ã©ããããã«ã¡ã¯ãððã¿ããððã§ããä»åã¯åãééããæã«æåããå®çããããŠãããã«å¯ŸããŠç³»ããããŠããããšæããŸããã¡ãªã¿ã«è¿œèšãããŠããå¯èœæ§ããããŸãã
蚌æã¯æžãããæžããªãã£ããããŸãã
$M$ãåãä»ãå¯èœãªåºåçã«æ»ãããªå€æ§äœãšãã$\omega$ã埮å圢åŒãšããããã®ãšããæ¬¡ã®çåŒãæãç«ã€ã
$$
\int_{\partial M}\omega=\int_{M}d\omega
$$
æè¿ç¥ã£ãè¶ åŒ·ãå®çã§ãã埮å圢åŒããã¡ããšåŠãã ç¶æ ã§ãã®å®çãåèŠééãããšæ¬åœã«éããã»ã©æåããŸããããããããã¿ãªããããæåã奪ã£ãŠããŸã£ããããããŸãããããã§ããã°ç³ãèš³ãªãã§ãããã®å®çã¯ãŸãã«ãäŒç·ååããšåŒã¶ã«çžå¿ãããšæããŸãã
äžã®åŒã§åºãŠãã$d$ã¯éåžžã®åŸ®åã§ã¯ãªãå€åŸ®åãšãããã®ã§ãèŠããã«å šåŸ®åãäžè¬ã®åŸ®å圢åŒã«æ¡åŒµãããã®ãšèããããšãã§ããŸãããããŸãã匷ãå®çãªã®ã§ãç³»ããããããããŸããç³»ã«åŒ·ã䞻匵ãããããæåããããã§ã¯ãªãã§ãã
åºé$[a,b]$äžã§åŸ®åå¯èœãªé¢æ°é¢æ°$f$ã«ã€ããŠã
$$
\int_a^bf'(x)dx=f(b)-f(a)
$$
Stokes-Cartanã®å®çããã
\begin{aligned}
\int_a^bf'(x)dx
&=\int_a^bdf\\
&=\int_{\partial[a,b]}f\\
&=f(b)-f(a)
\end{aligned}
FTCãç³»ãšããã®ã¯ãªããªãåãã§ãããã¡ãªã¿ã«ããã¯ãã«è§£æã®æ°å€ãã®å ¬åŒãå°ãããšãã§ããŸãã
$\R^3$ã§å®çŸ©ãããæ»ãããªãã¯ãã«å Ž$\bm F$ã«å¯ŸããŠã$V$ã$\R^3$ã«ãããŠæ»ãããªå¢ç$\partial V$ããã€æçé åãšãããšã
$$
\iiint_V\nabla\cdot\bm FdV=\iint_{\partial V}\bm F\cdot d\bm S
$$
ãæç«ããã
Stokes-Cartanã®å®çããã
\begin{aligned}
\int_{\partial V}\bm F\cdot (dx\wedge dy+dy\wedge dz+dz\wedge dx)
&=\int_V\left(\frac{\partial\bm F}{\partial x}+\frac{\partial\bm F}{\partial y}+\frac{\partial\bm F}{\partial z}\right)dx\wedge dy\wedge dz\\
&=\int_V\nabla\cdot\bm FdV
\end{aligned}
$\R^3$ã§å®çŸ©ãããæ»ãããªãã¯ãã«å Ž$\bm F$ã«å¯ŸããŠã$S$ã$\R^3$ã«ãããŠæ»ãããªå¢ç$\partial S$ããã€æçãªé¢ãšãããšã
$$
\iint_S(\nabla\times\bm F)\cdot d\bm S=\int_{\partial S}\bm F\cdot d\bm l
$$
Stokes-Cartanã®å®çãã
\begin{aligned}
\int_{\partial S}\bm F\cdot(dx+dy+dz)
&=\int_S\left(\frac{\partial\bm F}{\partial y}-\frac{\partial\bm F}{\partial z}\right)dy\wedge dz+\left(\frac{\partial\bm F}{\partial z}-\frac{\partial\bm F}{\partial x}\right)dz\wedge dx+\left(\frac{\partial\bm F}{\partial x}-\frac{\partial\bm F}{\partial y}\right)dx\wedge dy\\
&=\int_S(\nabla\times\bm F)\cdot d\bm S
\end{aligned}
ãç³»ãæãå端ãããªãã§ãããã¡ãªã¿ã«äžèšäºã€ã®å®çãšæ¬¡ã®å®çã¯ã»ãšãã©åãèšŒæææ³ãåã£ãŠããŠããããããããããããšæããŠããã®ã§æåãæ·±ãŸã£ããšããã®ã¯ãããšæããŸãã
$xy$å¹³é¢äžã®é å$D$ã«ãããŠã$D$ã®å
éšã§$C^1$çŽã§ããé£ç¶é¢æ°$P(x,y),Q(x,y)$ã«å¯ŸããŠæ¬¡ã®çåŒãæãç«ã€ã
$$
\int_{\partial D}Pdx+Qdy=\iint_D\left(\frac{\partial Q}{\partial y}-\frac{\partial P}{\partial x}\right)dxdy
$$
Stokes-Cartanã®å®çãã
\begin{aligned}
\int_{\partial D}Pdx+Qdy
&=\int_{D}\left(\frac{\partial Q}{\partial y}-\frac{\partial P}{\partial x}\right)dx\wedge dy\\
&=\int_D\left(\frac{\partial Q}{\partial y}-\frac{\partial P}{\partial x}\right)dxdy
\end{aligned}
éçšã®ã³ããšããŠã¯å€åŸ®åã®éãããã®ã¯å€§å€ãªã®ã§ãå¢çåŽããåŒå€åœ¢ãããšããã§ãã(èŠãã¡ãã£ãŠãããªããããªããšããªããŠãããã©)埮å圢åŒã圢åŒçã«åŠãã ã ãã®äººã¯ãæèšéåæžå®çããããã«ãªã£ãŠããŸããŸããããã¡ããšåŠã¶ãšãã®å®çã®çŸãããããããããšæããŸãã
ãã¯ãã«è§£æã§ã¯åŸé ãçºæ£ãå転ã®éãåºãŠããŸããããWedgeç©ãšå€åŸ®åãHodge starãªã©ãçšããããšã§èšè¿°ã§ããŸããããããåäžã®èæ¯ãæã£ãŠãããã§ãããGreenã®å®çãåºãŠããã®ã§Cauchyã®ç©åå®çã蚌æããŠãããŸãã
$D$ãé åã$f\colon\C\mapsto\C$ã$D$ã®å
éšã§æ£åãã€å¢çäžã§é£ç¶ã§ãã颿°ã§ãããšãã
$$
\oint_{\partial D}f(z)dz=0
$$
Greenã®å®çã«ããã$\partial D=C$ãšããŠã$f(z)=u(x,y)+iv(x,y)$ãšãããš$(u,v\colon\R^2\mapsto\R^2)$
\begin{aligned}
\oint_Cf(z)dz
&=\oint_C(u+v)(dx+idy)\\
&=\oint(udx-vdy)+i\oint(udy+vdx)\\
&=-\int_{D}\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)dxdy+i\int_{D}\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)dxdy\\
&=0
\end{aligned}
æåŸã®çåŒã§ã¯Cauchy-Riemannã®é¢ä¿åŒ$u_x=v_y, u_y=-v_x$ãçšããã
ãããå€éç©åçµè·¯åããŠãçæ°ã®å®çŸ©ãåœãŠã¯ãããšçæ°å®çãåºãŠããŸãããããããçºå±ããããšè€çŽ è§£æã®å€ãã®å®çããç³»ãã«ã§ããæ°ãããŠããŸããããStokes-Cartanãå å ããŠãããšããããã¯è€çŽ è§£æãåãã ããªæ°ãããã®ã§ããã®èŸºã§ãããšããŸãã(ããããããã¯è€çŽ é¢æ°è«ã®æç§æžã§ããªãã§ããã)
ããã«ã€ããŠã¯ 解説ããèšäº ãããã®ã§ãã¡ããèŠãŠã»ããã§ãã䞻匵ã®å 容ãšããŠã¯æ¬¡ã®éãã§ãã
$$
f(z)=\sum_{k=0}^\infty\frac{\varphi(k)}{k!}(-z)^k
$$
ãšå±éããããšãã
$$
\int_0^\infty x^{s-1}f(x)dx=\Gamma(s)\varphi(-s)
$$
ã§äžããããã
ããã«ã¯äž»åŒµã«äžåããããŸããããã®ãããã®è©±ãå«ããŠäžã§ç€ºããèšäºã§ããç³»ãšãããšãä»£å ¥ããŠèšç®ã§ãããã®ãšã...ïŒ
ããã¯æãå
¥ããæ·±ãçåŒã§ããRiemannã®ãŒãŒã¿é¢æ°
$$
\zeta(s)=\sum_{n>0}\frac1{n^s}
$$
ã«äŒŒãç©åãšçŽæ°ã§ãåãäžåŠäºå¹Žçåœæããããã£ãŠããªããã®ããããŸããããã®äžã€ã§ãã
$$ \sum_{n>0}\frac1{n^n}, \int_0^1\frac1{x^x},\int_0^\infty\frac1{x^x} $$
ãã®ãã¡ãåè äºã€ãçãããšäž»åŒµããã®ããã®ç¯ã§ã®äž»å®çã§ãã
çåŒ
$$
\sum_{n>0}\frac1{n^n}= \int_0^1\frac1{x^x}
$$
ãæç«ããã
çŽæ¥å€åœ¢ã«ãã£ãŠç€ºãã
\begin{aligned}
\int_0^1\frac1{x^x}
&=\int_0^1e^{-x\ln x}\\
&=\int_0^1\sum_{n\ge0}\frac{(-x\ln x)^n}{n!}dx\\
&=\sum_{n\ge0}\frac{(-1)^n}{n!}\int_0^1x^n\ln^n xdx\\
&=\sum_{n\ge0}\frac{(-1)^n}{n!}\int_0^\infty e^{-(n+1)t}(-t)^ndt&(x\mapsto e^{-t})\\
&=\sum_{n\ge0}\frac1{n!}\frac{\Gamma(n+1)}{(n+1)^{n+1}}\\
&=\sum_{n>0}\frac1{n^n}
\end{aligned}
åŒã®å¯Ÿç§°æ§ãå端ãããªãã§ãããè¿äŒŒå€ãæ±ããéãåè
äºã€ãã»ãŒåãå€ã§ãã£ãã®ã§åããããããªãããšããäºæ³ã¯ãã£ãã®ã§ãããå®éã«èšŒæãããšãã¯ããªãæåããŸãããåœæã®åã¯ã¬ã³ã颿°ãç¥ããŸããã§ããããããã®å°åºã®é
$$
\int_0^\infty t^{n-1}e^{-t}dt=n!
$$
ãæãç«ã€ã¯ãã ãšæã£ãŠããã®ã§ã蚌æèªäœã¯ãããŸã§å€§å€ã§ã¯ãªãã£ãã§ããåœæã¯äœãèšããçŽæ°ãšç©åã亀æããŠãŸããã...
ãã®å®çã¯èšŒæã奜ãã§ãã
æ£ã®å®è»žäžã§å¯Ÿæ°åžã§ããã$f(x+1)=xf(x), f(1)=1$ã§ããè§£æé¢æ°ã¯å¯äžã€ã§ããã
$0< x\le1, n\in\N$ãšãããšãäžãããã颿°çåŒãã
$$
f(x+n)=(x)_{n}f(x)
$$
ã§ããããŸãã$s(x_1, x_2)$ãäºç¹$(x_1,\ln f(x_1)),(x_2,\ln f(x_2))$ãçµãã çŽç·ã®åŸããšãã$x_1< x_2$ãšããŠãããšã察æ°åžã§ãããšããæ¡ä»¶ãã
$$
s(n-1,n)\le s(n,n+x)\le s(n,n+1)
$$
ãšãªããŸãããããå€åœ¢ããŠãããš
$$
s(n-1,n)\le s(n,n+x)\le s(n,n+1)\\
x(\ln f(n)-\ln f(n-1))\le \ln f(x+n)-\ln f(n)\le x(\ln f(n+1)-\ln f(n))\\
(n-1)^x(n-1)!\le f(x+n)\le n^x(n-1)!
$$
ãšãªããã¯ããã«äžããåŒãã
$$
\frac{(n-1)^x(n-1)!}{(x)_n}\le f(x)\le\frac{n^x(n-1)!}{(x)_n}
$$
å³ã¡
$$
\frac{n^xn!}{(x)_{n+1}}\le f(x)\le \frac{n^x(n-1)!}{(x)_n}\frac{x+n}{n}
$$
ãšãªããããã§$n\to\infty$ã®æ¥µéãèãããšãæã¿æã¡ã®åçã«ãã
$$
f(x)=\lim_{n\to\infty}\frac{n^xn!}{(x)_{n+1}}
$$
ãšãªãã$0< x\le1$ã«ãããŠã¯äžæçãªã®ã§ããããäžèŽã®å®çã«ããæ¡ä»¶ãæºããè§£æé¢æ°ã¯å¯äžã€ã§ããã
蚌æã«æã¿æã¡ã®åçã䜿ãããšãæã¿æã¡ã®åçã§æããã®ãçŽæ¥æ¥µéãããããªããŠããããšããæå³ã§æ°ããèŠç¹ãäžããããŸããããããèªãã ãšããããããããé ãã...!ããšæã£ãããšãããèŠããŠãŸãã
è€çŽ è§£æã«ãããŠããªã匷ãå®çã§ããå€ç«çæ§ç¹ç°ç¹è¿åã®åããé«ã äžã€ã®ç¹ãé€ã$\C$ãèŠãããšã䞻匵ããå®çã§ãæ£ç¢ºãªã¹ããŒãã¡ã³ãã¯ä»¥äžã®ããã«ãªããŸãã
$f(z)$ã$U_\delta(z_0)=\left\{z\in\C\colon0<\lvert z-z_0\rvert<\delta\right\}$ã§æ£åãã€$z_0$ã§çæ§ç¹ç°ç¹ãæã€ãšããããã®ãšãã
$$
\exists a\in\C, \forall b\in\C\setminus\{a\},\exists z\in U_\delta(z_0)\;s.t.\;f(z)=b
$$
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\begin{aligned}
X\cong Y\\
\exists!y\in Y[X\cong Y_y]\\
\exists!x\in X[Y\cong X_x]\\
\end{aligned}
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äžèŸºã®é·ãã$\sqrt{a},\sqrt{b},\sqrt{c}$ã®äžè§åœ¢ã®é¢ç©$S$ã¯
$$
S =\frac{\sqrt{\alpha\beta+\beta\gamma+\gamma\alpha}}{2}
$$
äœã
$$s=\displaystyle\frac{a+b+c}{2}, \alpha=s-a,\beta=s-b,\gamma=s-c$$
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