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Implicit equations

Theorem: for \(D \subseteq \mathbb{R}^n\) (\(n=2\) for simplicty), let \(f: D \to \mathbb{R}\) be continuously differentiable and \(\mathbf{a} \in D\). Assume

  • \(f(\mathbf{a}) = 0\),
  • \(\partial_2 f(\mathbf{a}) \neq 0\), nondegeneracy.

then there exists an \(I\) around \(a_1\) and an \(J\) around \(a_2\) such that \(\phi: I \to J\) is differentiable and

\[ \forall x \in I, y \in J: f(x,y) = 0 \iff y = \phi(x). \]

Now calculating \(\phi' (x)\) with the chain rule

\[ \begin{align*} f\big(x,\phi(x)\big) &= 0, \\ \partial_1 f\big(x,\phi(x)\big) + \partial_2 f\big(x,\phi(x)\big) \phi' (x) &= 0, \end{align*} \]

and we obtain

\[ \phi' (x) = - \frac{\partial_1 f\big(x,\phi(x)\big)}{\partial_2 f\big(x,\phi(x)\big)}. \]
Proof:

Will be added later.

General case

Theorem: Let \(\mathbf{F}: \mathbb{R}^{n+m} \to \mathbb{R}^m\) given by \(F(\mathbf{x},\mathbf{y}) = \mathbf{0}\) with \(\mathbf{x} \in \mathbb{R}^n\) and \(\mathbf{y} \in \mathbb{R}^m\). Suppose \(\mathbf{F}\) is continuously differentiable and assume \(D_2 \mathbf{F}(\mathbf{x},\mathbf{y}) \in \mathbb{R}^{m \times m}\) is nonsingular. Then there exists in neighbourhoods \(I\) of \(\mathbf{x}\) and \(J\) of \(\mathbf{y}\) with \(I \subseteq \mathbb{R}^n,\; J \subseteq \mathbb{R}^m\), such that \(\mathbf{\phi}: I \to J\) is differentiable and

\[ \forall (\mathbf{x},\mathbf{y}) \in I \times J: \mathbf{F}(\mathbf{x},\mathbf{y}) = \mathbf{0} \iff \mathbf{y} = \mathbf{\phi}(\mathbf{x}). \]

Now calculating \(D \mathbf{\phi}(\mathbf{x})\) with the generalized chain rule

\[ \begin{align*} \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big) &= \mathbf{0}, \\ D_1 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big) + D_2 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big) D \mathbf{\phi}(\mathbf{x}) &= \mathbf{0}, \\ \end{align*} \]

and we obtain

\[ D \mathbf{\phi}(\mathbf{x}) = - \Big(D_2 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big) \Big)^{-1} D_1 \mathbf{F}\big(\mathbf{x},\mathbf{\phi}(\mathbf{x})\big). \]
Proof:

Will be added later.