Let answer the two questions
- Is every smooth direction field in the plane is globally rectifiable ?
No, if the integral curves are circles, it would imply that circles are topologically equivalent to lines ?
- Consider the field of planes given by the one form : $y\,dx + dz = 0$. Why there is no surface tangent to the planes of this field ?
At a point $(a,b,c)$ of $\mathbb{R}^3$, the equation of the plane of the given field is : $bx + z=0$. And ?