I need help how to interpret the notation in the following IVP:
\begin{align} \dot x&=f(t,x), \quad \tag 1\\ x(t_o)&=x_0, \quad \tag 2 \end{align} We assume $f\in C(U,\mathbb R^n)$, where $U$ is an open subset of $\mathbb R^{n+1}$ and $(t_0,x_0)\in U$.
Q1: Does it mean I actually have the following:
Vector-valued functions: \begin{align} x&:\mathbb R \rightarrow \mathbb R^n, \quad x(t)=(x_1(t), x_2(t), \dots, x_n(t)) \tag 3 \\ \dot x&:\mathbb R \rightarrow \mathbb R^n, \quad \dot x(t)=(\dot x_1(t), \dot x_2(t), \dots, \dot x_n(t)) \tag 4 \\ f&:\mathbb R^{n+1} \rightarrow \mathbb R^n \tag 5 \end{align} We can write $f$ more precisely: $f:U \rightarrow \mathbb R^n$, $U\subset \mathbb R^{n+1}$.
Vector (constant vector): \begin{align} x_0\in \mathbb R^n, \quad x_0=(x_{0_1}, x_{0_2},\dots ,x_{0_n}) \tag 6 \end{align}
Scalars: \begin{align} t\in \mathbb R\\ t_0\in \mathbb R \end{align}
Q2: Is the explicit form of $f$ the following: \begin{align} f(t,x_1(t), x_2(t), \dots, x_n(t))= \large( &f_1(t,x_1(t), x_2(t), \dots, x_n(t)),\\ &f_2(t,x_1(t), x_2(t), \dots, x_n(t)),\\ &\qquad \qquad \qquad \vdots \\ &f_n(t,x_1(t), x_2(t), \dots, x_n(t))\large) \end{align}
Your interpretation is the way most people I know would interpret that, and I can't see how else you could interpret it.