Let $C^1(I,\mathbb{R}^n)$ be the space of $C^1$ curves. Give it the topology that satisfies that convergence of a sequence of curves $\gamma_n \to \gamma$ occurs iff these conditions hold:
a. $\gamma_n(t) \to \gamma(t)$ pointwise for every $t \in I$.
b. If $\omega$ is a $1$-form s.t. $\int_{\gamma} \omega$ is defined and $\int_{\gamma_n} \omega$ is defined for all $n \in \mathbb{N}$, then $\int_{\gamma_n} \omega \to \int_{\gamma} \omega$.
1. Is there a nice characterization of this topology in terms of modes of convergence?
I know that a sufficient conditions for $\int_{\gamma_n} \omega \to \int_{\gamma} \omega$ to hold is pointwise convergence and uniform boundness of derivatives ($|\gamma_n| \le M$ for all $n$). Note: this convergence is stronger than uniform convergence.
2. Is there an analogous treatment in the case of $n$-forms with convergence of hypersurfaces?