Implicit Function/Examples/x^3 + y^3 - 3 x y = 0
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Examples of Implicit Functions
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 $x \in \R$.
Proof
This is the Cartesian form of the equation for the folium of Descartes:
- $x^3 + y^3 - 3 a x y = 0$
for $a = 1$:
It is seen that when $x \le 0$, $y$ is uniquely determined by $x$.
Also, from Maximum Abscissa for Loop of Folium of Descartes, $y$ is also uniquely determined by $x$ when $x > 2^{2/3}$.
In between those values, for $y$ to be defined as a function of $x$ it is necessary to choose one of the $3$ possible values that $y$ can take for each $y$.
For example:
Thus we have:
- $y = \map f {x, \map g x}$
where $\map g x$ is not straightforward to define.
This demonstrates that $y$ is an implicit function of $x$.
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text D$: Implicit Function