Namespaces
Variants
Actions

Difference between revisions of "Implicit function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(added reference [a2] for the classical implicit function theorem, since the theorem cannot be found stated in this form in [2], [3], [5], or [a1]; [4] does not exist in an English-language translation, and [1] is out of print and difficult to find)
m (OldImage template added)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503101.png" /> given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i0503109.png" /> are certain sets, i.e. a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031011.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031015.png" /> are topological spaces and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031016.png" /> for some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031017.png" />, then under certain conditions the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031018.png" /> is uniquely solvable in one of the variables in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031019.png" />. Properties of the solution of this equation are described by implicit-function theorems.
+
<!--
 +
i0503101.png
 +
$#A+1 = 221 n = 0
 +
$#C+1 = 221 : ~/encyclopedia/old_files/data/I050/I.0500310 Implicit function
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The simplest implicit-function theorem is as follows. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031021.png" /> are subsets of the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031024.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031025.png" /> be an interior point of the plane set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031026.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031027.png" /> is continuous in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031028.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031029.png" /> and if there are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031030.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031032.png" />, for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031033.png" />, is strictly monotone on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031034.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031035.png" />, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031036.png" /> such that there is a unique function
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031037.png" /></td> </tr></table>
+
A function  $  f : E \rightarrow Y $
 +
given by an equation  $  F ( x , y ) = z _ {0} $,
 +
where  $  F: X \times Y \rightarrow Z $,
 +
$  x \in X $,
 +
$  y \in Y $,
 +
$  E \subset  X $,
 +
and  $  X $,
 +
$  Y $
 +
and  $  Z $
 +
are certain sets, i.e. a function  $  f $
 +
such that  $  F ( x , f ( x) ) = z _ {0} $
 +
for any  $  x \in E $.
 +
If  $  X $,
 +
$  Y $
 +
and  $  Z $
 +
are topological spaces and if  $  F ( x _ {0} , y _ {0} ) = z _ {0} $
 +
for some point  $  ( x _ {0} , y _ {0} ) \in X \times Y $,
 +
then under certain conditions the equation  $  F ( x , y ) = z _ {0} $
 +
is uniquely solvable in one of the variables in some neighbourhood of  $  ( x _ {0} , y _ {0} ) $.  
 +
Properties of the solution of this equation are described by implicit-function theorems.
  
for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031038.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031039.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031040.png" /> is continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031041.png" />.
+
The simplest implicit-function theorem is as follows. Suppose that  $  X $
 +
and  $  Y $
 +
are subsets of the real line  $  \mathbf R $,
 +
let  $  x _ {0} \in X $,
 +
$  y _ {0} \in Y $,
 +
and let  $  ( x _ {0} , y _ {0} ) $
 +
be an interior point of the plane set  $  X \times Y $;
 +
if  $  F $
 +
is continuous in some neighbourhood of  $  ( x _ {0} , y _ {0} ) $,
 +
if  $  F ( x _ {0} , y _ {0} ) = 0 $
 +
and if there are a  $  \delta > 0 $
 +
and an  $  \epsilon > 0 $
 +
such that  $  F ( x , y ) $,
 +
for any fixed  $  x \in ( x _ {0} - \delta , x _ {0} + \delta ) $,
 +
is strictly monotone on  $  ( y _ {0} - \epsilon , y _ {0} + \epsilon ) $
 +
as a function of  $  y $,
 +
then there is a  $  \delta _ {0} > 0 $
 +
such that there is a unique function
 +
 
 +
$$
 +
f :  ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} )  \rightarrow \
 +
( y _ {0} - \epsilon , y _ {0} + \epsilon )
 +
$$
 +
 
 +
for which $  F ( x , f ( x) ) = 0 $
 +
for all $  x \in ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} ) $;  
 +
moreover, $  f $
 +
is continuous and $  f ( x _ {0} ) = y _ {0} $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050310a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050310a.gif" />
Line 11: Line 65:
 
Figure: i050310a
 
Figure: i050310a
  
The hypotheses of this theorem are satisfied if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031042.png" /> is continuous in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031043.png" />, if the partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031044.png" /> exists and is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031045.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031046.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031047.png" />. If in addition the partial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031048.png" /> exists and is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031049.png" />, then the implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031050.png" /> is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031051.png" />, and
+
The hypotheses of this theorem are satisfied if $  F $
 +
is continuous in a neighbourhood of $  ( x _ {0} , y _ {0} ) $,  
 +
if the partial derivative $  F _ {y} $
 +
exists and is continuous at $  ( x _ {0} , y _ {0} ) $,  
 +
if $  F ( x _ {0} , y _ {0} ) = 0 $,  
 +
and if $  F _ {y} ( x _ {0} , y _ {0} ) \neq 0 $.  
 +
If in addition the partial derivative $  F _ {x} $
 +
exists and is continuous at $  ( x _ {0} , y _ {0} ) $,  
 +
then the implicit function $  f $
 +
is differentiable at $  x _ {0} $,
 +
and
 +
 
 +
$$
 +
 
 +
\frac{d f ( x _ {0} ) }{dx}
 +
  = -
 +
 
 +
\frac{F _ {x} ( x _ {0} , y _ {0} ) }{F _ {y} ( x _ {0} , y _ {0} ) }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031052.png" /></td> </tr></table>
+
This theorem has been generalized to the case of a system of equations, that is, when  $  F $
 +
is a vector function. Let  $  \mathbf R  ^ {n} $
 +
and  $  \mathbf R  ^ {m} $
 +
be  $  n $-
 +
and  $  m $-
 +
dimensional Euclidean spaces with fixed coordinate systems and points  $  x = ( x _ {1} \dots x _ {n} ) $
 +
and  $  y = ( y _ {1} \dots y _ {m} ) $,
 +
respectively. Suppose that  $  F $
 +
maps a certain neighbourhood  $  W $
 +
of  $  ( x _ {0} , y _ {0} ) \in \mathbf R  ^ {n} \times \mathbf R  ^ {m} $(
 +
$  x _ {0} \in \mathbf R  ^ {n} $,
 +
$  y _ {0} \in \mathbf R  ^ {m} $)
 +
into  $  \mathbf R  ^ {m} $
 +
and that  $  F _ {i} $,
 +
$  i = 1 \dots m $,
 +
are the coordinate functions (of the  $  n + m $
 +
variables  $  x _ {1} \dots x _ {n} , y _ {1} \dots y _ {m} $)
 +
of  $  F $,
 +
that is,  $  F = ( F _ {1} \dots F _ {m} ) $.
 +
If  $  F $
 +
is differentiable on  $  W $,
 +
if  $  F( x _ {0} , y _ {0} ) = 0 $
 +
and if the Jacobian
  
This theorem has been generalized to the case of a system of equations, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031053.png" /> is a vector function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031055.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031056.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031057.png" />-dimensional Euclidean spaces with fixed coordinate systems and points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031059.png" />, respectively. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031060.png" /> maps a certain neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031062.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031064.png" />) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031065.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031067.png" />, are the coordinate functions (of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031068.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031069.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031070.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031071.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031072.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031073.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031074.png" /> and if the Jacobian
+
$$
 +
\left .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031075.png" /></td> </tr></table>
+
\frac{\partial  ( F _ {1} \dots F _ {m} ) }{\partial  ( y _ {1} \dots y _ {m} ) }
  
then there are neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031079.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031080.png" />, and a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031082.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031083.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031085.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031086.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031087.png" />, then the explicit expression for the partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031090.png" />, can be found from the system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031091.png" /> linear equations in these derivatives:
+
\right | _ {( x _ {0, y _ {0} ) } \
 +
\neq  0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031092.png" /></td> </tr></table>
+
then there are neighbourhoods  $  U $
 +
and  $  V $
 +
of  $  x _ {0} \in \mathbf R  ^ {n} $
 +
and  $  y _ {0} \in \mathbf R  ^ {m} $,
 +
respectively,  $  U \times V \subset  W $,
 +
and a unique mapping  $  f : U \rightarrow V $
 +
such that  $  F ( x , f ( x) ) = 0 \in \mathbf R  ^ {m} $
 +
for all  $  x \in U $.
 +
Here  $  f ( x _ {0} ) = y _ {0} $,
 +
$  f $
 +
is differentiable on  $  U $,
 +
and if  $  f = ( f _ {1} \dots f _ {m} ) $,
 +
then the explicit expression for the partial derivatives  $  \partial  f _ {j} / \partial  x _ {i} $,
 +
$  i = 1 \dots n $,
 +
$  j = 1 \dots m $,
 +
can be found from the system of  $  m $
 +
linear equations in these derivatives:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031094.png" /> is fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031095.png" />. Sometimes the main assertion of the theorem is stated as follows: There are neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031096.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031097.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i05031099.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310100.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310102.png" />, and a unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310106.png" />. In other words, the conditions
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310107.png" /></td> </tr></table>
+
\frac{\partial  F _ {k} }{\partial  x _ {i} }
 +
+
 +
\sum_{j=}^ { m }
  
are equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310109.png" />. In this case one says that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310110.png" /> is uniquely solvable in the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310112.png" />.
+
\frac{\partial  F _ {k} }{\partial  y _ {j} }
  
The classical implicit-function theorem thus stated generalizes to the case of more general spaces in the following manner. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310113.png" /> be a topological space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310115.png" /> be affine normed spaces over the field of real or complex numbers, that is, affine spaces over the relevant field to which are associated normed vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310118.png" /> being complete, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310119.png" /> be the set of continuous linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310120.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310121.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310122.png" /> be an open set in the product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310126.png" />.
+
\frac{\partial  f _ {j} }{\partial  x _ {i} }
 +
  = 0 ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310127.png" /> be a continuous mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310128.png" />. If for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310130.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310131.png" /> has a partial [[Fréchet derivative|Fréchet derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310132.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310133.png" /> is a continuous mapping and if the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310134.png" /> has a continuous inverse linear mapping (that is, it is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310135.png" />), then there exist open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310136.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310139.png" />, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310140.png" /> there is a unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310141.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310142.png" />, satisfying the equations
+
$  k = 1 \dots m $,
 +
i $
 +
is fixed $  ( i = 1 \dots n ) $.  
 +
Sometimes the main assertion of the theorem is stated as follows: There are neighbourhoods  $  U $
 +
of  $  x _ {0} $
 +
in  $  \mathbf R  ^ {n} $
 +
and $  W _ {0} $
 +
of  $  ( x _ {0} , y _ {0} ) $
 +
in  $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} $,  
 +
$  W _ {0} \subset  W $,
 +
and a unique mapping  $  f : U \rightarrow \mathbf R  ^ {m} $
 +
such that  $  ( x , f ( x) ) \in W _ {0} $
 +
and  $  F ( x , f ( x) ) = 0 $
 +
for all  $  x \in U $.  
 +
In other words, the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310143.png" /></td> </tr></table>
+
$$
 +
( x , y )  \in  W _ {0} ,\ \
 +
F ( x , y )  = 0
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310144.png" /> thus defined is a continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310145.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310146.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310147.png" />.
+
are equivalent to  $  x \in U $,
 +
$  y = f ( x) $.
 +
In this case one says that the equation  $  F ( x , y ) = 0 $
 +
is uniquely solvable in the neighbourhood  $  W _ {0} $
 +
of  $  ( x _ {0} , y _ {0} ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310148.png" /> is also an affine normed space, then under certain conditions the implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310149.png" /> which satisfies the equation
+
The classical implicit-function theorem thus stated generalizes to the case of more general spaces in the following manner. Let  $  X $
 +
be a topological space, let  $  Y $
 +
and  $  Z $
 +
be affine normed spaces over the field of real or complex numbers, that is, affine spaces over the relevant field to which are associated normed vector spaces  $  \mathbf Y $
 +
and  $  \mathbf Z $,
 +
$  \mathbf Y $
 +
being complete, let  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $
 +
be the set of continuous linear mappings from  $  \mathbf Y $
 +
into  $  \mathbf Z $,
 +
and let  $  W $
 +
be an open set in the product space $  X \times Y $,
 +
$  ( x _ {0} , y _ {0} ) \in W $,
 +
$  x _ {0} \in X $,  
 +
$  y _ {0} \in Y $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310150.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Let  $  F : W \rightarrow Z $
 +
be a continuous mapping and  $  F ( x _ {0} , y _ {0} ) = z _ {0} $.
 +
If for every fixed  $  x $
 +
and  $  ( x , y ) \in W $
 +
the mapping  $  F $
 +
has a partial [[Fréchet derivative|Fréchet derivative]]  $  F _ {y} \in {\mathcal L} ( \mathbf Y , \mathbf Z ) $,
 +
if  $  F _ {y} ( x , y ) : W \rightarrow {\mathcal L} ( \mathbf Y , \mathbf Z ) $
 +
is a continuous mapping and if the linear mapping  $  F _ {y} ( x _ {0} , y _ {0} ) : \mathbf Y \rightarrow \mathbf Z $
 +
has a continuous inverse linear mapping (that is, it is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $),
 +
then there exist open sets  $  U \subset  X $
 +
and  $  V \subset  Y $,
 +
$  x _ {0} \in U $,
 +
$  y _ {0} \in V $,
 +
such that for any  $  x \in U $
 +
there is a unique element  $  y \in V $,
 +
denoted by  $  y = f ( x) $,
 +
satisfying the equations
  
is also differentiable. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310152.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310153.png" /> be affine normed spaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310154.png" /> be an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310155.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310158.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310159.png" /> be the implicit mapping given by (1), taking a certain neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310160.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310161.png" /> into an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310162.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310164.png" />. Thus, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310165.png" />,
+
$$
 +
f ( x)  \in  V \ \
 +
\textrm{ and } \  F ( x , f ( x) ) = z _ {0} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310166.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
The function  $  y = f ( x) $
 +
thus defined is a continuous mapping from  $  U $
 +
into  $  V $,
 +
and  $  y _ {0} = f ( x _ {0} ) $.
  
Suppose also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310167.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310168.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310169.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310170.png" /> is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310171.png" />, if its partial Fréchet derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310173.png" /> are continuous linear operators taking the vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310174.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310175.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310177.png" /> into the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310178.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310179.png" />, and if the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310180.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310181.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310182.png" /> is differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310183.png" /> and its Fréchet derivative is given by
+
If $  X $
 +
is also an affine normed space, then under certain conditions the implicit function  $  f : x \mapsto y $
 +
which satisfies the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310184.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
F ( x , y )  = z _ {0}  $$
  
This is obtained as a result of formally differentiating (2):
+
is also differentiable. Namely, let  $  X $,
 +
$  Y $
 +
and  $  Z $
 +
be affine normed spaces, let  $  W $
 +
be an open set in  $  X \times Y $,
 +
let  $  F :  W \rightarrow Z $,
 +
$  F ( x _ {0} , y _ {0} ) = z _ {0} $,
 +
$  x _ {0} \in Y $,
 +
and let  $  f $
 +
be the implicit mapping given by (1), taking a certain neighbourhood  $  U $
 +
of $  x _ {0} $
 +
into an open subset  $  V $
 +
of  $  Y $,
 +
$  U \times V \subset  W $.
 +
Thus, for all  $  x \in U $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310185.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
f ( x)  \in  V ,\ \
 +
F ( x , f ( x) )  = z _ {0} .
 +
$$
  
and multiplying this equality on the left by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310186.png" />.
+
Suppose also that  $  f $
 +
is continuous at  $  x _ {0} $
 +
and that  $  f ( x _ {0} ) = y _ {0} $.  
 +
If  $  F $
 +
is differentiable at  $  ( x _ {0} , y _ {0} ) $,
 +
if its partial Fréchet derivatives  $  F _ {x} ( x _ {0} , y _ {0} ) $
 +
and  $  F _ {y} ( x _ {0} , y _ {0} ) $
 +
are continuous linear operators taking the vector spaces  $  \mathbf X $
 +
and  $  \mathbf Y $
 +
associated with  $  X $
 +
and  $  Y $
 +
into the vector space  $  \mathbf Z $
 +
associated with  $  Z $,
 +
and if the operator  $  F _ {y} ( x _ {0} , y _ {0} ) $
 +
is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $,
 +
then  $  f $
 +
is differentiable at  $  x _ {0} $
 +
and its Fréchet derivative is given by
  
If in addition the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310187.png" /> is continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310188.png" />, if the implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310189.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310191.png" />, and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310192.png" /> the partial Fréchet derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310193.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310194.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310195.png" /> is a continuously-differentiable mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310196.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310197.png" />.
+
$$
 +
f ^ { \prime } ( x _ {0} )  = \
 +
- F _ {y} ^ { - 1 }
 +
( x _ {0} , y _ {0} )
 +
\circ F _ {x} ( x _ {0} , y _ {0} ) .
 +
$$
  
In the general case one can also indicate conditions for the existence and the uniqueness of the implicit function in terms of the continuity of the Fréchet derivative: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310198.png" /> is complete, if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310199.png" /> is continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310200.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310201.png" />, and if the partial Fréchet derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310202.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310203.png" />, then (1) is uniquely solvable in a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310204.png" />, i.e. there exist neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310205.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310206.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310208.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310210.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310211.png" />, and a unique implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310212.png" /> satisfying (2). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310213.png" /> is also continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310214.png" />. In this form the implicit-function theorem for normed spaces is a direct generalization of the corresponding classic implicit-function theorem for a single scalar equation in two variables.
+
This is obtained as a result of formally differentiating (2):
  
Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310215.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310216.png" />-times continuously-differentiable mapping in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310217.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310218.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310219.png" /> then the implicit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310220.png" /> is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050310/i050310221.png" /> times continuously differentiable.
+
$$
 +
F _ {x} ( x _ {0} , y _ {0} ) +
 +
F _ {y} ( x _ {0} , y _ {0} )
 +
\circ f ^ { \prime } ( x _ {0} )  = \
 +
0 \in  {\mathcal L} ( \mathbf X , \mathbf Y )
 +
$$
  
Far-reaching generalizations of the classic implicit-function theorem to differential operators were given by J. Nash (see [[Nash theorems (in differential geometry)|Nash theorems (in differential geometry)]]).
+
and multiplying this equality on the left by  $  F _ {y} ^ { - 1 } ( x _ {0} , y _ {0} ) $.
  
====References====
+
If in addition the mapping  $  F :  W \rightarrow Z $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.N. Kolmogorov,   S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.A. Lyusternik,  V.I. Sobolev,   "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977(Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L. Schwartz,   "Cours d'analyse" , '''1''' , Hermann (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1''' , MIR (1982) (Translated from Russian)</TD></TR></table>
+
is continuously differentiable on $  W $,  
 +
if the implicit function $ f : U \rightarrow V $
 +
is continuous on  $  U $,  
 +
$  U \times X \subset  W $,  
 +
and if for any  $ x \in U $
 +
the partial Fréchet derivative  $ F _ {y} ( x , f ( x) ) $
 +
is an invertible element of $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $,  
 +
then $ f $
 +
is a continuously-differentiable mapping of  $  U $
 +
into  $ V $.
  
 +
In the general case one can also indicate conditions for the existence and the uniqueness of the implicit function in terms of the continuity of the Fréchet derivative: If  $  Z $
 +
is complete, if the mapping  $  F :  W \rightarrow Z $
 +
is continuously differentiable on  $  W $,
 +
if  $  F ( x _ {0} , y _ {0} ) = z _ {0} $,
 +
and if the partial Fréchet derivative  $  F _ {y} ( x _ {0} , y _ {0} ) $
 +
is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $,
 +
then (1) is uniquely solvable in a sufficiently small neighbourhood of  $  ( x _ {0} , y _ {0} ) $,
 +
i.e. there exist neighbourhoods  $  U $
 +
of  $  x _ {0} $
 +
in  $  X $
 +
and  $  V $
 +
of  $  y _ {0} $
 +
in  $  Y $,
 +
$  U \times V \subset  W $,
 +
and a unique implicit function  $  f :  U \rightarrow V $
 +
satisfying (2). Here  $  f $
 +
is also continuously differentiable on  $  U $.
 +
In this form the implicit-function theorem for normed spaces is a direct generalization of the corresponding classic implicit-function theorem for a single scalar equation in two variables.
  
 +
Furthermore, if  $  F :  W \rightarrow Z $
 +
is a  $  k $-
 +
times continuously-differentiable mapping in a neighbourhood  $  W $
 +
of  $  ( x _ {0} , y _ {0} ) $,
 +
$  k = 1 , 2 \dots $
 +
then the implicit function  $  f :  U \rightarrow V $
 +
is also  $  k $
 +
times continuously differentiable.
  
====Comments====
+
Far-reaching generalizations of the classic implicit-function theorem to differential operators were given by J. Nash (see [[Nash theorems (in differential geometry)|Nash theorems (in differential geometry)]]).
  
 +
{{OldImage}}
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Addison-Wesley  (1965)</TD><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Flett,  "Differential Analysis" ,  Cambridge University Press  (1980)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elemente der Funktionalanalysis" , Akademie Verlag  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Schwartz,  "Cours d'analyse" , '''1''' , Hermann  (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1''' , MIR  (1982)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Fleming,  "Functions of several variables" , Addison-Wesley  (1965)</TD><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Flett,  "Differential Analysis" ,  Cambridge University Press  (1980)</TD></TR></table>

Latest revision as of 08:00, 21 January 2024


A function $ f : E \rightarrow Y $ given by an equation $ F ( x , y ) = z _ {0} $, where $ F: X \times Y \rightarrow Z $, $ x \in X $, $ y \in Y $, $ E \subset X $, and $ X $, $ Y $ and $ Z $ are certain sets, i.e. a function $ f $ such that $ F ( x , f ( x) ) = z _ {0} $ for any $ x \in E $. If $ X $, $ Y $ and $ Z $ are topological spaces and if $ F ( x _ {0} , y _ {0} ) = z _ {0} $ for some point $ ( x _ {0} , y _ {0} ) \in X \times Y $, then under certain conditions the equation $ F ( x , y ) = z _ {0} $ is uniquely solvable in one of the variables in some neighbourhood of $ ( x _ {0} , y _ {0} ) $. Properties of the solution of this equation are described by implicit-function theorems.

The simplest implicit-function theorem is as follows. Suppose that $ X $ and $ Y $ are subsets of the real line $ \mathbf R $, let $ x _ {0} \in X $, $ y _ {0} \in Y $, and let $ ( x _ {0} , y _ {0} ) $ be an interior point of the plane set $ X \times Y $; if $ F $ is continuous in some neighbourhood of $ ( x _ {0} , y _ {0} ) $, if $ F ( x _ {0} , y _ {0} ) = 0 $ and if there are a $ \delta > 0 $ and an $ \epsilon > 0 $ such that $ F ( x , y ) $, for any fixed $ x \in ( x _ {0} - \delta , x _ {0} + \delta ) $, is strictly monotone on $ ( y _ {0} - \epsilon , y _ {0} + \epsilon ) $ as a function of $ y $, then there is a $ \delta _ {0} > 0 $ such that there is a unique function

$$ f : ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} ) \rightarrow \ ( y _ {0} - \epsilon , y _ {0} + \epsilon ) $$

for which $ F ( x , f ( x) ) = 0 $ for all $ x \in ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} ) $; moreover, $ f $ is continuous and $ f ( x _ {0} ) = y _ {0} $.

Figure: i050310a

The hypotheses of this theorem are satisfied if $ F $ is continuous in a neighbourhood of $ ( x _ {0} , y _ {0} ) $, if the partial derivative $ F _ {y} $ exists and is continuous at $ ( x _ {0} , y _ {0} ) $, if $ F ( x _ {0} , y _ {0} ) = 0 $, and if $ F _ {y} ( x _ {0} , y _ {0} ) \neq 0 $. If in addition the partial derivative $ F _ {x} $ exists and is continuous at $ ( x _ {0} , y _ {0} ) $, then the implicit function $ f $ is differentiable at $ x _ {0} $, and

$$ \frac{d f ( x _ {0} ) }{dx} = - \frac{F _ {x} ( x _ {0} , y _ {0} ) }{F _ {y} ( x _ {0} , y _ {0} ) } . $$

This theorem has been generalized to the case of a system of equations, that is, when $ F $ is a vector function. Let $ \mathbf R ^ {n} $ and $ \mathbf R ^ {m} $ be $ n $- and $ m $- dimensional Euclidean spaces with fixed coordinate systems and points $ x = ( x _ {1} \dots x _ {n} ) $ and $ y = ( y _ {1} \dots y _ {m} ) $, respectively. Suppose that $ F $ maps a certain neighbourhood $ W $ of $ ( x _ {0} , y _ {0} ) \in \mathbf R ^ {n} \times \mathbf R ^ {m} $( $ x _ {0} \in \mathbf R ^ {n} $, $ y _ {0} \in \mathbf R ^ {m} $) into $ \mathbf R ^ {m} $ and that $ F _ {i} $, $ i = 1 \dots m $, are the coordinate functions (of the $ n + m $ variables $ x _ {1} \dots x _ {n} , y _ {1} \dots y _ {m} $) of $ F $, that is, $ F = ( F _ {1} \dots F _ {m} ) $. If $ F $ is differentiable on $ W $, if $ F( x _ {0} , y _ {0} ) = 0 $ and if the Jacobian

$$ \left . \frac{\partial ( F _ {1} \dots F _ {m} ) }{\partial ( y _ {1} \dots y _ {m} ) } \right | _ {( x _ {0} , y _ {0} ) } \ \neq 0 , $$

then there are neighbourhoods $ U $ and $ V $ of $ x _ {0} \in \mathbf R ^ {n} $ and $ y _ {0} \in \mathbf R ^ {m} $, respectively, $ U \times V \subset W $, and a unique mapping $ f : U \rightarrow V $ such that $ F ( x , f ( x) ) = 0 \in \mathbf R ^ {m} $ for all $ x \in U $. Here $ f ( x _ {0} ) = y _ {0} $, $ f $ is differentiable on $ U $, and if $ f = ( f _ {1} \dots f _ {m} ) $, then the explicit expression for the partial derivatives $ \partial f _ {j} / \partial x _ {i} $, $ i = 1 \dots n $, $ j = 1 \dots m $, can be found from the system of $ m $ linear equations in these derivatives:

$$ \frac{\partial F _ {k} }{\partial x _ {i} } + \sum_{j=}^ { m } \frac{\partial F _ {k} }{\partial y _ {j} } \frac{\partial f _ {j} }{\partial x _ {i} } = 0 , $$

$ k = 1 \dots m $, $ i $ is fixed $ ( i = 1 \dots n ) $. Sometimes the main assertion of the theorem is stated as follows: There are neighbourhoods $ U $ of $ x _ {0} $ in $ \mathbf R ^ {n} $ and $ W _ {0} $ of $ ( x _ {0} , y _ {0} ) $ in $ \mathbf R ^ {n} \times \mathbf R ^ {m} $, $ W _ {0} \subset W $, and a unique mapping $ f : U \rightarrow \mathbf R ^ {m} $ such that $ ( x , f ( x) ) \in W _ {0} $ and $ F ( x , f ( x) ) = 0 $ for all $ x \in U $. In other words, the conditions

$$ ( x , y ) \in W _ {0} ,\ \ F ( x , y ) = 0 $$

are equivalent to $ x \in U $, $ y = f ( x) $. In this case one says that the equation $ F ( x , y ) = 0 $ is uniquely solvable in the neighbourhood $ W _ {0} $ of $ ( x _ {0} , y _ {0} ) $.

The classical implicit-function theorem thus stated generalizes to the case of more general spaces in the following manner. Let $ X $ be a topological space, let $ Y $ and $ Z $ be affine normed spaces over the field of real or complex numbers, that is, affine spaces over the relevant field to which are associated normed vector spaces $ \mathbf Y $ and $ \mathbf Z $, $ \mathbf Y $ being complete, let $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $ be the set of continuous linear mappings from $ \mathbf Y $ into $ \mathbf Z $, and let $ W $ be an open set in the product space $ X \times Y $, $ ( x _ {0} , y _ {0} ) \in W $, $ x _ {0} \in X $, $ y _ {0} \in Y $.

Let $ F : W \rightarrow Z $ be a continuous mapping and $ F ( x _ {0} , y _ {0} ) = z _ {0} $. If for every fixed $ x $ and $ ( x , y ) \in W $ the mapping $ F $ has a partial Fréchet derivative $ F _ {y} \in {\mathcal L} ( \mathbf Y , \mathbf Z ) $, if $ F _ {y} ( x , y ) : W \rightarrow {\mathcal L} ( \mathbf Y , \mathbf Z ) $ is a continuous mapping and if the linear mapping $ F _ {y} ( x _ {0} , y _ {0} ) : \mathbf Y \rightarrow \mathbf Z $ has a continuous inverse linear mapping (that is, it is an invertible element of $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $), then there exist open sets $ U \subset X $ and $ V \subset Y $, $ x _ {0} \in U $, $ y _ {0} \in V $, such that for any $ x \in U $ there is a unique element $ y \in V $, denoted by $ y = f ( x) $, satisfying the equations

$$ f ( x) \in V \ \ \textrm{ and } \ F ( x , f ( x) ) = z _ {0} . $$

The function $ y = f ( x) $ thus defined is a continuous mapping from $ U $ into $ V $, and $ y _ {0} = f ( x _ {0} ) $.

If $ X $ is also an affine normed space, then under certain conditions the implicit function $ f : x \mapsto y $ which satisfies the equation

$$ \tag{1 } F ( x , y ) = z _ {0} $$

is also differentiable. Namely, let $ X $, $ Y $ and $ Z $ be affine normed spaces, let $ W $ be an open set in $ X \times Y $, let $ F : W \rightarrow Z $, $ F ( x _ {0} , y _ {0} ) = z _ {0} $, $ x _ {0} \in Y $, and let $ f $ be the implicit mapping given by (1), taking a certain neighbourhood $ U $ of $ x _ {0} $ into an open subset $ V $ of $ Y $, $ U \times V \subset W $. Thus, for all $ x \in U $,

$$ \tag{2 } f ( x) \in V ,\ \ F ( x , f ( x) ) = z _ {0} . $$

Suppose also that $ f $ is continuous at $ x _ {0} $ and that $ f ( x _ {0} ) = y _ {0} $. If $ F $ is differentiable at $ ( x _ {0} , y _ {0} ) $, if its partial Fréchet derivatives $ F _ {x} ( x _ {0} , y _ {0} ) $ and $ F _ {y} ( x _ {0} , y _ {0} ) $ are continuous linear operators taking the vector spaces $ \mathbf X $ and $ \mathbf Y $ associated with $ X $ and $ Y $ into the vector space $ \mathbf Z $ associated with $ Z $, and if the operator $ F _ {y} ( x _ {0} , y _ {0} ) $ is an invertible element of $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $, then $ f $ is differentiable at $ x _ {0} $ and its Fréchet derivative is given by

$$ f ^ { \prime } ( x _ {0} ) = \ - F _ {y} ^ { - 1 } ( x _ {0} , y _ {0} ) \circ F _ {x} ( x _ {0} , y _ {0} ) . $$

This is obtained as a result of formally differentiating (2):

$$ F _ {x} ( x _ {0} , y _ {0} ) + F _ {y} ( x _ {0} , y _ {0} ) \circ f ^ { \prime } ( x _ {0} ) = \ 0 \in {\mathcal L} ( \mathbf X , \mathbf Y ) $$

and multiplying this equality on the left by $ F _ {y} ^ { - 1 } ( x _ {0} , y _ {0} ) $.

If in addition the mapping $ F : W \rightarrow Z $ is continuously differentiable on $ W $, if the implicit function $ f : U \rightarrow V $ is continuous on $ U $, $ U \times X \subset W $, and if for any $ x \in U $ the partial Fréchet derivative $ F _ {y} ( x , f ( x) ) $ is an invertible element of $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $, then $ f $ is a continuously-differentiable mapping of $ U $ into $ V $.

In the general case one can also indicate conditions for the existence and the uniqueness of the implicit function in terms of the continuity of the Fréchet derivative: If $ Z $ is complete, if the mapping $ F : W \rightarrow Z $ is continuously differentiable on $ W $, if $ F ( x _ {0} , y _ {0} ) = z _ {0} $, and if the partial Fréchet derivative $ F _ {y} ( x _ {0} , y _ {0} ) $ is an invertible element of $ {\mathcal L} ( \mathbf Y , \mathbf Z ) $, then (1) is uniquely solvable in a sufficiently small neighbourhood of $ ( x _ {0} , y _ {0} ) $, i.e. there exist neighbourhoods $ U $ of $ x _ {0} $ in $ X $ and $ V $ of $ y _ {0} $ in $ Y $, $ U \times V \subset W $, and a unique implicit function $ f : U \rightarrow V $ satisfying (2). Here $ f $ is also continuously differentiable on $ U $. In this form the implicit-function theorem for normed spaces is a direct generalization of the corresponding classic implicit-function theorem for a single scalar equation in two variables.

Furthermore, if $ F : W \rightarrow Z $ is a $ k $- times continuously-differentiable mapping in a neighbourhood $ W $ of $ ( x _ {0} , y _ {0} ) $, $ k = 1 , 2 \dots $ then the implicit function $ f : U \rightarrow V $ is also $ k $ times continuously differentiable.

Far-reaching generalizations of the classic implicit-function theorem to differential operators were given by J. Nash (see Nash theorems (in differential geometry)).


🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian)
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[4] L. Schwartz, "Cours d'analyse" , 1 , Hermann (1967)
[5] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1 , MIR (1982) (Translated from Russian)
[a1] W. Fleming, "Functions of several variables" , Addison-Wesley (1965)
[a2] T.M. Flett, "Differential Analysis" , Cambridge University Press (1980)
How to Cite This Entry:
Implicit function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Implicit_function&oldid=40209
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article