Partial Differential Equation of Spheres in 3-Space

Theorem
The set of spheres in real Cartesian $3$-dimensional space can be described by the system of partial differential equations:


 * $\dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_x} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$

and if the spheres are expected to be real:
 * $z_{xx} z_{yy} > z_{xy}$

Proof
From Equation of Sphere, we have that the equation defining a general sphere $S$ is:
 * $\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$

where $a$, $b$ and $c$ are arbitrary constants.

We use the technique of Elimination of Constants by Partial Differentiation.

Taking the partial first derivatives $x$ and $y$ and simplifying, we get:

$2$ equations are insufficient to dispose of $3$ constants, so the process continues by taking the partial second derivatives $x$ and $y$:

Eliminating $z - c$:


 * $\dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_x} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$

Let $\lambda = \dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_y} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$.

Then:

and so:
 * $z_{xx} z_{yy} > z_{xy}$