I am doing some transformations of vectors (wind vectors) between different cartographic transformations, for instance, from Mercator to Lambert Conical Conformal, from Plate Carree to stereographic, and so on.
I want to know if the modulus of the vector is an invariant in all those coordinate reference systems. Note that these cartographic spaces are projections from the sphere to the plane, with more or less distortion.
It seems to me that the modulus has to be an invariant. My reasoning is that the modulus of a vector is a scalar, so a tensor of order 0, and tensors are invariant in all reference frames (I am reasoning in the "relativity" style). So, for a vector (again, relativity writing)
$$V'^{\mu}=\frac{\partial x'^{\mu}}{\partial x^{\rho}}V^{\rho}$$
so the modulus is
$$V'^{\mu}V'^{\nu}g'_{\mu\nu}=\frac{\partial x'^{\mu}}{\partial x^{\rho}}V^{\rho}\frac{\partial x'^{\nu}}{\partial x^{\sigma}}V^{\sigma}g'_{\mu\nu}$$
But precisely
$$g_{\rho\sigma}=g'_{\mu\nu}\frac{\partial x'^{\mu}}{\partial x^{\rho}}\frac{\partial x'^{\nu}}{\partial x^{\sigma}}$$
So $\mod{V'}=\mod{V}$. Is this correct as a proof that the modulus of a vector is an invariant quantity between coordinate reference systems, in particular for my case of different cartographic projections?