Where is the Phasor Form of Maxwells Equations used?

I've studied and understand the differential and integral forms of Maxwell's equations, and understand the basic math and logic behind phasor notation for circuit analysis. Still, I'm confused as to the significance of Maxwell's equations in phasor form. They seem only to be functions of frequency, which from scattered research I gather is helpful in analyzing fields that only vary based on sinusoidal expressions with respect to time, What is the added intuition and/or computational value of the phasor form vs. the differential and integral forms (Ex: specific applications like in circuit analysis, wave propagation, etc.)? Or if any of my initial assumptions are wrong and I'm just rambling right now, what would be the correct interpretation?

First remember that Maxwell's differential equations are linear in the vector variables $$\mathbf {E,D,H,B}$$. As long as the constitutive (material) relationship between $$\mathbf {E,D}$$ and between $$\mathbf {H,B}$$ is linear complex amplitudes more than just dominate, they rule. Cases where it cannot be used are materials with strong nonlinear relationship between these variables, e.g., saturation, hysteresis, etc., the kind you find in the magnetostatics of ferromagnetism.
• do you mean the practical difference between using a complex number $z$ as $x=Re^{\mathfrak j \phi}$ or as $z=x+\mathfrak j y$? Commented May 18, 2023 at 13:40
A electromagnetic wave oscillates and it can be represented a sum of sines or cosines.However doing math with sines and cosines is really hard because the trigonometric functions are not linear.However you can do a trick.You can write the components of a EM field in their complex form using Euler's identity $$e^{ia}=cos(a)+isin(a)$$ perform all the hard math there,then take the real part of what you have found and you get your result which would be much MUCH harder if you stuck with sines and cosines.