Orientation of manifold given by external normal field 

Consider the unit sphere $S^1$ of $\mathbb{R}^2$. This is a 1-dimensional manifold. And an orientation $\sigma$ of $S^1$ is given by the orientated atlas $\left\{\phi_1,\phi_2\right\}$ with the maps
    $$
\phi_1\colon (-\pi,\pi)\to S^1\setminus (-1,0), t\mapsto (\cos t,\sin t)\\
\phi_2\colon (0,2\pi)\to S^1\setminus (1,0), t\mapsto (\cos t,\sin t)
$$
    Moreover an orientation of $S^1$ is given by the external normal field
    $$
\nu\colon S^1\to\mathbb{R}^2.
$$
    Show that the orientation given by the external normal field is identical with the orientation $\sigma$ above.


Do I have to show that the external normal field $\nu$ is positive orientated related to $\sigma$? Or something different?

I think you want to show that, if you pick a point $p\in S^{1}\subset\mathbb{R}^{2}$, then  $\nu(p),v(p)$ is positively oriented in $T_{p}\mathbb{R}^2$, where $v(p)$ is the oriented basis for $T_{p}S^{1}$ you get from the atlas $\{\phi_1,\phi_2\}$. Is this what you meant by "the external normal field $\nu$ is positive oriented related to $\sigma$"?
