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implicit function theorem (Theorem)
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 $$ Dg(x) = - (D_y f(x,g(x))) ^ {-1} D_x f(x,g(x)). $$




"implicit function theorem" is owned by azdbacks4234. [ full author list (2) | owner history (1) ]
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See Also: rectification theorem, derivative as parameter for solving differential equations


Attachments:
proof of implicit function theorem (Proof) by paolini
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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 MSC26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)

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Claim D_yf(x,g(x)) invertible by kfgauss70 on 2009-01-04 14:28:22
Immediately after claiming f(x,g(x))=f(x0,y0), should one claim also claim that D_y f(x,g(x)) is invertible for all x \in U (as it actually is)? In this way the expression of the derivative D_x g(x) can be given.
Regards.
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Bigger Ideas by pzadunaisky on 2006-07-10 08:40:34
This theorem can be generalized to arbitrary finite dimensional Banach Spaces... It'd be great to have that version too
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What about mapping from less to more dimensions? by spuzzzzzzz on 2006-02-21 18:15:26
There is a version of this for a function f : R^n -> R^{n+m} also. I have always called both versions the implicit function theorem -- maybe there is a different name for it? But if there isn't, it would be nice to see both versions here.
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Meaning of D_j by lha on 2004-05-24 22:02:15
Could you please define D_j? I assume it is the derivative with resepect to the jth component of the argument, but it would be nice to have it defined or cross-referenced. Thanks, Lachlan
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