??? ?? ???? ?????? ?????????. ?? ???????? ?????? ? ???????? ??????? (variety), ??? ????? ??????? .

? ??????? , ????? ??? ???? ????? ????? ??? ???? ????? (??? ?????) ???? ????? ??? ???????? ???????? , ?? ???? ???? ??? ??? ?????? ??????? ????. ??? ??? ???? ???? ??? ????? ??????; ????? ??? ???? ???? ?? ???? ???? ????? ????? ??????? . ??????? ????????? ????? ?????? ????? ???? ???? ????? ?????? ??? ??? ??? ??? ???? ????? ????? ???????? ?????? ?? ?????. ?? ?? ?? ?????? ????? ?????? ???? ? ???? ??????? ??????? ??????? ??? ?????? ????? ???? ???????. ??????, ???? ? ?????? ???????? ???? ??????? ?????? ??? 180 ?????, ?? ??? ???? (????? ?????? ????? ?????) ???? ????? ??????? ?? ???? ?????? ?????.

?????? ????? [ ????? ??? ???? | ????? ]

???? ?????? ?????? ?????????? ???????, ???? ????? ??? ????? ?? ????? ????? ????, ??? ???????? ?????? ???????? ( ????????? ?????? , ?????? ?????? ????), ????????? ???? ????? ???? ?????? ????? ????. ???? ?????, ??? ???? ??????? ???? ????? ??? ???? ???????? ???? ???- ???? ?????? ??? ??????? ????. ????? ??? ???? "?????" ?? ????? ?? ??? ????? ????? (?????? ???????? ) ?? ????? ???? ??? ??? ??? ???? ??? ????? ????????? ????? (????? ?????????? ????). ?? ??? ??????? ????, ??? ?? ????????????? ??? ????? ??????, ???? ??? , ????? ?? ????? ???? ???? . ???? ???? ?????? ??? ???? ????? ????????? ????? ????? ?? ?????? ?? ????? ?????? ??? ????? ???????? ???? ?????. ????? ??? ????? ???? ????? , ???? ?????? ???? ????? ??? ????? ???????? ????? ?????? ?????????? ????? ?????. ????? ???? ?????? ????????? ?? ???????????? ?? ??? ?? ?? ??? ?? ???? ???- ??? ?????? ???? ????? ????.

???? ???? ???? ??? ???? ????? ????????? ????? ?????? ????? ?? ????? ?????? ??? ???? ?????? ???? ???? ??? ??? ??????? ???????? . ????? ??? ?????? ????? ????? ?????? ????? ??? ??? ????? ???????? ????? ??? ???????. ?? ????? ?????? ???? n ??? ?? ????? ????? ????? ???? 2n.

????? ????????? [ ????? ??? ???? | ????? ]

??? ????? ? ????? ?????????

????? ??? ???? ???????? , ?????? ?? ??????? ?????? ?????? , ??? ?????? ???? ????? ????? ?? ????? ? ???????????? ?- ${\displaystyle \mathbb {R} ^{n}}$. ???? n ????? ????? ??? ??????.
????? ???? ???? ????? ????? ???? ?? ?????? ?????? ? ???? ?? ?????, ????? ?????? ?? ?? ??????? ????? ??? ????? ????, ?? ??? ????? ?????? ???????????? ?????? ????? ?? ????? ?????? ?- ${\displaystyle \mathbb {R} ^{n}}$, (?? ???????, ?? ?? ????? ???????? ???? ?- ${\displaystyle \mathbb {R} ^{n}}$ ?? ?- ${\displaystyle \mathbb {R} ^{n}}$ ????). ?? ????? ??????, ??? ?? ????????????? ???, ????? ??? , ????? ?? ????? ???? ???? .

????? ??????? [ ????? ??? ???? | ????? ]

???? ???????? M ???? ????? ????????? ?? ??? ???? ???????? , ????? ?? ??????? ?????? ?????? ????? ?? ????? ???? ${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$ (???? ${\displaystyle \ A}$ ????? ????????) ?? ??? ${\displaystyle \ U_{\alpha }}$ ??????????? ?????? ????? ${\displaystyle \ V_{\alpha }}$, ${\displaystyle \ V_{\alpha }\subset \mathbb {R} ^{n}}$. ???????? ????????????? ??? ${\displaystyle \ U_{\alpha }}$ ?- ${\displaystyle \ V_{\alpha }}$ ?????? ?- ${\displaystyle \varphi _{\alpha }}$.
?? ????? ???? ${\displaystyle \alpha {\mbox{ ,}}\beta \in A}$ ???????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$ ???? ??????? ?- ${\displaystyle \mathbb {R} ^{n}}$ ????? ??? ??????? ????? (????? ??? ?????? ??????). ???? ${\displaystyle \left(\ U_{\alpha }{\mbox{,}}\varphi _{\alpha }\right)}$ ???? ??? ????? ?? ????? ???? ????. ??? ${\displaystyle \!\ \alpha }$, ${\displaystyle \varphi _{\alpha }=\left(x_{1},x_{2},\dots ,x_{n}\right)}$ ???? ${\displaystyle \!\ x_{i}}$ ??? ??????? ?? ?? ????? ?-M ?? ${\displaystyle \mathbb {R} }$. ????????? ${\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)}$ ?????? ??????????? ??????? ?? M.

????? ????? ?????? ???? ?????? ??????? ??? ???? ?????. ????? ?????? ????? M ?????? ?? ??? ???? ${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$ ?? ????? , ?? ???? ${\displaystyle \ U_{\alpha }}$ ????? ??????? ??-??-????? ??? ${\displaystyle \varphi _{\alpha }}$ ??????? ?? ${\displaystyle \ U_{\alpha }}$ ?????? ????? ?- ${\displaystyle \mathbb {R} ^{n}}$, ?? ???? ????? ?? M ???? ?? ????? ????????? ?? ?? ????????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$ ?? ???????? ?????? (????? ??? ?????? ??????), ????????? ?- ${\displaystyle \mathbb {R} ^{n}}$ ?????. ????? ?????? ????? ? ????????? ????? ?? ??? ????? ???? ?????????? ????? ?? ??????? ??????? ?? ??????? ??????? ??????? ?- ${\displaystyle \ U_{\alpha }}$, ???? ${\displaystyle \alpha \in A}$ ?????.

??? ?????? ?????? ???? ?? ???? ???????? ???? ????? ????? ???? ??? ?????? ????? ????? ????? ?????? ?????? ?? ?????, ?? ??????? ??? ?????? ???? ?????? ?? ??????? ??????? ????? ?? ?????? ??????? ??????? ?????? ????? ?? ?????? ??????.

????, ????? ?? ?????? ????? ?? ??? ???? ??????? ????? ???? ???????? ?????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$. ?? ????????? ???????? ??????? ?? ?????? ?? ???? ?????? (???? ??????) ?? ????? ????????? ?? ${\displaystyle \mathbb {R} ^{n}}$ ??????? ?????? ???? ???????? ?? M.
?? ????? ?????????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$ ????? ?? ?????? , ????? ?? ???????? ?? ????????? ${\displaystyle \varphi _{\alpha }}$ ?????? ???? ?????????? ????? ?? M, ????? ????? ?????????.

?? ????? ????? (??? ???? ) ?? ?????? ????? ????? ?????, ????? ??? ??? ?????? ???????. ????? ?? ?????? ????? ??????? ?? ??? ????? ????? ???????? ????? ????? ????? (?????????? ????? ?????????).

?????? ????? [ ????? ??? ???? | ????? ]

??? ????? ? ????? ????

????? ???? (?? ????? ??????????? ) ??? ????? ????????? ??? ?????? ?????????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$ ????? ???????? ????? . ????? ?? ?????? ????? ???? ???? ?????????? . ??? ?????? ?????? ?????? ?? ?????? ???? ???? ??? ???? ??????????, ????? ?????? ???????? ?? ??? ?? ??? ???????? ?? "???????" ??? ????. (???? ????? ????? ???? ???? ??? ?????? ?? ????? ?? ???????.)
?????? ????? ??? ???? ???? ??????? ?????? ?????? ?? ?????? ??????? (??? ???? ????????? ??????? , ???? ????? ???????? ???????? ???????????? ?????? ???') ?????? ????????.
????? ??????? ???? ?????? ?????? ??????????? ?? ????? ???? ?????? ?? ????? ?????? ??????????? ???? ?? ????? (?? ??????? ???? ??????? ?? ?????) ???? ????? ?? , ?? ???? ?? ?????? ?????? ???? - ???? ????? ?????? ??? ????.

??????? ??????? ????? [ ????? ??? ???? | ????? ]

• ${\displaystyle \mathbb {R} ^{n}}$ ??? ????? ???? ???? n, ?? ??? ??? ????? ?????? ?? ?? ?????, ??? ?? ????? ?????.
• ?? ???? ?????? ???? ???? n, ??? ????? ???? ???? n ?? ??? ??? ????? ????? ?? ?? ?????, ?? ??????????? ???????? ?? ??? ???? ?????.
• ?????? ?-n ?????, ${\displaystyle \mathbb {S} ^{n}}$, ???? ??? ???? ?????? ?-n+1 ????, ??? ????? ???? ???? n.

?????? ?? ${\displaystyle \mathbb {S} ^{1}}$, ?????? ???-????? (??????? ??????), ???? ???? ????? ??? ?????? ?????. ???? ?????? ?????? ????? ?????? ?? ??? ????:
${\displaystyle \left(\{\ ({\sqrt {1-y^{2}}},y):y\in (-1,1)\}\,(x,y)\rightarrow y\right)}$
???? ????? ????? ???? ??? ?????? ????? ???????? ????- ??? ????? ?? ?????, ?????? ?????? ??? ????????, ???? ???? ???? ?? ?????? ?? ??? ?-Y.
?? ??? ????? ${\displaystyle \mathbb {S} ^{1}}$, ?? ???? ?????? ?- ${\displaystyle \mathbb {C} }$ ???? ????? ??- ${\displaystyle \mathbb {S} ^{1}}$ ?? ?? ???? ?? ?????, ????? ????? ????????.

???????? ????? ?? ?????? [ ????? ??? ???? | ????? ]

?????? ?????? ????? ?? ??????? ?????? ??? ?????? ?????? ????? ?????? ?? ?????? ?- ????? ??? ???? ?????? ???????????? ???????, ?? ??? ????? ?????? ????????? ?????.
??????? ${\displaystyle \ f:M\rightarrow \mathbb {R} }$ ????? "??????? ???? ?????? p" ?? ????? ??? ${\displaystyle \ (U,\varphi )}$ ?? ????????? ${\displaystyle \ f\circ \varphi ^{-1}:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ ???? . ??? ????? ???????? ?? ?? ?????? ????? ????????? ?????? ?????? ????? ??? . ???? ????? ?????? ???? ??? ???????? ${\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}$ ??? ??????? ????, ?? ?? ???? ?? ????? ?????? ??????? ???? ????, ???????? ?????? ??? ??? ??? ???? ????? ?? ??? ?? ??? ???? (??: ${\displaystyle \ f\circ \varphi _{\beta }^{-1}=\left(f\circ \varphi _{\alpha }^{-1}\right)\circ \left(\varphi _{\alpha }\circ \varphi _{\beta }^{-1}\right)}$, ????? ?? ????? ???????? ????? ???? ???? ??????? ????). ??????? ??? ????? ?? ??????? ???? ??? ??????????? ???? ?????? ???????????.

??? ???? ????? ??????? ${\displaystyle \ F:M\rightarrow N}$ ??? ??????? ???? ??? ??????? M ?-N ?? ??? ??? ???? ${\displaystyle (U,\varphi ),(W,\psi )}$ ???????? ${\displaystyle \psi \circ F\circ \varphi ^{-1}}$ ??? ???? ????? ??????.

?? ?????? ???? ????? ${\displaystyle \ (a,b)\subset \mathbb {R} }$ ??? ????? (?? ????? ????? ?????? ????? ????? ????????- ???? ????? ????) ?? ? ????? ${\displaystyle \gamma :(a,b)\rightarrow M}$ ??? ????? ???? (?? ????? ????) ?? ??? ??????? ???? ??? ??????? (a,b) ?-M.

??? ??????? ??????? ?? ?????? ????? ??? ??????? ????? ???????? ????? ????? ?-1 ?? ????? ????? ????? ???????? ???? ?????? ????? ?????? ???? (bump function). ???????? ???????? ???? ?? ?????? ??? ?? ?????? ????????, ?????? ?? ????? ????????, ?? ?? ??????? ??????? ?????? ?? ??? ????? - ??? ?????? ?? ?? ????.

????? ????? [ ????? ??? ???? | ????? ]

??? ????? ? ????? ?????

????? ????? ??? ???? ?????? ????? ?? ????? ???? ?????? ?" ????? ??????? " ?? ???? ????? ????? ?????, ????? ???? ???? ?? ???????? ?????? ???? ???? ?????? ?? ?????? ????????? (????? ???? ???? ??????? ?????). ????? ????? ??? ???? ???? ??????? ??? ? ????? ??????? ??, ???? ????? ?? ????????? ????? ????? ?????? ?? ? ?????? ????????? .
??? ????? p, ????? ????? ?????? ?????? ${\displaystyle T_{p}M}$, ??? ???? ?????? ??????? ?? ??????????? ??????? ?? ????????? ?????? ??? ?????? ???????? ?????? ????? ?? ????????? ???????? ?? ??? ?????? (" ??? ??????? "):

${\displaystyle \ v(fg)=v(f)g(p)+f(p)v(g)}$ ???? ${\displaystyle \ f,g}$ ?? ???????? ????? ?- ${\displaystyle \ v}$ ??? ????? ????? ?????.

????????? ?? ????? ????? ?????? ??????? ??????.

????? ??????? [ ????? ??? ???? | ????? ]

????? ?????? ?????? ?????- ????? ??????? ??? ????? ????????? ?????? ??? ??????? ????? ?????? ??????? ??????? ????? ??? ??? ???? ???????? ${\displaystyle \varphi \circ \psi ^{-1}}$ ??? ??????? ???????. ?? ?????? ???? ???? ?????? ??? ???????? ?? ???????? ???????? ????? ????? ???? ?????? ????????????.

????? ?????? ??? ??????? ?????? ?????????? ???? ???-????? ?? ????? ????? ??????? ????? ????, ???? ?????? ????? ???? ???? ?????.

??????? ??????? [ ????? ??? ???? | ????? ]

• ??? ???? (????)
• ????? ?????
• ????? ?????

??? ?? [ ????? ??? ???? | ????? ]

• ???????? ???????????
• ????????????
• ???????????

??????? ???????? [ ????? ??? ???? | ????? ]

???? ?????? ????? ????? ? ??????????

• ????? , ???? ??????????? ???????? (???????)
• ????? , ???? MathWorld (???????)