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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?
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