Fourier transform in curvilinear coordinates

[FredFred]

Hello, can you suggest a good book reference to find this:

I have a 3D coordinate system where the axis are:
1) locally tangential to a spiral in the equatorial plane;
2) perpendicular to 1 in the equatorial plane;
3) colatitude.

The direction of axes 1 and 2 changes with position.
I need to do the Fourier transform of a quantity and it's convenient that the 3D scalar product in the exponential exp(i k x) has one axis parallel to axis 1.
How the scalar product in exp(i k x) for the Fourier transform is expressed in the coordinates 1-2-3?
How the volume element dk_x dk_y dk_z reads in the coordiantes 1-2-3?
Thanks

 

[jvicens]

Hello,

Thank you for your inquiry. I would recommend the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. This book covers a variety of mathematical methods used in physics, including coordinate systems and Fourier transforms.

In particular, Chapter 9 covers Fourier transforms and their applications, including the use of different coordinate systems. It also includes examples and exercises that can help you understand how to express scalar products and volume elements in different coordinate systems.

I hope this helps and best of luck with your research!