For example, I have two vectors A and B in 2D rectangular coordinates (x,y). I can calculate the scalar (dot) product as (A,B)=Ax Bx + Ay By. In polar coordinates (r,phi), it will be (A,B) = Ar Br + Aphi Bphi, since these coordinates are ortogonal and normalized. If I want to make transition what should I write?
1) Ar Br + Aphi Bphi = (Ax2+Ay2)0.5(Bx2+By2)0.5+atan(Ay/Ax)atan(By/Bx) without Lame coefficient or
2) Ar Br + Aphi Bphi =(Ax2+Ay2)0.5(Bx2+By2)0.5(1+atan(Ay/Ax)atan(By/Bx)) with Lame coefficient.
And finally, both these cases are not the same as (A,B)=Ax Bx + Ay By. How to explain this inconsistency?
The scalar product is (geometrically) defined as:
(1) (A,B) = |A| |B| cos().
In cartesian coordinates it has the well known form.
(2) (A,B) = Ax Bx + Ay By.
In polar coordinates, you can either first transfer to cartesian coordinates and then use (2) or alternatively directly use (1). Remembering that the length of the vector is given by the radial component (1) has the simple form
(3) (A,B) = Ar Br cos(Aphi - Bphi).
A more general definition would be (see Yuri's answer)
(4) (A,B) = Ai Gij Bj
(summed over i,j). Gij in (4) is a diagonal matrix in cartesian coordinates, but takes a more complicated form for different coordinate systems.
(See also: https://physics.stackexchange.com/questions/258943/dot-product-of-two-vectors-in-spherical-polar-coordinates-do-i-have-to-convert)
This:
" In polar coordinates (r,phi), it will be (A,B) = Ar Br + Aphi Bphi, since these coordinates are ortogonal and normalized. "
is not true.
You can have a look at this:
https://physics.stackexchange.com/questions/258943/dot-product-of-two-vectors-in-spherical-polar-coordinates-do-i-have-to-convert
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