Implicit Function Theorem

The implicit function theorem is a strong theorem that allows us to express a variable as a function of another variable. For instance, if x^2y+y^3x+9xy=0, can we make y the subject, i.e. write y as a function of x? The implicit function theorem allows us to answer such questions, though like most Pure Math theorems, it only guarantees existence, the theorem does not explicitly tell us how to write out such a function.

The below material are taken from Wikipedia.

Implicit function theorem

Let f:\mathbb{R}^{n+m}\to\mathbb{R}^m be a continuously differentiable function, and let \mathbb{R}^{n+m} have coordinates (\mathbf{x},\mathbf{y})=(x_1,\dots,x_n,y_1,\dots,y_m). Fix a point (\mathbf{a},\mathbf{b})=(a_1,\dots,a_n,b_1,\dots,b_m) with f(\mathbf{a},\mathbf{b})=\mathbf{c}, where \mathbf{c}\in\mathbb{R}^m. If the matrix \displaystyle [(\partial f_i/\partial y_j)(\mathbf{a},\mathbf{b})] is invertible, then there exists an open set U containing \mathbf{a}, an open set V containing \mathbf{b}, and a unique continuously differentiable function g:U\to V such that \displaystyle \{(\mathbf{x},g(\mathbf{x}))\mid\mathbf{x}\in U\}=\{(\mathbf{x},\mathbf{y})\in U\times V\mid f(\mathbf{x},\mathbf{y})=\mathbf{c}\}.


Abbreviating (a_1,\dots,a_n,b_1,\dots,b_m) to (\mathbf{a},\mathbf{b}), the Jacobian matrix is
\displaystyle (Df)(\mathbf{a},\mathbf{b})=\begin{pmatrix}  \frac{\partial f_1}{\partial x_1}(\mathbf{a},\mathbf{b}) & \dots &\frac{\partial f_1}{\partial x_n}(\mathbf{a},\mathbf{b}) & \frac{\partial f_1}{\partial y_1}(\mathbf{a},\mathbf{b}) & \dots & \frac{\partial f_1}{\partial y_m}(\mathbf{a},\mathbf{b})\\  \vdots & \ddots &\vdots & \vdots & \ddots &\vdots\\  \frac{\partial f_m}{\partial x_1}(\mathbf{a},\mathbf{b}) & \dots & \frac{\partial f_m}{\partial x_n}(\mathbf{a}, \mathbf{b}) & \frac{\partial f_m}{\partial y_1}(\mathbf{a}, \mathbf{b}) & \dots & \frac{\partial f_m}{\partial y_m}(\mathbf{a}, \mathbf{b})  \end{pmatrix}  =(X\mid Y)
where X is the matrix of partial derivatives in the variables x_i and Y is the matrix of partial derivatives in the variables y_j.

The implicit function theorem says that if Y is an invertible matrix, then there are U, V, and g as desired.

Example (Unit circle)

In this case n=m=1 and f(x,y)=x^2+y^2-1.

\displaystyle (Df)(a,b)=(\frac{\partial f}{\partial x}(a,b)\ \frac{\partial f}{\partial y}(a,b))=(2a\ 2b).

Note that Y=(2b) is invertible iff b\neq 0. By the implicit function theorem, we see that we can locally write the circle in the form y=g(x) for all points where y\neq 0.

Author: mathtuition88

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