Partial Differential Equation of Planes in 3-Space

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

Proof
From Equation of Plane, we have that the equation defining a general plane $P$ is:
 * $\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$

which can be written as:
 * $z = a x + b y + c$

by setting:

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

We see we have:
 * $1$ dependent variable, that is: $z$
 * $2$ independent variables, that is: $x$ and $y$
 * $3$ constants, that is: $a$, $b$ and $c$.

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

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

and the system of partial differential equations has been established.

Also defined as
Some older sources suggest that it is "customary" to assign a standard system of labels to these partial differential equations:

but this is a technique which is rarely emphasised in more modern works.