17] L(t X and Y be sinooth vector fields on R”. Define a map IXYLC”R”) →…

17] L(t X and Y be sinooth vector fields on R”. Define a map IXYLC”R”) → C”R”) by a Show that X, Y is a derivation on Co (R”), hence represents a smooth vector field on R”. This is called the Lie bracket of X and Y lb] If we write X = Xia and Y = Ya,, then IX, Y-Zkak for some suooth functions Zk. Find an explicit expression for Zk in terms of the X’s and Y”s. Ic] It is clear that IY,X)–(X,YL Prove that This identity is called the Jacobi identity and it says that the space of smooth vector fields on R” is an infinite-dimensional Lie algebra.