I know the position vector in spherical coordinates:
$\vec r = rsin\theta cos\phi \hat r + rsin \theta sin\phi \hat \theta + rcos \theta \hat\phi$
But I do not know the derivation of it. How is the derivation of a position vector for any coordinate system. For instance, I want to write the position vector in elliptic coordinates. I know these definitions:
x = acosh(u)cos(v)
y=asinh(u)sin(v)