Implicit Function/Examples/x^3 + y^3 - 3 x y = 0

From ProofWiki
Jump to navigation Jump to search

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 $a = 1$:

FoliumOfDescartes.png


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:

FoliumOfDescartes-section1.png $\quad$ FoliumOfDescartes-section2.png $\quad$ FoliumOfDescartes-section3.png


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