Transforming integral to spherical coordinates

[Nkj]

Hello everyone,
I am studying a book about derivation of Green's function for three dimensional wave equation. In a part of this book it is stated as follows:



I don't understand how this transformation is done. What does it mean geometrically that

in such a manner that the polar axis ([imath]\theta=0[/imath]) lies along the half line from the origin to ([imath]x-\zeta[/imath], [imath]y-\eta[/imath], [imath]z-\xi[/imath])

and how [imath]x-\zeta=0[/imath], [imath]y-\eta=0[/imath] and [imath]z-\xi=R[/imath] are obtained?

 

[blamocur]

I don't quite understand that one particular line ("... so that [imath]x-\xi = 0[/imath]..."), but the rest makes sense to me. I am guessing that [imath]R[/imath] is the length of vector [imath](x-\xi, y-\eta, z-\zeta)[/imath], i.e.,
[math]R = \sqrt{(x-\xi)^2 + (y-\eta)^2 + (z-\zeta)^2}[/math]We also know that [imath]\kappa[/imath] is the length of vector [imath](k,l,m)[/imath], and since [imath]\theta[/imath] is the angle between those two vectors we get the following equality for their dot products:
[math]k(x-\xi)+l(y-\eta)+m(z-\zeta) = \kappa R \cos ( \theta)[/math]To complete the transition from 4.5.6 to 4.5.7 we remember the expression for volume differential in spherical coordinates:
[math]dk\,dl\,dm = \kappa^2 \sin (\theta) \kappa\, d\phi\, d\theta\, d\kappa[/math]Hope this helps, but let us know if it doesn't.

 

[blamocur]

I hope these additional details make my previous post somewhat more rigorous, but I also hope it doesn't make the whole thing more confusing. So here goes: using, as before,
[math]R = \sqrt{(x-\xi)^2 + (y-\eta)^2 + (z-\zeta)^2}[/math]define the ``new vertical'' unit vector:
[math]\mathbf e_1 = \left(\frac{x-\xi}{R}, \frac{y-\eta}{R}, \frac{z-\zeta}{R} \right)[/math]and then complete the orthonormal basis with vectors [imath]\mathbf e_2, \mathbf e_3[/imath], which don't need to be defined as long as the basis [imath](\mathbf e_1,\mathbf e_2, \mathbf e_3)[/imath] is orthonormal. (One can use Gram-Schmidt process to get exact expressions for the basis, but those exact expressions do not matter).

If [imath]\mathbf f = (k,l,m)[/imath] then [imath]|\mathbf f| = \kappa[/imath], and [imath]\mathbf f[/imath] can be represented through [imath]\mathbf e_1,\mathbf e_2,\mathbf e_3[/imath] as follows:
[math]\mathbf f = \kappa\left(\mathbf e_1 \cos\theta + \mathbf e_2 \sin\theta \cos\phi + \mathbf e_3 \sin\theta \sin\phi\right)[/math]The corresponding dot product then becomes:
[math]k(x-\xi)+l(y-\eta)+m(z-\zeta) = \langle \mathbf f, R \mathbf e_1\rangle = R \kappa \cos\theta[/math]

 

[blamocur]

and how x−ζ=0x-\zeta=0x−ζ=0, y−η=0y-\eta=0y−η=0 and z−ξ=Rz-\xi=Rz−ξ=R are obtained?

On third thought, I think what they mean is that vector [imath]x-\xi, y-\eta, z-\zeta[/imath] gets transformed to [imath]0, 0, R[/imath] in the new basis from post #3.

 

[Nkj]

On third thought, I think what they mean is that vector [imath]x-\xi, y-\eta, z-\zeta[/imath] gets transformed to [imath]0, 0, R[/imath] in the new basis from post #3.

Thank you for your useful answers. They really helped me.