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implicit function theorem (Theorem)
ImplicitFunctionTheorem

"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 29956 times total.

Classification:
AMS MSC: 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)
Pending Errata and Addenda
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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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