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Geometry guy (talk | contribs) m Avoid disambiguation page for pushforward |
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The conclusion of the theorem is that the system of ''n'' equations ''y''<sub>''i''</sub> = ''f''<sub>''j''</sub>(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) can be solved for ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> in terms of''y''<sub>1</sub>,...,''y''<sub>''n''</sub> if we restrict ''x'' and ''y'' to small enough neighborhoods of ''p''.
The inverse function theorem can be generalized to differentiable maps between [[differentiable manifold]]s. In this context the theorem states that for a differentiable map ''F'' : ''M'' → ''N'', if the [[pushforward (differential)|derivative]] of ''F'',
:(''DF'')<sub>''p''</sub> : T<sub>''p''</sub>''M'' → T<sub>''F''(''p'')</sub>''N''
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