Definition:Implicit Function/General

Definition

Let:

$f: \R^{n + 1} \to \R, \tuple {x_1, x_2, \ldots, x_n, z} \mapsto \map f {x_1, x_2, \ldots, x_n, z}$

be a real-valued function on $\R^{n + 1}$, where:

$\tuple {x_1, x_2, \ldots, x_n} \in \R^n, z \in \R$

Let a relation between $x_1, x_2, \ldots, x_n$ and $z$ be expressed in the form:

$\map f {x_1, x_2, \ldots, x_n, z} = 0$

defined on some subset $S \subseteq \R^n$.

If there exists a function $g: S \to \R$ such that:

$\forall \tuple {x_1, x_2, \ldots, x_n} \in S: z = \map g {x_1, x_2, \ldots, x_n} \iff \map f {x_1, x_2, \ldots, x_n, z} = 0$

then the relation $\map f {x_1, x_2, \ldots, x_n, z} = 0$ defines $z$ as an implicitly defined function of $x_1, x_2, \ldots, x_n$.

Examples

Example: $u^2 + x^2 + y^2 = a^2$

Consider the equation:

$(1): \quad u^2 + x^2 + y^2 = a^2$

where $u, x, y \in \R$ are real variables.

Then $(1)$ defines $u$ as an implicit function of $x$ and $y$ on the region $x^2 + y^2 \le a^2$.