I am currently reading Arnold's book on ordinary differential equations and I came across the definition of the integral of a differential 1-form on a closed segment of a curve. Arnold says that if $\gamma : I \rightarrow \mathbb{R}^2$ is a smooth mapping, $I \subseteq \mathbb{R}$ is an interval, $\Gamma = \gamma(I)$ and $\omega = adx + bdy$ is a differential 1-form (where $a, b$) are smooth functions of the plane, then the integral of $\omega$ is defined as $$\int_{\Gamma} \omega = \int_{I} \omega(\gamma ')du$$ However, adopting the definition I saw in another analysis textbook, the integral would be $$\int_{\Gamma} \omega = \int_{I} \left(a(\gamma(u))\gamma_1'(u) + b(\gamma(u))\gamma_2'(u)\right)du$$ where $\gamma_i, i = 1, 2$ are the components of $\gamma$.
These two definitions seem different to me... or am I misreading Arnold's definition?