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Let $$F:U⊂ℝ^{r+1}→ℝ^{r+1}$$ $$(s₁,s₂,...,s_{r},s_{r+1})→F(s₁,s₂,...,s_{r},s_{r+1})=(f(s₁),f(s₂),....,f(s_{r}),f(s_{r+1}))$$ be a continuously differentiable function defined from an open set $U⊂ℝ^{r+1}$ into $ℝ^{r+1}$ and invertible at a point $z∈U$. Hence the function $F$ has a continuously differentiable inverse function defined in some neighborhood $V_{z}$ of $F(z)$ to a neighborhood $U_{z}$ of $z$.

My question is : How I can find the inverse function $F⁻¹$?

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If the function $f$ has a continuously differentiable inverse then $$ F^{-1}(u_1,u_2,...,u_{r+1}) = (f^{-1}(u_1),f^{-1}(u_2),...,f^{-1}(u_{r+1}))$$

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