HW4 MA5311. Due February 24

Exercise 1. Let $M$ be a smooth manifold and $V: C^\infty (M,R) \to C^\infty (M,R)$ be a linear operator such that for every smooth functions $f,g: M \to R$ , $V(fg)=fVg+gVf$ . Show that there exists a vector field $U$ on $M$ such that for every smooth function $g$ , $Vg(x)=dg_x (U(x))$ .

Exercise 2. Let $B_n$ be the open unit ball in $R^n$ . Let $y$ in $B_n$ . Show that there exists a smooth vector field on $R^n$ , such that $e^V(0)=y$ and $V(x)=0$ if $x$ is not in $B_n$ .