Implicit function theorem

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Suppose LaTeX: h:R^n \rightarrow R^m, where LaTeX: n > m, and LaTeX: h is in smooth. Further, suppose we can partition the variables, LaTeX: x = (y, z), such that LaTeX: y is m-dimensional with LaTeX: \nabla_y h(x) nonsingular at LaTeX: x^* = (y^*, z^*). Then, there exists LaTeX: \varepsilon > 0 for which there is an implicit function, LaTeX: f, on the neighborhood, LaTeX: N_{\varepsilon}(z*) = \{z: \|z-z*\| < e\} such that LaTeX: h(f(z), z)=0 for all LaTeX: z \in N_{\varepsilon}(z*). Further, LaTeX: f is smooth with LaTeX: \nabla_z f(z^*) = -\left( \nabla_y h(x^*) \right)^{-1} \nabla_z h(z^*).


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