Nov 18, 2021

How to interpret equations with differentials?

Consider the following equation:

$x_{2}+y_{2}+z_{2}=4.$

Geometrically, this equation represents a sphere centered at $(0,0,0)$ with the radius $r=2$. The technical term for this is **implicit surface** because the relationship between $x$, $y$, and $z$ is defined implicitly.

The surface above can be also defined as:

$xdx+ydy+zdz=0,x=2,y=0,z=0.$

This equation has the following geometric interpretation.

Consider a tangent plane touching the implicitly defined surface at some point $(x_{0},y_{0},z_{0})$. The differential equation states that for any point $(x,y,z)$ on that plane, the following must hold:

$x_{0}(x−x_{0})+y_{0}(y−y_{0})+z_{0}(z−z_{0})=0,$

or more concisely,

$x_{0}Δx+y_{0}Δy+z_{0}Δz=0.$

All such planes turn out to be tangent planes to spheres centered at $(0,0,0)$, and the $x=2,y=0,z=0$ initial condition allows us to select the "correct" sphere of radius $r=2$.