# 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.

## Sources

- 1926: E.L. Ince:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.211$ The Partial Differential Equations of All Planes and All Spheres