Let $f$ be a $C^r$ vector field in an open set $\Delta\subseteq\mathbb{R}^n$. Let $F$ be defined by
$$F(\varphi,\dot{\varphi})(t,x)=I_d+\int_0^t Df(\varphi(s,x)) \dot{\varphi}(s,x)\ ds$$ where $\varphi:I\times \mathbb{R}^n\to \overline{B_b}\subset\mathbb{R}^n$ is continuous and $\dot{\varphi}\in X=\{\dot{\varphi}\in C(I\times \mathbb{R}^n:\mathcal{L}(\mathbb{R}^n,\mathbb{R}^n))\ | \ \dot{\varphi} \ \mbox{is limited}\}$ and $I_d$ is the identity of $\mathcal{L}$.
How to prove that $F\in X$? i.e. $F(\varphi,\dot{\varphi})(t,x)$ is a linear transformtion.
I don't understand how $\int Df(\varphi(s,x)) \dot{\varphi}(s,x)\ ds$ is a linear transformation. For $F$, applied at a point $(t, x)$, to be a linear application, the integral should be...