnetic current source does not exist. Taking the volume integral of both sides over volume V and surface S gives
We now apply divergence theorem as described in Section 1.3.5 for the left‐hand side of the equation in (1.108) as
From (1.108) and (1.109), we can write
Equation (1.110) is the integral form of the equation given in (1.91). Similarly, the integral form of Eq. (1.90) can be found as
(1.111)
In summary, the integral forms of Maxwell's equations are
(1.112)
(1.113)
(1.114)
(1.115)
1.5 Time Harmonic Fields
Let's assume we have a sinusoidal function that changes in position and time. This function can be expressed as
Equation (1.116) can rewritten as
In (1.117),
(1.118)
This can be applied for vectorial function as
(1.119)
The representation of the time harmonic functions in phasor form provides several advantages. They convert the time domain differential equations to frequency domain algebraic equations. This can be better understood by studying the derivative property as follows. Let's take derivative function g(r,t) with respect to time as