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.

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