Partial Differential Equation of Planes in 3-Space

From ProofWiki
Jump to navigation Jump to search

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

Retrieved from "https://proofwiki.org/w/index.php?title=Partial_Differential_Equation_of_Planes_in_3-Space&oldid=482971"