Let $\gamma:[a;b] \to U, \gamma(t)=\big(x(t),y(t)\big)$ be a path. The integral of $\omega$ along $\gamma$ is defined as $$ \int_\gamma \omega := \int_a^b \omega_x\big(x(t),y(t)\big)\dot{x}(t)+\omega_y\big(x(t),y(t)\big)\dot{y}(t)~ dt$$ where $\dot{x}=\frac{dx}{dt}$, $\dot{y}=\frac{dy}{dt}$.
My Question: What does $\dot{x}=\frac{dx}{dt}$ exactly mean? And could I replace it with $\dot{x}=\frac{\partial x}{\partial t}$?