It is stated in a physics book that I am reading that the identity $\vec \nabla \mathrm{ln}(|\vec \theta|) = \frac{\vec \theta}{|\vec \theta|^2}$ is valid for any two dimensional vector $\vec \theta$.
This is easy enough to show in cartesian coordinates:
$\vec \nabla \mathrm{ln}(|\vec \theta|) = \frac{\partial}{\partial x}[\sqrt{x^2 + y^2}]\frac{1}{\sqrt{x^2 + y^2}} \hat i + \frac{\partial}{\partial y}[\sqrt{x^2 + y^2}]\frac{1}{\sqrt{x^2 + y^2}} \hat j$
$\vec \nabla \mathrm{ln}(|\vec \theta|) = \frac{x}{x^2 + y^2} \hat i + \frac{y}{x^2 + y^2} \hat j = \frac{\vec \theta}{|\vec \theta|^2}$
and presumably easy enough to show in other coordinate systems too, but I would like to write this in a general form that covers all coordinate systems.
I think I can write that like this using Einstein notation:
$\vec \nabla \mathrm{ln}(|\vec \theta|) = \frac{\partial}{\partial x^i} \mathrm{ln}(|\vec \theta|) \sqrt{g^{ii}} \hat e^i = \frac{\partial |\vec \theta|}{\partial x^i} \frac{\sqrt{g^{ii}}}{|\vec \theta|} \hat e^i$
But i'm not sure how to express the partial derivative of the magnitude in this notation.