*Context: I'm following the textbook of Barrett O'Neill about differential geometry. In order to calculate the connection forms of the torous I need the relation between a toroidal frame and the natural frame. *
I'm trying to construct a toridal frame system in terms of the natural frame system $\{\hat{e}_x,\,\hat{e}_y,\,\hat{e}_z\}$, where $\hat{e}_x=(1,\,0,\,0)$, etc. The same as with an spherical frame $\{\hat{e}_r,\,\hat{e}_\theta,\,\hat{e}_\phi\}$, where:
$\hat{e}_r=\cos(\phi)\cos(\theta)\hat{e}_x+\cos(\phi)\sin(\theta)\hat{e}_y+\sin(\phi)\hat{e}_z$
$\hat{e}_\theta=-\sin(\theta)\hat{e}_x+\cos(\theta)\hat{e}_y$
$\hat{e}_\phi=-\sin(\phi)\cos(\theta)\hat{e}_x-\sin(\phi)\sin(\theta)\hat{e}_y+\cos(\phi)\hat{e}_z$
My attempt was to construct parametric curves, then take its derivative to make a tangent vector and finally make it a unit vector (this seems to work fine por cylindrical and spherical frames). For example, i'll take the next curve on the tourus with its standar coordinates
$\alpha(\phi)=((R+\rho_o\,\cos(\phi))\,\cos(\theta_o),\,(R+\rho_o\,\cos(\phi))\,\sin(\theta_o),\,\rho_o\,\\\sin(\phi))$
where $R$, $\rho_o$ and $\theta_o$ are fixed (i'll just take a point on the torous with its standar coordinates and fix two of them). Then:
$\alpha'(\phi)=(-\rho_o\,\sin(\phi)\,\cos(\theta_o),\,-\rho_o\,\sin(\phi)\,\sin(\theta_o),\,\rho_o\,\cos(\phi))$
where it's easy to see that $|\alpha'(\phi)|=\rho_o$. Finally: $\hat{e}_\phi=-\sin(\phi)\cos(\theta)\hat{e}_x-\sin(\phi)\sin(\theta)\hat{e}_y+\cos(\phi)\hat{e}_z$ wich is the same as in spherical frame and this happens with the rest. So, I came to this: Toroidal frame is same as spherical frame...And this jus't feels wrong.
So, I would love and I will appreciate some help here