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implicit function theorem
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(Theorem)
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Theorem Let $\Omega$ be an open subset of $\mathbb{R}^n \times \mathbb{R}^m$ and let $f\in C^1(\Omega,\mathbb{R}^m)$ Let $(x_0,y_0)\in \Omega \subset \mathbb{R}^n\times\mathbb{R}^m$ If the matrix $D_y f(x_0,y_0)$ defined by $$ D_y f(x_0,y_0) = \left( \frac{\partial f_j}{\partial y_k}(x_0,y_0)\right)_{j,k} \quad j=1,\ldots,m\quad k=1,\ldots,m $$ is invertible, then there exists a neighborhood $U\subset
\mathbb{R}^n$ of $x_0$ and a function $g \in C^1(U,\mathbb{R}^m)$ such that $$ f(x,g(x)) = f(x_0,y_0) \qquad \forall x \in U. $$
Moreover $$ Dg(x) = - (D_y f(x,g(x))) ^ {-1} D_x f(x,g(x)). $$
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"implicit function theorem" is owned by azdbacks4234. [ full author list (2) | owner history (1) ]
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Cross-references: function, neighborhood, invertible, matrix, open subset
There are 9 references to this entry.
This is version 9 of implicit function theorem, born on 2002-08-24, modified 2008-12-13.
Object id is 3347, canonical name is ImplicitFunctionTheorem.
Accessed 29112 times total.
Classification:
| AMS MSC: | 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables) |
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Pending Errata and Addenda
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