${\displaystyle {}x\in \mathbb {R} }$

${\displaystyle {}x>-1}$

${\displaystyle x\longmapsto \operatorname {Fak} \,(x):=\int _{0}^{\infty }t^{x}e^{-t}\,dt}$

die Fakultätsfunktion.

${\displaystyle {}d\colon M\times M\to \mathbb {R} }$

${\displaystyle {}x,y,z\in M}$

1. ${\displaystyle {}d\left(x,y\right)=0\Longleftrightarrow x=y}$ (Definitheit),
2. ${\displaystyle {}d\left(x,y\right)=d(y,x)}$ (Symmetrie), und
3. ${\displaystyle {}d\left(x,y\right)\leq d(x,z)+d(z,y)}$ (Dreiecksungleichung).

${\displaystyle {}f}$

${\displaystyle {}c\geq 0}$

${\displaystyle {}d{\left(f(x),f(y)\right)}\leq c\cdot d{\left(x,y\right)}\,}$

für alle ${\displaystyle {}x,y\in L}$ gibt.

${\displaystyle {}\operatorname {Jak} (f)_{P}:={\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial f_{1}}{\partial x_{n}}}(P)\\\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(P)&\ldots &{\frac {\partial f_{m}}{\partial x_{n}}}(P)\end{pmatrix}}\,}$

heißt die Jacobi-Matrix zu ${\displaystyle {}f}$ im Punkt ${\displaystyle {}P}$.

${\displaystyle {}V}$

${\displaystyle {}I\subseteq \mathbb {R} }$

${\displaystyle {}U\subseteq V}$

${\displaystyle f\colon I\times U\longrightarrow V,\,(t,v)\longmapsto f(t,v),}$

ein Vektorfeld auf ${\displaystyle {}U}$. Dann nennt man

${\displaystyle {}v'=f(t,v)\,}$

die gewöhnliche Differentialgleichung zum Vektorfeld ${\displaystyle {}f}$.