From 2e0661ec21bed6a1722f74933a623929ae081a15 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Vely=C4=8Dko?= Date: Thu, 15 May 2025 07:27:37 +0200 Subject: [PATCH] oprava chybneho vzorce --- prvni-integral.tex | 2 +- skripta.pdf | Bin 194710 -> 194399 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/prvni-integral.tex b/prvni-integral.tex index 8b66abc..c4ebc9f 100644 --- a/prvni-integral.tex +++ b/prvni-integral.tex @@ -36,7 +36,7 @@ $$ \odv*{V(x'(t), x(t))}{t} = x'\cdot x'' + kxx' = x'(x'' + kx) = 0. $$ Je dobré si uvědomit, že prvních integrálů, které jsou lineárně nezávislé v bodě $x_0$, může být nejvýše $n - 1$, což plyne z Věty \ref{thm-first-integral}, a to tak, že pro každý první integrál platí $\nabla U_i(x_0) \perp f(x_0) \neq 0$ a v prostoru dimenze $n$ existuje přesně $n - 1$ lineárně nezávislých vektorů kolmých na daný vektor. Metodu prvních integrálů budeme používat k redukci počtu rovnic v soustavách ODR. Skutečně, mějme soustavu -$$ \begin{brace}x' = f(x, y)\\y' = g(x, y)\end{brace},$$ +$$ \begin{cases}x' = f(x, y);\\y' = g(x, y),\end{cases}$$ a nechť $V(x, y) = K$ je její první integrál. Ve většině případů z toho můžeme vyjádřit $x$ jakožto funkci $x = h(y, K)$, což po dosazení do druhé rovnice nám dává $$ y' = g(h(y, K), y), $$ čímž jsme zredukovali počet rovnic na jednu. K přesné formulaci právě popsaného postupu použijeme následující větu. diff --git a/skripta.pdf b/skripta.pdf index 0364fb789256b2b03f41637886e849cdcce1b5eb..9dff3e1dc7613d54921b6dcaa0d2f50834acd9c9 100644 GIT binary patch delta 24556 zcmV)FK)=71@eAMX3xKo%D|ml3H8?UhRFWnpa!c-rM$TW=i6b$-vUn7l~{rd@UF zE^(P4K@z}zvMu{1n}?;i<2V}yaIiqU|32TTdsSBthcqSG&MpL!J=JwNbuQnz&FK#( zc@p?vo*bQ=3|rv;P9Hzv+3)c0Ur+z|;q&TCIqSIE= zr|Dfqbw=L)BYs};%nyJ4r~HW%_lA#yxybvt0{Of2+@|N4o8O@F*JHQZWB&QjktfY9 z(p>JY5-;wP&!{Pq=AS>LYmvGq5J|NOEww!14$LNu4W23W=8tbPJUtf^)4C0N z)OJ4YuzuLcIwR}qmU!O#bL17i_4tdPn9)8Kc90({vP@zl zb!oG^fmPEuZSQ~cb`p~sKeCzX57dNTYP|_1b&*HQ~PS4CpSKWcNqs*vyIH%kVBz>sp8Gv2&=dp63x`Bq3O zqun51VzXA1oxSgUjM z3(9!gxLVKpmfr{ANmrd)^zFT2-J5joy_1f!4Ht&tPNpSY+FQ)>aq{1I3KNm1dgjbXuF`BFj!x5EsJxtS(&w&$tI?W?_qu! zzFVtUf*}h~vYAvrE@|#I395xzF!dzs24mz^g#dq&d#_*LG~F`MkIRX$>)@=eHty&? z&+R6=o-_-{pWYoX{u!7#l!kQ37dL!AOn_UtT4!>UNj2GVcV zV8MSvrH>VPGF@ab=B#@7IiCfJ>r9UK&B{Nj<$`UC$tND+Z$pYZDZrMnY_S_pLBn33 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