Definition:Implicit Function
Definition
Consider a (real) function of two independent variables $z = \map f {x, y}$.
Let a relation between $x$ and $y$ be expressed in the form $\map f {x, y} = 0$ defined on some subset of $\R^2$.
If there exists a function:
- $y = \map g x$
defined on some real interval $\mathbb I$ such that:
- $\forall x \in \mathbb I: \map f {x, \map g x} = 0$
then the relation $\map f {x, y} = 0$ defines $y$ as an implicit function of $x$.
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: $x^2 + y^2 - r^2 = 0$
Consider the equation:
- $(1): \quad x^2 + y^2 - r^2 = 0$
where:
- $x, y \in \R$ are real variables
- $r \in \R_{>0}$ is a strictly positive real constant.
Then $(1)$ defines $y$ as an implicit function of $x$ on the closed interval $\closedint {-r} r$.
Example: $x^2 + y^2 + 1 = 0$
Consider the equation:
- $(1): \quad x^2 + y^2 + 1 = 0$
where $x, y \in \R$ are real variables.
Solving for $y$, we obtain:
- $y = \pm \sqrt {-1 - x^2}$
and it is seen that no $y \in \R$ can satisfy this equation.
Hence $(1)$ does not define a real function.
Example: $x^3 + y^3 - 3 x y = 0$
Consider the equation:
- $(1): \quad x^3 + y^3 - 3 x y = 0$
where $x, y \in \R$ are real variables.
Then $(1)$ defines $y$ as an implicit function of $x$ for all $x \in \R$.
Also see
For sufficient conditions for the existence of such functions:
- Results about implicit functions can be found here.
Sources
- 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text I$: Differentiation: Implicit functions
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: $1.2$ Implicit Functions
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text D$: Implicit Function: Definition $2.81$
- 1978: Garrett Birkhoff and Gian-Carlo Rota: Ordinary Differential Equations (3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction: Example $1$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): implicit