Local diffeomorphism

Calculus Level 5

Let f : X = { ( ρ , ϕ , θ ) ∈ R 3 ; ρ > 0 } ⟶ R 3 f : X = \{(\rho,\phi,\theta) \in \mathbb{R}^3 \text{ ; } \rho > 0 \} \longrightarrow \mathbb{R}^3 defined as f ( ρ , ϕ , θ ) = ( ρ cos ⁡ ϕ sin ⁡ θ , ρ sin ⁡ ϕ sin ⁡ θ , ρ cos ⁡ θ ) f(\rho,\phi,\theta) = (\rho \cos \phi \sin \theta,\rho \sin \phi \sin \theta, \rho \cos \theta) Find where f f is a local diffeomorphism

Assumption.-

f f is a local diffeomorphism at a point x ∈ X ⊆ R 3 x \in X \subseteq \mathbb{R}^3 if there exists a neighborhood U U of x x such that f ∣ U f_{|U} has an inverse function f ∣ f ( U ) − 1 ∈ C 1 ( f ( U ) ) f^{-1}_{|f(U)} \in \mathcal{C}^1 (f(U))

{ ( ρ , ϕ , θ ) ∈ X ; θ ≠ π / 2 + k π ( k ∈ Z ) } \{(\rho,\phi,\theta) \in X \text{ ; }\space \theta \neq \pi/2 + k\pi \space(k\in\mathbb{Z})\} { ( ρ , ϕ , θ ) ∈ X ; ϕ ≠ π / 2 + k π ( k ∈ Z ) } \{(\rho,\phi,\theta)\in X\text{ ; }\space \phi \neq \pi/2 + k\pi \space (k \in \mathbb{Z})\} { ( ρ , ϕ , θ ) ∈ X ; θ ≠ k π ( k ∈ Z ) } \{(\rho,\phi,\theta) \in X \text{ ; } \space \theta \neq k\pi \space (k \in \mathbb{Z})\} { ( ρ , ϕ , θ ) ∈ X ; ϕ ≠ k π ( k ∈ Z ) } \{(\rho, \phi, \theta) \in X \text{ ; } \space \phi \neq k\pi \space (k \in \mathbb{Z})\}

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

Applying Inverse function theorem and its reciprocal, f f will be a local diffeomorphism at the point a a if the determinant of Jacobian matrix of f f at a a is different to 0 0 , i. e, if det (Jf(a)) ≠ 0 \text{ det (Jf(a)) } \neq 0 det (Jf( ρ , ϕ , θ ) ) = ∣ cos ⁡ ϕ sin ⁡ θ − ρ sin ⁡ ϕ sin ⁡ θ ρ cos ⁡ ϕ cos ⁡ θ sin ⁡ ϕ sin ⁡ θ ρ cos ⁡ ϕ sin ⁡ θ ρ sin ⁡ ϕ cos ⁡ θ cos ⁡ θ 0 − ρ sin ⁡ θ ∣ = − ρ 2 sin ⁡ θ . \text{ det (Jf(}\rho,\phi, \theta)) = \left| \begin{matrix} \cos \phi \sin \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos\theta\\ \sin \phi \sin \theta & \rho \cos \phi \sin \theta & \rho \sin \phi \cos \theta \\ \cos \theta & 0 & -\rho \sin \theta \end{matrix} \right| = - \rho^2 \sin \theta. Therefore, f f is a local diffeomorphism at { ( ρ , ϕ , θ ) ∈ X ; θ ≠ k π ( k ∈ Z ) } \{(\rho, \phi, \theta) \in X \text{ ; } \theta \neq k\pi \space (k\in \mathbb{Z})\}

Other example .-

Let y = x 3 , ∀ x ∈ R y = x^3, \space \forall x \in \mathbb{R} . This function is a homeomorphism,i.e, it is a continuous function with an inverse continuous function x = y 3 x = \sqrt[3]{y} , and hence a local homeomorphism ( y y has a local inverse continuous function ∀ x ∈ R \forall x \in \mathbb{R} ). Furthemore, y y is an infinite differentiable function, but it is only a local diffeomorphism where y ′ ≠ 0 y' \neq 0 because its inverse function x = y 3 x = \sqrt[3]{y} has a finite derivative function in R − { 0 } \mathbb{R} - \{0\} ,i.e, x = y 3 x = \sqrt[3]{y} has a infinite slope at y = 0 y = 0 .

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