申論題

(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 neighborhoodsof O in (the domain) R³, and a diffeomorphism φ :which maps O to O, ●open neighborhoodsof O in (the target) R³, and a diffeomorphism V which maps O to O,
 
 What follows is the diagram for your reference.