Implicit Function Theorem - Statement of The Theorem

Statement of The Theorem

Let f : Rn+m → Rm be a continuously differentiable function. We think of Rn+m as the Cartesian product Rn × Rm, and we write a point of this product as (x,y) = (x1, ..., xn, y1, ..., ym). Starting from the given function f, our goal is to construct a function g : Rn → Rm whose graph (x, g(x)) is precisely the set of all (x, y) such that f(x, y) = 0.

As noted above, this may not always be possible. We will therefore fix a point (a,b) = (a1, ..., an, b1, ..., bm) which satisfies f(a, b) = 0, and we will ask for a g that works near the point (a, b). In other words, we want an open set U of Rn, an open set V of Rm, and a function g : U → V such that the graph of g satisfies the relation f = 0 on U × V. In symbols,

To state the implicit function theorem, we need the Jacobian matrix of, which is the matrix of the partial derivatives of . Abbreviating (a1, ..., an, b1, ..., bm) to (a, b), the Jacobian matrix is

\begin{matrix} (Df)(\mathbf{a},\mathbf{b}) & = & \left[\begin{matrix} \frac{\partial f_1}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a},\mathbf{b})\\ \vdots & \ddots & \vdots\\ \frac{\partial f_m}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a},\mathbf{b}) \end{matrix}\right|\left. \begin{matrix} \frac{\partial f_1}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial y_m}(\mathbf{a},\mathbf{b})\\ \vdots & \ddots & \vdots\\ \frac{\partial f_m}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial y_m}(\mathbf{a},\mathbf{b})\\ \end{matrix}\right]\\ & = & \begin{bmatrix} X & | & Y \end{bmatrix}\\ \end{matrix}

where is the matrix of partial derivatives in the 's and is the matrix of partial derivatives in the 's. The implicit function theorem says that if is an invertible matrix, then there are, and as desired. Writing all the hypotheses together gives the following statement.

Let f : Rn+m → Rm be a continuously differentiable function, and let Rn+m have coordinates (x, y). Fix a point (a1,...,an,b1,...,bm) = (a,b) with f(a,b)=c, where c∈ Rm. If the matrix is invertible, then there exists an open set U containing a, an open set V containing b, and a unique continuously differentiable function g:U → V such that

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