Equations of Surfaces in 3-Dimensions

Recall that in 2-dimensional space, an equation containing the variables $x$ and $y$ represents a curve. We will now extend this to look at equations involving three variables, commonly $x$, $y$, and $z$ (though sometimes $x_1$, $x_2$, and $x_3$) which happen to form surfaces in 3-dimensional space.

We will first outline some examples of equations and their corresponding surfaces in 3-dimensions.

$x = 1$	$z = 2$	$x + y = z$
$y = x^2 - z^2$	$z = x^2$	$x^2 + y^2 = 1$
$x^2 + z^2 = 3$	$x^2 + y^2 + z^2 = 1$	$x^2 + y^2 = z$