(4) [20 分] Let F be a smooth map from R3 to R3. Denote by (x1, x2, x3) the coordinate for the domain R3. Denote the origin (0,0,0) by 0. Suppose that

Prove that there exist
●open neighborhoods
of O in (the domain) R3, and a diffeomorphism φ :
which maps O to O, ●open neighborhoods
of O in (the target) R3, and a diffeomorphism
V which maps O to O,

What follows is the diagram for your reference.
