Definition:Implicit Function/General
< Definition:Implicit Function(Redirected from Definition:General Implicit Function)
Jump to navigation
Jump to search
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: $y x_1 + y^2 x_2 + {x_3}^2$
The following equation:
- $y x_1 + y^2 x_2 + {x_3}^2 = 0$
is an example of an implicit function of $x_1$, $x_2$ and $x_3$.
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 $x^2 + y^2 \le a^2$.
Also see
- Results about implicit functions can be found here.
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): implicit function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): implicit function