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

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

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).

${\displaystyle {}f}$

${\displaystyle L(f)={\operatorname {sup} \,(L(f(t_{0}),\ldots ,f(t_{k})),a=t_{0}\leq t_{1}\leq \ldots \leq t_{k-1}\leq t_{k}=b{\text{ Unterteilung}},\,k\in \mathbb {N} )}.}$

${\displaystyle {}\int _{\gamma }F:=\int _{a}^{b}\left\langle F(\gamma (t)),\gamma '(t)\right\rangle dt\,.}$

${\displaystyle {}\varphi }$

${\displaystyle {}P}$

${\displaystyle {}\mathbb {R} }$

${\displaystyle {}L\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$

${\displaystyle {}\varphi (P+v)=\varphi (P)+L(v)+\Vert {v}\Vert r(v)\,}$

gibt, wobei ${\displaystyle {}r\colon U{\left(0,\delta \right)}\rightarrow W}$ eine in ${\displaystyle {}0}$ stetige Abbildung mit ${\displaystyle {}r(0)=0}$ ist und die Gleichung für alle ${\displaystyle {}v\in V}$ mit ${\displaystyle {}P+v\in U{\left(P,\delta \right)}\subseteq G}$ gilt.

${\displaystyle {}f}$

${\displaystyle {}c}$

${\displaystyle {}N_{c}={\left\{x\in \mathbb {R} ^{n}\mid f(x)=c\right\}}\,.}$

${\displaystyle {}P}$

${\displaystyle {}\operatorname {rang} \,\left(D\varphi \right)_{P}={\min {\left(\dim _{}{\left(V\right)},\dim _{}{\left(W\right)}\right)}}\,}$

ist.

${\displaystyle {}\leq k}$

${\displaystyle {}f}$

${\displaystyle {}P}$

${\displaystyle \sum _{r={\left(r_{1},\ldots ,r_{n}\right)},\,\vert {\,r\,}\vert \leq k}{\frac {1}{r!}}D^{r}f(P)\cdot (x_{1}-a_{1})^{r_{1}}\cdots (x_{n}-a_{n})^{r_{n}}.}$

${\displaystyle V\longrightarrow V,\,P\longmapsto \operatorname {Grad} \,f(P),}$

heißt Gradientenfeld zu ${\displaystyle {}f}$.