Transversality on manifolds Let be $M^{n}$ a differentiable manifold and $N_{1}$,$N_{2}$ two submanifolds of $M$. We say that $N_{1}$ and $N_{2}$ intersect transversely if $N_{1} \cap N_{2} = \emptyset$ or for all $p \in N_{1} \cap N_{1} $, then $T_{p}N_{1}+ T_{p}N_{2} = T_{p}M$. Show that $S^{1}$ is transversal to any straight line in $\mathbb{R^{2}}$ passing thought the origin.
My idea is to parameterize the two submanifolds, for example for the line denote 
$l$ = $ \{ y- ax : a \in  \mathbb{R} \}$ and $\frac{{\partial l}}{{\partial y}} = 1$ and $\frac{{\partial l}}{{\partial x}} = a$ and then consider the spam of the derivates, and do the same for $S^{1}$.
I appreciate your ideas,
Thank you
 A: Hints and ideas: 
The geometric point of view: the tangent space of the line at a point $p$ should (and is) be a vector parallel to the line. By hypothesis the line and the circle intersect in a least one point $p.$ You can calculate the tangent spaces of the circle $S^1$ and the line $l$ at $p$ directly and show that they span $\mathbb R^2.$ One way to do this is to notice that a rotation of the circle through $\theta$ radians is a diffeomorphism, so without loss of generality, assume $p=(1,0)$. Then, $T_pS^1=\frac{\partial}{\partial y}$, but the line passes through the origin and so can not be tangent to the circle. To do the computation, you might use the canonical isomorphism that sends $a\frac{\partial}{\partial x}+b\frac{\partial }{\partial y }\in T_pl$ to $(a,b)\in \mathbb R^2$ and similarly for $T_pS^1$ and then observe that if $p$ is a point on the circle, the tangent space can be computed using the dot product. On the other hand, the tangent space of the line at $p$, as we mentioned, should be a vector parallel to the line and so is not tangent to the circle. 
Here is a hint for a slick way if you know about differentials: with $f(x,y)=y-mx+b$ and $g(x,y)=x^2-y^2-1$, observe that $T_pl=\text{ker}\ f_*$ and $T_pS^1=\text{ker}\ g_*$. 
Finally, if you know about tangent vectors as differentials of curves, you can compute the tangent spaces using them. 
In any case, the basic idea is that you can actually calculate the tangent spaces and show they span $\mathbb R^2.$
