Theorem of the implicit function

The theorem of the implicit function is an important theorem in analysis . It contains a relatively simple criterion as to when an implicit equation or a system of equations can be solved uniquely (locally).

The theorem specifies the condition under which an equation or system of equations implicitly defines a function for which applies. Such a function can generally only be found locally in a neighborhood of a location . However, under more rigorous assumptions, a global version of the theorem also exists. ${\displaystyle F(x,y)=0}$ ${\displaystyle y=f(x)}$ ${\displaystyle F(x,f(x))=0}$ ${\displaystyle x_{0}}$

If the condition of the proposition is fulfilled, the derivative can be obtained as a function of and without knowledge of the explicit function ; this is also called implicit differentiation . ${\displaystyle {\tfrac {\mathrm {d} f}{\mathrm {d} x}}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle y=f(x)}$

Definition

An implicitly defined function ( implicit function for short ) is a function that is not given by an explicit assignment rule, but whose function values ​​are implicitly defined by an equation . Here is a vector-valued function that contains as many individual functions as there are components. If you fix, a system of equations results in with as many equations as there are unknowns. The theorem about the implicit function describes conditions under which the following statement applies: ${\displaystyle y=f(x)}$ ${\displaystyle F(x,y)=0}$ ${\displaystyle F}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$

If a solution for a parameter vector is known, then a uniquely determined solution of the system of equations can also be found for each parameter vector from a sufficiently small neighborhood of , which is in a neighborhood of the original solution . ${\displaystyle y_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle x\approx x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle y\approx y_{0}}$ ${\displaystyle F(x,y)=0}$ ${\displaystyle y_{0}}$

This statement makes it possible to define a function that assigns the solution vector to each parameter vector , so that this function fulfills the equation on its domain of definition . The implicit function theorem also ensures that this allocation to certain conditions and limitations , and is well-defined - in particular, that it is unique. ${\displaystyle f}$ ${\displaystyle x\approx x_{0}}$ ${\displaystyle y=f(x)\approx y_{0}}$ ${\displaystyle F(x,f(x))=0}$ ${\displaystyle x\mapsto f(x)}$ ${\displaystyle F}$ ${\displaystyle x}$ ${\displaystyle y}$

example

The unit circle is described as the set of all points that satisfy the equation with . In a neighborhood of the point A can as a function of be expressed . This is not possible with point B. ${\displaystyle (x,y)}$ ${\displaystyle F(x,y)=0}$ ${\displaystyle F(x,y)=x^{2}+y^{2}-1}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y=f(x)={\sqrt {1-x^{2}}}}$

If one sets , the equation describes the unit circle in the plane. The unit circle cannot be written as a graph of a function , because for each of the open interval there are two possibilities for , namely . ${\displaystyle F(x,y)=x^{2}+y^{2}-1}$ ${\displaystyle F(x,y)=0}$ ${\displaystyle y=f(x)}$ ${\displaystyle x}$ ${\displaystyle (-1,1)}$ ${\displaystyle y}$ ${\displaystyle y=\pm {\sqrt {1-x^{2}}}}$

However, it is possible to display parts of the circle as a function graph. The upper semicircle is obtained as a graph of the function

${\displaystyle f_{1}\colon (-1,1)\to \mathbb {R} ,f_{1}(x)={\sqrt {1-x^{2}}}}$,

the lower one as a graph of

${\displaystyle f_{2}\colon (-1,1)\to \mathbb {R} ,f_{2}(x)=-{\sqrt {1-x^{2}}}}$.

The theorem of the implicit function gives criteria for the existence of functions like or . It also guarantees that these functions are differentiable. ${\displaystyle f_{1}}$ ${\displaystyle f_{2}}$

Theorem of the implicit function

statement

Be and open sets and ${\displaystyle U\subseteq \mathbb {R} ^{m}}$ ${\displaystyle V\subseteq \mathbb {R} ^{n}}$

${\displaystyle F\colon U\times \,V\to \mathbb {R} ^{n},\quad (x,y)=(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})\mapsto F(x,y)=(\,F_{1}(x,y),\dots ,F_{n}(x,y)\,)}$

a continuously differentiable map. The Jacobian matrix

${\displaystyle \mathrm {D} F={\frac {\partial F}{\partial (x,y)}}={\frac {\partial (F_{1},\dots ,F_{n})}{\partial (x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})}}={\begin{pmatrix}{\frac {\partial F_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial F_{1}}{\partial x_{m}}}&{\frac {\partial F_{1}}{\partial y_{1}}}&\cdots &{\frac {\partial F_{1}}{\partial y_{n}}}\\\vdots &&\vdots &\vdots &&\vdots \\{\frac {\partial F_{n}}{\partial x_{1}}}&\cdots &{\frac {\partial F_{n}}{\partial x_{m}}}&{\frac {\partial F_{n}}{\partial y_{1}}}&\cdots &{\frac {\partial F_{n}}{\partial y_{n}}}\end{pmatrix}}}$

then consists of two partial matrices

${\displaystyle {\frac {\partial F}{\partial x}}={\frac {\partial (F_{1},\dots ,F_{n})}{\partial (x_{1},\dots ,x_{m})}}={\begin{pmatrix}{\frac {\partial F_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial F_{1}}{\partial x_{m}}}\\\vdots &&\vdots \\{\frac {\partial F_{n}}{\partial x_{1}}}&\cdots &{\frac {\partial F_{n}}{\partial x_{m}}}\end{pmatrix}}}$

and

${\displaystyle {\frac {\partial F}{\partial y}}={\frac {\partial (F_{1},\dots ,F_{n})}{\partial (y_{1},\dots ,y_{n})}}={\begin{pmatrix}{\frac {\partial F_{1}}{\partial y_{1}}}&\cdots &{\frac {\partial F_{1}}{\partial y_{n}}}\\\vdots &&\vdots \\{\frac {\partial F_{n}}{\partial y_{1}}}&\cdots &{\frac {\partial F_{n}}{\partial y_{n}}}\end{pmatrix}},}$

the latter being square.

The theorem of the implicit function now says:

If the equation is fulfilled and the second sub-matrix can be inverted at the point , then there are open environments of and from and a clear, continuously differentiable mapping ${\displaystyle (x_{0},y_{0})\in U\times \ V}$ ${\displaystyle F(x_{0},y_{0})=0}$ ${\displaystyle {\tfrac {\partial F}{\partial y}}}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle U_{0}\subseteq U}$ ${\displaystyle x_{0}}$ ${\displaystyle V_{0}\subseteq V}$ ${\displaystyle y_{0}}$

${\displaystyle f\colon U_{0}\to V_{0}}$

with so for all , the following applies: ${\displaystyle f(x_{0})=y_{0}}$ ${\displaystyle x\in U_{0}}$ ${\displaystyle y\in V_{0}}$

${\displaystyle F(x,y)=0\;\Leftrightarrow \;y=f(x)}$.

example

Now apply this theorem to the example of the circular equation given at the beginning: To do this, consider the partial derivatives according to the -variables. (In this case , therefore, results in the a matrix, so just a real function): The partial derivative of the function according to results . The reciprocal of this term exists if and only if is. With the help of the theorem one concludes that this equation is solvable for local to . The fall occurs only at the points or . So these are the problem points. In fact, you can see that the formula branches into a positive and negative solution precisely at these problem points. In all other points the resolution is locally unique. ${\displaystyle y}$ ${\displaystyle n=1}$ ${\displaystyle 1\times 1}$ ${\displaystyle F(x,y)=x^{2}+y^{2}-1}$ ${\displaystyle y}$ ${\displaystyle {\tfrac {\partial F(x,y)}{\partial y}}=2y}$ ${\displaystyle y\neq 0}$ ${\displaystyle y\neq 0}$ ${\displaystyle y}$ ${\displaystyle y=0}$ ${\displaystyle x=-1}$ ${\displaystyle x=1}$ ${\displaystyle y=\pm {\sqrt {1-x^{2}}}}$

Evidence Approach

The classical approach considers the initial value problem of the ordinary differential equation to solve the equation ${\displaystyle F(x,y)=0}$

${\displaystyle \phi _{v}'(t)=-G(x_{0}+tv,\phi _{v}(t))\,v\;{\text{ mit }}\;G(x,y)=\left({\tfrac {\partial F}{\partial y}}(x,y)\right)^{-1}{\tfrac {\partial F}{\partial x}}(x,y)\;{\text{ und }}\;\phi _{v}(0)=y_{0}}$.

As in can be inverted, this is also in a small neighborhood of the case d. That is, for small vectors the differential equation and its solution exist for all . The solution to the implicit equation is now through ${\displaystyle {\tfrac {\partial F}{\partial y}}}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle v}$ ${\displaystyle t\in [0,1]}$

${\displaystyle f(x)=\phi _{x-x_{0}}(1)}$

given, the properties of this solution given above result from the properties of the solutions of parameter-dependent differential equations.

The modern approach formulates the system of equations with the help of the simplified Newton method as a fixed point problem and applies Banach's fixed point theorem to it. For the associated fixed point mapping, the inverse of the partial matrix of the Jacobi matrix is ​​formed from in the given solution point . To the picture ${\displaystyle F(x,y)=0}$ ${\displaystyle A}$ ${\displaystyle {\tfrac {\partial F}{\partial y}}(x_{0},y_{0})}$ ${\displaystyle F}$ ${\displaystyle (x_{0},y_{0})}$

${\displaystyle T(y)=y-A\,F(x,y)}$

one can now show that it is contractive for parameter vectors close to a neighborhood of . This follows from the fact that continuously differentiable is and applies. ${\displaystyle x}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle T}$ ${\displaystyle {\tfrac {\partial T}{\partial y}}(x_{0},y_{0})=0}$

Summary

The advantage of the theorem is that you don't have to know the function explicitly in order to be able to make a statement about its existence and uniqueness. Often the equation cannot be resolved at all using elementary functions , but only with numerical methods. It is interesting that the convergence of such procedures usually requires the same or similar conditions as the theorem of the implicit function (the invertibility of the matrix of the derivatives). ${\displaystyle f}$ ${\displaystyle y}$ ${\displaystyle y}$

Another valuable conclusion of the theorem is that the function is differentiable if it is what is assumed when applying the theorem about implicit functions . The derivative can even be explicitly specified by the equation according to the multi-dimensional chain rule derived ${\displaystyle f}$ ${\displaystyle F(x,y)}$ ${\displaystyle F(x,f(x))=0}$

${\displaystyle {\frac {\partial F}{\partial x}}(x,f(x))+{\frac {\partial F}{\partial y}}(x,f(x))\cdot {\frac {\partial f}{\partial x}}(x)=0}$

and then resolves to: ${\displaystyle {\frac {\partial f}{\partial x}}(x)}$

${\displaystyle {\frac {\partial f}{\partial x}}(x)=-\left({\frac {\partial F}{\partial y}}(x,f(x))\right)^{-1}\cdot {\frac {\partial F}{\partial x}}{\big (}x,f(x){\big )}}$.

A similar conclusion applies to higher derivatives. If one replaces the prerequisite “ is continuously differentiable” with “ is -time continuously differentiable” (or any number of times differentiable or analytical), one can conclude that -time is differentiable (or any number of times differentiable or analytical). ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle k}$ ${\displaystyle f}$ ${\displaystyle k}$

Set of the reverse illustration

A useful corollary to the theorem of implicit function is the theorem of inverse mapping . It gives an answer to the question of whether one can find a (local) inverse function :

Be open and ${\displaystyle U\subseteq \mathbb {R} ^{n}}$

${\displaystyle f\colon U\to \mathbb {R} ^{n}}$

a continuously differentiable map. Be and . The Jacobian matrix is invertible. Then there is an open environment of and an open environment of , so that the set maps bijectively to and the inverse function ${\displaystyle a\in U}$ ${\displaystyle b:=f(a)}$ ${\displaystyle \mathrm {D} f(a)}$ ${\displaystyle U_{a}\subseteq U}$ ${\displaystyle a}$ ${\displaystyle V_{b}}$ ${\displaystyle b}$ ${\displaystyle f}$ ${\displaystyle U_{a}}$ ${\displaystyle V_{b}}$

${\displaystyle g=f^{-1}\colon V_{b}\to U_{a}}$

is continuously differentiable, or in short: is a diffeomorphism . The following applies: ${\displaystyle f|_{U_{a}}}$

${\displaystyle \mathrm {D} (f^{-1})(b)=\mathrm {D} g(b)=(\mathrm {D} f(a))^{-1}=(\mathrm {D} f(g(b)))^{-1}}$

literature

• Herbert Amann, Joachim Escher: Analysis II. Birkhäuser, Basel, 1999, ISBN 3-7643-6133-6 , p. 230 ff.
• Otto Forster : Analysis 2. Differential calculus in R ⁿ , ordinary differential equations. 8 edition. Vieweg + Teubner Verlag, Wiesbaden 2008, ISBN 3-7643-6133-6 , pp. 86-99 ( pp. 90 ff. ).

Individual evidence

1. ↑ D. Gale, Nikaido: The Jacobian matrix and Global Univalence of mappings. . Mathematische Annalen 159 (1965), pp. 81-93.