The Implicit and Inverse Function Theorems

These fundamental theorems establish when a system of equations can be locally solved for some variables in terms of others, and when a multivariable function has a local inverse.

The Implicit Function Theorem

Theorem8.5Implicit Function Theorem

Let $F : \mathbb{R}^{n+m} \to \mathbb{R}^m$ be continuously differentiable, and let $(\mathbf{a}, \mathbf{b}) \in \mathbb{R}^n \times \mathbb{R}^m$ satisfy $F(\mathbf{a}, \mathbf{b}) = \mathbf{0}$ . If the $m \times m$ matrix of partial derivatives $\frac{\partial(F_1, \ldots, F_m)}{\partial(y_1, \ldots, y_m)}(\mathbf{a}, \mathbf{b})$ is invertible, then there exist open neighborhoods $U$ of $\mathbf{a}$ and $V$ of $\mathbf{b}$ and a unique $C^1$ function $\mathbf{g} : U \to V$ such that $F(\mathbf{x}, \mathbf{g}(\mathbf{x})) = \mathbf{0}$ for all $\mathbf{x} \in U$ . Moreover, the Jacobian of $\mathbf{g}$ is $J_\mathbf{g} = -\left(\frac{\partial F}{\partial \mathbf{y}}\right)^{-1} \frac{\partial F}{\partial \mathbf{x}}$

ExampleImplicit differentiation of a curve

For $F(x, y) = x^2 + y^2 - 1 = 0$ near $(0, 1)$ : $F_y = 2y = 2 \neq 0$ at $(0, 1)$ , so $y$ can be written as $y = g(x) = \sqrt{1-x^2}$ locally, with $g'(x) = -F_x/F_y = -x/y$ .

The Inverse Function Theorem

Theorem8.6Inverse Function Theorem

Let $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ be continuously differentiable in a neighborhood of $\mathbf{a}$ , and suppose the Jacobian matrix $J_\mathbf{f}(\mathbf{a})$ is invertible (equivalently, $\det J_\mathbf{f}(\mathbf{a}) \neq 0$ ). Then there exists a neighborhood $U$ of $\mathbf{a}$ such that:

$\mathbf{f}$ is a bijection from $U$ to $\mathbf{f}(U)$
$\mathbf{f}(U)$ is open
The inverse $\mathbf{f}^{-1} : \mathbf{f}(U) \to U$ is continuously differentiable
$J_{\mathbf{f}^{-1}}(\mathbf{f}(\mathbf{a})) = (J_\mathbf{f}(\mathbf{a}))^{-1}$

ExamplePolar coordinate transformation

The map $\mathbf{f}(r, \theta) = (r\cos\theta, r\sin\theta)$ has Jacobian determinant $r$ . For $r > 0$ , the inverse function theorem guarantees a local inverse, confirming that polar coordinates provide a valid coordinate system away from the origin.

RemarkRelationship between the two theorems

The inverse function theorem and implicit function theorem are equivalent: each can be derived from the other. The implicit function theorem for $F(\mathbf{x}, \mathbf{y}) = \mathbf{0}$ follows from applying the inverse function theorem to the map $(\mathbf{x}, \mathbf{y}) \mapsto (\mathbf{x}, F(\mathbf{x}, \mathbf{y}))$ . Both theorems are fundamentally about the local solvability of nonlinear systems.