The orientation preserving of folliation.

The orientation preserving of folliation.

A foliation $\mathcal{F}$ on a compact, oriented manifold $X$ is a
decomposition into $1-1$ immersed oriented manifolds $Y_\alpha$ (not
necessarily compact) that is locally given (preserving all orientations)
by the canonical foliation in a suitable chart at each point.
So I first define $A_\alpha: U_\alpha \to T_x(X)$ by $A_\alpha(t,x) =
(t,x,1,0)$ using the chart $\phi_\alpha$. Let $\psi_\alpha$ be a partition
of unity subordinate to $U_\alpha$ and define the vector field $$A =
\sum_{\alpha \in \mathcal{A}} \psi_\alpha A_\alpha.$$
Here $A_\alpha$ is just constant function on the vector field, hence
smooth. And $\psi_\alpha$ guarantees the existance of smooth function over
the whole vector field $A$.
Then I show $A$ has no zeros. First by partition of unity, $A$ can be
covered by finitely many subcovers. Since the folliation is orientation
preserving, $A_\alpha$s will not flip back and change the orientation to
$-1$, hence every point is $+1$ and will not cancel out. If $A_\alpha$s
have no zero, $A$ is nonvanishing as well.
Is this good?