# 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:

 $\ds \dfrac {\partial^2 z} {\partial x^2}$ $=$ $\ds 0$ $\ds \dfrac {\partial^2 z} {\partial x \partial y}$ $=$ $\ds 0$ $\ds \dfrac {\partial^2 z} {\partial y^2}$ $=$ $\ds 0$

## 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:

 $\ds a$ $=$ $\ds \dfrac {-\alpha_1} {\alpha_3}$ $\ds b$ $=$ $\ds \dfrac {-\alpha_2} {\alpha_3}$ $\ds c$ $=$ $\ds \dfrac {-\gamma} {\alpha_3}$

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 with respect to $x$ and $y$, we get:

 $\ds \dfrac {\partial z} {\partial x}$ $=$ $\ds a$ $\ds \dfrac {\partial z} {\partial y}$ $=$ $\ds b$

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

 $\ds \dfrac {\partial^2 z} {\partial x^2}$ $=$ $\ds 0$ $\ds \dfrac {\partial^2 z} {\partial x \partial y}$ $=$ $\ds 0$ $\ds \dfrac {\partial^2 z} {\partial y^2}$ $=$ $\ds 0$

and the system of partial differential equations has been established.

$\blacksquare$

## Also defined as

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

 $\ds p$ $:=$ $\ds \dfrac {\partial z} {\partial x}$ $\ds q$ $:=$ $\ds \dfrac {\partial z} {\partial y}$ $\ds r$ $:=$ $\ds \dfrac {\partial^2 z} {\partial x^2}$ $\ds s$ $:=$ $\ds \dfrac {\partial^2 z} {\partial x \partial y}$ $\ds t$ $:=$ $\ds \dfrac {\partial^2 z} {\partial y^2}$

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