Functional Derivative of Long Range Dipole Interaction Energy

[Poligon]

Homework Statement

Dear all, Good day.

I am currently working on the phase field modeling of ferroelectrics. For this reason, I need to find functional derivative of an expression as presented in attached picture 1.

Then in picture 2, it shows the final form of equation that I am supposed to arrive at.

Homework Equations

I am stuck at line P in attached picture 5. Looking at the equation in line P, if I can include only the exponentials in the integral d^3g, then I can change them to 2*Dirac's delta, and the multiplication of dirac's delta with P' leads to P and performing Fourier transform, I can get the form as shown in attached picture 2. However, I doubt if I can do so because n is a function of g. Can anyone please help me on this? I have been breaking my head trying to figure this out for two days.

The Attempt at a Solution

My attempt on the question is as attached in picture 3, 4 and 5.

As shown in attached picture 3, line A, B, C, D is how I started. I bring the unit vector n into the integral of P because it is not a function of r, then I separate exponential into cos real part and sin imaginary part. This is to get rid of the magnitude operator.

Then in attached picture 4 and 5, from line E to P is how I am trying to find the functional derivative of F dip wrt P1. I am stuck at line P.

 

[lippyka]

If I can include only the exponentials in d^3g, then I can change them to 2*Dirac's delta, and the multiplication of dirac's delta with P' leads to P and performing Fourier transform, I can get the form as shown in attached picture 2. However, I doubt if I can do so because n is a function of g. Can anyone please help me on this? I have been breaking my head trying to figure this out for two days.