Question on current equation in small loop

This is from p233 of Balanis Antenna Theory book. This is part of the derivation of vector magnetic potential of a small current loop center at origin and on xy plane. The current is stated to assumed to be constant along the loop. Attach is the scanned copy of the page in question.



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I cannot understand why the author equate the current like this. It is constant along the loop, every other book I"ve seen just use I0 as constant and take it out of the integration.

Furthermore, in equation (5-5), the xyz components are represented in cylindrical co and using

to show it"s the SOURCE coordinates. Then equation ( 5-6) shows the general transformation of Spherical coordinates to xyz coordinates. (5-6) does not have

. But then the author combine (5-5) and (5-6) into (5-3) and the result is (5-7) that contain both

together with

! This is not correct to me as this is only concerning the current along the loop, there is no involvement of any FIELD POINT in this equation. I don"t even see why the author put source and field angles into the current equation.

to me

Where all angles are source angle with ". My point is there should never be any FIELD angles in the current loop equation as this is only about the current.

Please help.

I posted this same post in Physics Forum when I don't have response here and I got a lot of responses. Basically, they all said it does not make sense at all in the point of view of classical physics and electromagnetics. I have other books that explained this very clearly using a different approach. So I guess this question is solved..........Stay away from this section. I posted this more because I am puzzled rather than getting the answer as I have it from another book and the result is the same.

This make me question I should study the advanced electromagnetics by Balanis as this part is consider quite out there.