General Implicit Function/Examples

Examples of General Implicit Functions

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 $x^2 + y^2 \le a^2$.