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implicit function theorem (Theorem)

Let $ \mathbf{f} = (f_{1}, ..., f_{n})$ be a continuously differentiable, vector-valued function mapping an open set $ E \subset \mathbb{R}^{n + m}$ into $ \mathbb{R}^{n}$. Let $ (\mathbf{a}, \mathbf{b}) = (a_{1}, ..., a_{n}, b_{1}, ..., b_{m})$ be a point in $ E$ for which $ \mathbf{f}(\mathbf{a}, \mathbf{b}) = 0$ and such that the $ n \times n$ determinant

$\displaystyle \vert D_{j} f_{i}(\mathbf{a}, \mathbf{b}) \vert \neq 0$
for $ i, j = 1, ..., n$, where $ D_j f_i = \frac{\partial f_i}{\partial x_j}$. Then there exists an $ m$-dimensional neighbourhood $ W$ of $ \mathbf{b}$ and a unique continuously differentiable function $ \mathbf{g}: W \to \mathbb{R}^{n}$ such that $ \mathbf{g}(\mathbf{b}) = \mathbf{a}$ and
$\displaystyle \mathbf{f}(\mathbf{g}(\mathbf{t}), \mathbf{t}) = 0$
for all $ \mathbf{t} \in W$.

Simplest case

When $ n = m = 1$, the theorem reduces to: Let $ F$ be a continuously differentiable, real-valued function defined on an open set $ E \subset R^{2}$ and let $ (x_{0}, y_{0})$ be a point on $ E$ for which $ F(x_{0}, y_{0}) = 0$ and such that
$\displaystyle \frac{\partial F}{\partial x} \vert _{x_{0}, y_{0}} \neq 0$
Then there exists an open interval $ I$ containing $ y_{0}$, and a unique function $ f:I \to \mathbb{R}$ which is continuously differentiable and such that $ f(y_{0}) = x_{0}$ and
$\displaystyle F(f(y), y) = 0$
for all $ y \in I$.

Note

The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.



"implicit function theorem" is owned by vypertd.
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See Also: rectification theorem


Attachments:
proof of implicit function theorem (Proof) by paolini
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Cross-references: variable, dimension, inverse function theorem, open interval, function, neighbourhood, determinant, point, open set, mapping, vector-valued function, continuously differentiable
There are 8 references to this entry.

This is version 8 of implicit function theorem, born on 2002-08-24, modified 2004-05-25.
Object id is 3347, canonical name is ImplicitFunctionTheorem.
Accessed 25871 times total.

Classification:
AMS MSC: 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)
Pending Errata and Addenda
None.
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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
[ reply | up ]
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.
[ reply | up ]
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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