What does the inverse function theorem say about the mapping F : R^3 R^3 defined by F (, , ) = (sin() sin(), sin() sin(), cos()). Fixing =1 gives a...

What does the inverse function theorem say about the mapping F : R^3 → R^3 defined by

F (ρ, θ, φ) = ρ(sin(θ) sin(φ), sin(θ) sin(φ), cos(φ)).

Fixing ρ=1 gives a mapping G:R^2 →R^3 defined by

G(θ, φ) = (sin(θ) cos(φ), sin(θ) sin(φ), cos(φ)).

What is the image of G?