Implicit function theorem

Submitted by Structure on Sun, 01/20/2008 - 17:01.

Implicit function theorem can be considered an extension to the non linear case of a well known problem of solving under determinated system of linear equations having more unknown then equations.

Let $F=(F_1,F_2,...F_j,...F_m):D\subset R^{n+m}\rightarrow R^m$ a differentiable function of class $\mathcal{C}^1(D)$
Let $(a,b)=(a_1,a_2,...a_n,b_1,b_2,..b_m)\in D$ a point such as $F(a,b)=0$ and rank $\frac{\partial F}{\partial y}(a,b)=m$ .
This means that

$det \frac{\partial F}{\partial y}(a,b)=det \left | \begin{array}{cccc} \frac{\partial F_1}{\partial y_1}&\frac{\partial F_1}{\partial y_2}&..&\frac{\partial F_1}{\partial y_m}\\ \frac{\partial F_2}{\partial y_1}&\frac{\partial F_2}{\partial y_2}&..&\frac{\partial F_2}{\partial y_m}\\ .&.&.&.\\ \frac{\partial F_m}{\partial y_1}&\frac{\partial F_m}{\partial y_2}&..&\frac{\partial F_m}{\partial y_m} \end{array}\right|(a,b)\ne 0$

Then there are U a neighborhood of a, $V=V_1\times V_2\times...\times V_m$ a neighborhood of b and a function $f=(f_1,f_2,...f_m):U\rightarrow V$ with properties:
(1)f(a)=b, i.e. $f_i(a)=f_i(a_1,a_2,...a_n)=b_i,\:1\leq i\leq m$
(2)for all $x\in U,\: F(x,f(x))=0, \;$ i.e. for all $1\leq j\leq m$

$F_1(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n), ...,f_m(x_1,x_2,...x_n))=0$

$F_2(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n), ...,f_m(x_1,x_2,...x_n))=0$

$F_j(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n), ...,f_m(x_1,x_2,...x_n))=0$

$F_m(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n), ...,f_m(x_1,x_2,...x_n))=0$

(3) f is continuous on U;
Function with these three properties is local unique, any two functions having these properties are identical on the intersection of their domains of definition.
In plus f is differentiable and

$\frac{\partial f}{\partial x}(x)=-\frac{\partial F}{\partial y}(x,f(x))^{-1}\frac{\partial F}{\partial x}(x,f(x))$

By the above relation we mean

$\left [ \begin{array}{cccc} \frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}&..&\frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}&..&\frac{\partial f_2}{\partial x_n}\\ .&.&.&.\\ \frac{\partial f_m}{\partial x_1}&\frac{\partial f_m}{\partial x_2}&..&\frac{\partial f_m}{\partial x_n} \end{array}\right]=$

$- \left [ \begin{array}{cccc} \frac{\partial F_1}{\partial y_1}&\frac{\partial F_1}{\partial y_2}&..&\frac{\partial F_1}{\partial y_m}\\ \frac{\partial F_2}{\partial y_1}&\frac{\partial F_2}{\partial y_2}&..&\frac{\partial F_2}{\partial y_m}\\ .&.&.&.\\ \frac{\partial F_m}{\partial y_1}&\frac{\partial F_m}{\partial y_2}&..&\frac{\partial F_m}{\partial y_m} \end{array}\right]^{-1} \left [ \begin{array}{cccc} \frac{\partial F_1}{\partial x_1}&\frac{\partial F_1}{\partial x_2}&..&\frac{\partial F_1}{\partial x_n}\\ \frac{\partial F_2}{\partial x_1}&\frac{\partial F_2}{\partial x_2}&..&\frac{\partial F_2}{\partial x_n}\\ .&.&.&.\\ \frac{\partial F_m}{\partial x_1}&\frac{\partial F_m}{\partial x_2}&..&\frac{\partial F_m}{\partial x_n} \end{array}\right]$

After a short calculation or solving a system of linear equations

$\frac{\partial F_1}{\partial x_j }+\frac{\partial F_1 }{\partial y_1 }\frac{\partial f_1 }{\partial x_j }+\frac{\partial F_1 }{\partial y_2 }\frac{\partial f_2 }{\partial x_j }+...\frac{\partial F_1 }{\partial y_m }\frac{\partial f_m }{\partial x_j }=0$

$\frac{\partial F_2}{\partial x_j }+\frac{\partial F_2 }{\partial y_1 }\frac{\partial f_1 }{\partial x_j }+\frac{\partial F_2 }{\partial y_2 }\frac{\partial f_2 }{\partial x_j }+...\frac{\partial F_2 }{\partial y_m }\frac{\partial f_m }{\partial x_j }=0$

.....

$\frac{\partial F_m}{\partial x_j }+\frac{\partial F_m }{\partial y_1 }\frac{\partial f_1 }{\partial x_j }+\frac{\partial F_m }{\partial y_2 }\frac{\partial f_2 }{\partial x_j }+...\frac{\partial F_m }{\partial y_m }\frac{\partial f_m }{\partial x_j }=0$

we have more explicitly

$\frac{\partial f_i}{\partial x_j}=-\frac{\left |\begin{array}{cccccc} \frac{\partial F_1}{\partial y_1}&\frac{\partial F_1}{\partial y_2}&.&\frac{\partial F_1}{\partial x_j}&.&\frac{\partial F_1}{\partial y_m}\\ \frac{\partial F_2}{\partial y_1}&\frac{\partial F_2}{\partial y_2}&.&\frac{\partial F_2}{\partial x_j}&.&\frac{\partial F_2}{\partial y_m}\\ .&.&.&.&.&.\\ \frac{\partial F_m}{\partial y_1}&\frac{\partial F_m}{\partial y_2}&.&\frac{\partial F_m}{\partial x_j}&.&\frac{\partial F_m}{\partial y_m} \end{array}\right |}{\left |\begin{array}{cccccc} \frac{\partial F_1}{\partial y_1}&\frac{\partial F_1}{\partial y_2}&.&\frac{\partial F_1}{\partial y_i}&.&\frac{\partial F_1}{\partial y_m}\\ \frac{\partial F_2}{\partial y_1}&\frac{\partial F_2}{\partial y_2}&.&\frac{\partial F_2}{\partial y_i}&.&\frac{\partial F_2}{\partial y_m}\\ .&.&.&.&.&.\\ \frac{\partial F_m}{\partial y_1}&\frac{\partial F_m}{\partial y_2}&.&\frac{\partial F_m}{\partial y_i}&.&\frac{\partial F_m}{\partial y_m} \end{array}\right |}$

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Implicit function theorem