Let $\omega = \mathrm{d}y-A(t,y)\,\mathrm{d}t$ and $\gamma(t)=(t,y(t))^t$ be given. I want to compute $\gamma^*\omega(t)=\omega_{\gamma(t)}(\dot\gamma(t))$.
Now the actual problem I have, is to understand the notation of the pullback. In my script the definition of the pullback of a $k$-form by $\Phi$
$\alpha=\sum_{1\le l_1<...<l_k\le N}\alpha^{l_1...l_k}\,\mathrm{d}y_{l_1}\wedge ... \wedge\mathrm{d}y_{l_k} $
on an $N$-dimensional manifold is given by
$\Phi^*\alpha=\sum_{1\le l_1<...<l_k\le N}(\Phi^*\alpha^{l_1...l_k})\,\mathrm{d}\Phi_{l_1}\wedge ... \wedge\mathrm{d}\Phi_{l_k} $
which is pretty self-explanatory. I just don't understand what is meant by the first pullback notation given.
When I use the definition of a $k$-form I get
$\gamma^*\omega(t)=\dot \gamma (t)\,\mathrm{d}t - A(t,\gamma(t))\,\mathrm{d}t$
But what is meant by $\omega_{\gamma(t)}(\dot\gamma(t))$?