I'm reading the chapter about dynamical systems in J. E. Marden and T. Ratiu's Manifolds, Tensor Analysis and Applications.
After stating and proving the existence theorems for autonomous ODEs (i.e., every vector field $ X\colon U\subset E\to E $ has a unique in-some-sense-local integral curve, where $ E $ is any Banach space), the authors remark that the same theorem holds if the vector field $ X $ depends on "time" and on another parameter $ \rho $.
I'm familiar with non-autonomous vector fields. But let's suppose that $ X\colon U\times \mathbb R\times V\to E $ is a vector field defined on some open sets $ U\subset E $, $ V\subset F $, where $ E $ is the ambient Banach space and $ F $ is another Banach space of "parameters" $ \rho $.
What is an integral curve for $ X $? Is it just a curve $ \gamma\colon J\times V \to U $ defined on some open interval such that $$ \dot\gamma(t,\rho) = X(\gamma(t,\rho),t,\rho) $$ for any $ t\in J $ and $ \rho\in V $?