2 KiB
Implicit equations
Theorem: for D \subseteq \mathbb{R}^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.