@tomleslie I saw the part that whatever coordinate system the PositionVector function is called in, it will be converted to cartesian coordinate system with [0,0,0] being the root.
I think (or I am unable to grasp) my problem is still there. Let me explain:
Look at the function,
If these are the components, the vector to be converted is (in spherical coordinates)
r*e_r + 0*e_th + 0*e_ph
Now, the function converts this to cartesian coordinate system with [0,0,0] being the root.
The conversion of this vector to cartesian coordinate system must be
[r*sin(th)*cos(ph), r*sin(th)*cos(ph), r*cos(th)]
by the virtue of the conversion of the unit vectors from spherical to cartesian
e_r= sin(th)*cos(ph)*e_x + sin(th)*sin(ph)*e_y + cos(th)*e_z
e_th=cos(th)*cos*ph)*e_x + cos(th)*sin(ph)*e_y - sin(th)*e_z
e_ph=-sin(ph)*e_x + cos(ph)*e_y + 0*e_z
But PositionVector function gives
0*e_x + 0*e_y + r*e_z
Basically, [0,0,r]. So, this is NOT the conversion of the components from spherical to cartesian coordinate system.
This result is the result of the conversion of r*e_r with the coordinates given as [r,th,ph] to cartesian:
This can easily be seen by
r*e_r=r*(sin(th)*cos(ph)*e_x + sin(th)*sin(ph)*e_y + cos(th)*e_z)
When r!=0, th=0, and ph=0, this becomes [0,0,r].
So, the term "components" used in the help file still dilutes me :(