> [!definition] > > Let $J \subset \mathbb R$ be an open interval with $0 \in J$, $E$ be a [[Banach Space|Banach space]], and $U \subset E$ be [[Open Set|open]]. A **time-dependent [[Vector Field|vector field]]** of class $C^{p}$ is a [$C^p$](Space%20of%20Continuously%20Differentiable%20Functions) mapping $f: J \times U \to E$.