91Ó°ÊÓ

The Implicit Function Theorem states that if we have an equation \(F(x, y) = 0\), where \(F\) is a continuously differentiable function in a neighborhood of the point \((a, b)\) and \(\frac{\partial F}{\partial y}(a, b) \neq 0\), then there exists a unique differentiable function \(g(x)\) defined in a neighborhood of \(a\) such that \(F(x, g(x)) = 0\) for any \(x\) in that neighborhood, and \(g(a) = b\).

From the statement, we need to satisfy two conditions to apply the theorem: 1. \(F\) is a continuously differentiable function in a neighborhood of the point \((a, b)\). 2. \(\frac{\partial F}{\partial y}(a, b) \neq 0\).

We can apply the Inverse Function Theorem to the function \(F(x, y)\), with variables \((x, y)\) and values \((a, b)\). Consider the continuously differentiable function \(G: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n\) defined as \(G(x, y) = (F(x, y), x)\), where \(G(a, b) = (0, a)\). We will show that the Jacobian matrix of \(G\) with respect to the variables \((x, y)\), denoted by \(D_{(x, y)}G(x, y)\), is invertible. Notice that the Jacobian matrix has the form: \[ D_{(x, y)}G(x, y) = \begin{pmatrix} \frac{\partial F}{\partial x}(x, y) & \frac{\partial F}{\partial y}(x, y) \\ I_n & 0 \end{pmatrix}, \] where \(I_n\) is the \(n \times n\) identity matrix, and \(0\) is the \(n \times n\) zero matrix. Using the block matrix determinant formula, the determinant of the Jacobian matrix is given by \(\det(D_{(x, y)}G(x, y)) = \det(\frac{\partial F}{\partial y}(x, y)) \cdot \det(I_n) = \det(\frac{\partial F}{\partial y}(x, y))\). Since \(\frac{\partial F}{\partial y}(a, b) \neq 0\), the Jacobian matrix \(D_{(x, y)}G(a, b)\) is invertible.

Applying the Inverse Function Theorem to \(G\) with the invertible Jacobian matrix, there exists a continuously differentiable inverse function \(G^{-1}\) defined in a neighborhood of \((0, a)\) in the range of \(G\), and \(G^{-1}(0, a) = (a, b)\).

Define the function \(g: \mathbb{R}^n \rightarrow \mathbb{R}^n\) as the second component of \(G^{-1}(x, y)\), that is, \(g(x) = \pi_2(G^{-1}(x, y))\), where \(\pi_2\) is the projection onto the second component. By construction, we have: 1. \(g\) is differentiable, as it is the composition of the differentiable functions \(G^{-1}\) and the projection \(\pi_2\). 2. For any \(x\) in a neighborhood of \(a\), we have \(F(x, g(x)) = \pi_1(G^{-1}(x, g(x))) = \pi_1(G(a, b)) = 0\), where \(\pi_1\) is the projection onto the first component. 3. \(g(a) = \pi_2(G^{-1}(0, a)) = \pi_2(G(a, b)) = b\). Thus, the function \(g\) satisfies the conclusion of the Implicit Function Theorem, and the theorem is proven.