Let $\Delta$ a open subset of $\mathbb{R}^n$. A vector field of class $C^k$ in $\Delta$ is an application $X:\Delta\to\mathbb{R}^n$ of class $C^k$. Defining $F$ as
$$F(\phi(t,x))=\int_{0}^{t}X(\phi(s,x))\ ds$$ where $\phi: \mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$.
How do I integrate a vector field?
It also made me think about the solutions of an ODE. An application $\varphi:\mathbb{R}\to\mathbb{R}^n$ is a solution of the Cauchy Problem $x'=f(t,x), x(t_0)=x_0$ if
$$\varphi(t)=x_0 + \int_{t_0}^{t} f(s,\varphi(s))\ ds$$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$. Again we are integrating something that has $\mathbb{R}^n$ as an image. In this case, I think that to integrate, you must integrate each component of $f$? Is the same for the vector field?