I am trying to understand this classical example of a nowhere vanishing differential one form from Tu's Introduction to Manifolds
Let $\omega = -ydx + xdy$ be restricted to $S^1 \subset \mathbb{R}^2$. Then $\omega(X)$ is nowhere vanishing for every vector field $X \in \mathfrak{X}(S^1)$.
Now I understand that if $\iota: S^1 \hookrightarrow \mathbb{R}^2$ is the inclusion map and $\frac{d}{dt} \in \mathfrak{X}(S^1)$ is the vector field $\frac{d}{dt} = c'(t) = (-\sin t, \cos t)$, then $\iota_*(\frac{d}{dt}) = -y \partial_x + x \partial_y$, and we have $\omega(\iota_*(\frac{d}{dt})) = 1$. But if $z(t) \in C^{\infty}(S^1, \mathbb{R})$ I believe that $z(t) \frac{d}{dt} \in \mathfrak{X}(S^1)$ as well. Now my question, is the pushforward $\iota_*(z(t) \frac{d}{dt}) = z(t) (-y \partial_x + x \partial_y)?$ and if it is, then couldn't I set $z(t_0) = 0$ for some $t_0$ and thus $\omega$ wouldn't be non vanishing?