Implicit function theorem

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

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

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 $

$$F_1(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),<br />
...,f_m(x_1,x_2,...x_n))=0$$
$$F_2(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),<br />
...,f_m(x_1,x_2,...x_n))=0$$
$$F_j(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),<br />
...,f_m(x_1,x_2,...x_n))=0$$
$$F_m(x_1,x_2,...x_n,f_1(x_1,x_2,...x_n),f_2(x_1,x_2,...x_n),<br />
...,f_m(x_1,x_2,...x_n))=0$$

(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

$$\left [ \begin{array}{cccc}<br />
\frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}&..&\frac{\partial f_1}{\partial x_n}\\<br />
\frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}&..&\frac{\partial f_2}{\partial x_n}\\<br />
.&.&.&.\\<br />
\frac{\partial f_m}{\partial x_1}&\frac{\partial f_m}{\partial x_2}&..&\frac{\partial f_m}{\partial x_n}<br />
\end{array}\right]=$$
$$- \left [ \begin{array}{cccc}<br />
\frac{\partial F_1}{\partial y_1}&\frac{\partial F_1}{\partial y_2}&..&\frac{\partial F_1}{\partial y_m}\\<br />
\frac{\partial F_2}{\partial y_1}&\frac{\partial F_2}{\partial y_2}&..&\frac{\partial F_2}{\partial y_m}\\<br />
.&.&.&.\\<br />
\frac{\partial F_m}{\partial y_1}&\frac{\partial F_m}{\partial y_2}&..&\frac{\partial F_m}{\partial y_m}<br />
\end{array}\right]^{-1} \left [ \begin{array}{cccc}<br />
\frac{\partial F_1}{\partial x_1}&\frac{\partial F_1}{\partial x_2}&..&\frac{\partial F_1}{\partial x_n}\\<br />
\frac{\partial F_2}{\partial x_1}&\frac{\partial F_2}{\partial x_2}&..&\frac{\partial F_2}{\partial x_n}\\<br />
.&.&.&.\\<br />
\frac{\partial F_m}{\partial x_1}&\frac{\partial F_m}{\partial x_2}&..&\frac{\partial F_m}{\partial x_n}<br />
\end{array}\right]$$

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

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