From 6013e6d1605ed2c12172b9e4387bf08ebd7c6056 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 28 Feb 2025 20:57:58 +0100 Subject: [PATCH] better wording --- jednoznacnost-reseni.tex | 2 +- skripta.pdf | Bin 81444 -> 81438 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex index c9770b7..419827b 100644 --- a/jednoznacnost-reseni.tex +++ b/jednoznacnost-reseni.tex @@ -44,7 +44,7 @@ Zavedeme značení $f \in C_x^1(\Omega)$, jestliže $\pdv{f}{x_i}$ existují a j \begin{proof} Mějme $(x_0, t_0) \in \Omega$. Nechť $\delta > 0$ je takové, že množina - $$M = \overline{U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)}$$ je podmnožinou $\Omega$. Z kompaktnosti $M$ máme, že její parciální derivace $\pdv{f}{x_i}$ jsou omezené konstantou $K$. + $$M = \overline{U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)}$$ je podmnožinou $\Omega$. Z kompaktnosti $M$ máme, že parciální derivace $\pdv{f}{x_i}$ jsou omezené konstantou $K$. Dále mějme dva body $(x, t),(y,t) \in M$. Potom $|f(x,t) - f(y, t)| = |f(x + 0(y - x), t) - f(x + 1(y - x)t)| = |\left[f(x+s(y - x), t)\right]^1_0| = |\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds|$. Pro derivaci $f$ platí $\odv*{f(x + s(y - x), t)}{s} = \sum_{i=1}^n \pdv{f}{x_i}(x + s(y - x), t) (y_i - x_i)$. Z toho máme, že $$\left|\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds\right| \leq \int_0^1 \sum_{i=1}^n K|y_i - x_i| ds = \sum_{i=1}^n K \max_i |y_i - x_i| = $$ diff --git a/skripta.pdf b/skripta.pdf index 17da5e437b62481150ecf2c46e059a269c6f9821..d01c532d63132eb95f21e2ac946340a20289caf3 100644 GIT binary patch delta 5625 zcmVPapqBgGbrEKkHgg}4`wT){0n5udiA3yS?UwPsi5sZT)}rH~M)! zdgpaS!j+RgVeRYFG~#3_(Dj!u|A3d0Z{M1ni6ME=gqL02KAXl!JBF&?kDTksS68nr@vCRk0~>!vlQ-FWOO;Zvj8DqEfN!{57#8&}<9}r)TTZF4xJ##Z zCfP)THy%G!UjM6SW2A|jzkNA;I~bQS+zJW{TR zFWgwT6HkR>s#+8Opr-g1Cr;|PKss)|U#|G;vFNZqR=)UoUd(xWGtU53?@61WjTD94 zq-ZMVu9V3H7pz~T9F!tkUaP}L$6!ec%)^MWIRq$u9qD5+pCWZNGSJl1coyhU;7O1A z%yeKPOlKt`T~9huh50`We!$VU$NuXGP|QCiglDO!b5_g3a5+ckd3#Q=z|1@ExXRp0c?e?sP+ao z>2t21zONTf^}U*rY$*G*YG*~B`Av#D5R-%_;ZmO!L!vRY#O3Qlk!wOVN?O*g|)W6oSD`J4^lh?jlF$lCdOEpuLS9Sa&ky zVN~a~y`5OVy>M<;BL(BU7oe#6Tn;>25dYJ10rYCxl|8e)Y%Lb1he2+Y%s-Ux6H48* zeqcknW02tZ&5T^)h)(4?}4E8Cq9AdRsyv?HHC8!5)C z|Mn+`KsbQI{mJ89lP^}Y#ej&#lk6cN9z(zpYfnE6if@|(w%eN;XE?_%mUv=b z*-f=>*lW4tc@E*aD;ou>mLSDpWEd5EbPYN_>o3fiKq=0$`Lq|4P)6`3F~IYX%O*&S z8k7L;4B6OTY-5rwAT(3>giy%28k79BOBJ9V;Y;{njHReYUu)U%Ov0h^Lv`3Mnr4p- zK)=#M)_i|p!w_jkhH$XvyAabs{rOE1XnH z)ZJH@nG%$lk~v}FewIguy_K1w&$7auExYAc{s2SR>YF1+>HlkZXN= z`8dsLXAy@eI>z{vc^1+}%ey>lmp4tB3X(j{dCsD!SuIL-mX>6{WKolEX%*dykQgn*1R`PfsAfNW6`RV8MgRBjh*O?Omv zR&~*aZ5=j+1g@K1Son0_>=39lbOK0xk!i>+4LM>KYdH`uZAe;#moaLB4 zj*!b;?yDEOT}a>h@L`PvavK$xWEb43gXe#_(}oro48npi>)O{{YP38ib3(f&rMtW@ z`<%l&khSb*6j>vIzMn-_9e}k@y?`gvi0!mrAJ7+^7rzNBJB7nQaY+*2tuzlNidlG` z=o5P|6~QLph10^=-s<2lec(HV4ImEsU&*-;xq(E!gF=mG9|S2E^mfkoE(#o0=K_D- zwUTVaW}n`Tq{PJF;~cSkf+gm7MMv^VCd5GbqPzs2nWH>xl}{$5zI5hD{=J&x1yA(F z`!`uJ*)EoNpKRf~gLTEIdoT~e)4m$PL8W^bOH0%HvGp(eDO+{;ZqO25w%pWN?J$Df zStMUfDG}d_mQ-^}HE?};r36NIl(K&;Fx2q>&zBa5kiAxX+bS|LRHXk}j6}GK$TMyh z>2`T78S)%NI-$yRruYZ%9G+B4*~jWAMHkXWzQfzEA9bl!Lx7-|4&VBC~%tlCe^7##v6KehMTAcm<3Vi`O!dRxsDg$Hv;u!dmc&F?^iH znh?j*vY*;FR-iQJFemtL-&%hqW#YT6bw|{DGS|^XT728qZ6Pd8tmP{B5Mf&j;(zMz zXjVT+Ve7cwY}|6@O*v`j`55+Vv>$H}^uZZ~`zQ?JV3}BgPfINL!?S+_ge<#4o2`Ve zjxyHht-Ig8#t~og-8)HN9i(lJbkC!^Gnx+>dG2sSXJ=g6Jw>gR=`MY7zsqjnzARtv zY`c`UNTdO;)15r!uuS0c3idnW3%rSFBKo7QgDq^63?#;San-{A-bbrSTlM2D^_y0?^ocKd`pwA;D-Oojrw^SsW)#pqqbc!{7Q7!^1F_IK?0) zA@`VIY3nzz)TBKhfM|hM$YujguZ#}=?U zGE3Okiq2IB5}_iQ;3g0A@u|*!=PWtj?3~pWPwltu=aB*jVlnj?4K7WH(3cYFhPPsN zJM<^6ocAeFZg+oh7_bIm0I(vYJ~KkZp9@7w5@K?jLvG2G1kq5AOij>ne^0bkGug@eo5DA^SrNb8GkhAj8~Dy*tcMFe&mAHoKAb{AFG`klq{| zd7-?X@PHc8Q87Q!(joGY#<>L&g6=NnWIPDV+6w6OEFlc%o6Th^*kIiR2-{~PMfPQvD>lG8SOS!##=~>oOZZ_oIv&-fEvpPt!Bfo#8xxKujxq=w?k3=Fq*7F=R?}vul z>vB-096_7AQOkjqQZUp0*?K=$Z7Sn#YQtc?6T;MNd0Ks3y`%oCXozjElo`|~#@?wB zV&F@T)Ep|*Gg>)Z+Qixw6=1I^F14ZqgM;T-1IxRQ(tY)vvrQ3Q=gNfV2rzHgFEjuQ>h1|sxW*I5eVv&4Q;?;vc{3A}7?JS(793*Oo z9yFK(e6+Z&>A@Ci4|uV_)%FC*sRm)qpZj!CUOKX@ zQO}+*)tajwWz4k+TL}v@R(}4A{{uQXm}rwBTNJaJP#4w#IJdGU0dxoiWVGJcw<<3I zVgeONr8-WbJa`sKAx-(TjF9DI%EES}x1lisngV|=j)eIcr23f2faBWjt735dCDqyc z^ZL`Ek25DM2Y0nkr_@~oTehp{B_K^Nhk^Lr{20f)!R6lQdRSy4KN|L~ar{QV--ZQb zdDk0_r^QcV{+|`K|6Y`HqvPdf{ZWmh!LZQuMpHV!L!-rv?w%W_y@2U~`0UQrzWXZw zd31j{?gNUvl9v?L??H$$UqPh9`}z38Xn@)UDdIjcn%T1d`S|1M=Y=%tIoT3c*phBr z9>5k?Ws51<;%tpAuEG{wV+*_>*iycW3MKicWXneVRoN12Y;kf6wruKuTekRhw&?BI za{SxT-lq?9X~Hv-B|T@YlEplREZ%J-OHhARvcQ7vDtKpU=q<>nHZ2>}SK*4&b*|WL zEK76u+fpU2Q^jmgmGfWEU%h`l7sfp&Rop67!egis^d_n}XDU>&(h^kyQS<4kvPpea zs`xrp;ucid(EYYlN!Q+WuGsCl^83#pKD<3x7&-BbWJxbcRLJ7hqnuFOXje?E+7*9I zTNZi|*+~E7PH3b4Dr`ybgDlwM^%iW|)c@9WaqDz(+tX$5g>gT>98=X{F`eeu^mMnF zfqcAMt9S^H<^t8nve|n?PPFMRPuPkXTo+ZYyQL!Eygl7}{}!XYZ=$+; zYO0In9&5fis@?(3NiO0#$$edqZ*Pk0UVck2Q4EW#S&Ee!9YVu@7n6T+J{}eM?5{Cs zPMoK=UWfU`q<8s8G2=#MJT%&LD2gQNY6VM+}~SA!8;MP4M2o!^ZHz;r-^l&^;|oW%mzA~n3g zo*7?sgy8az>99BbQh~On?FVC9zDPdDR;v|kgIa=%_yXGx=X(clKK|qMZ2z8scBtD^?gNMwFx|2U8b%sd#`^huB?Tl&xdbbov~AC zu=DX*J#mnY#=f*sl7B3|qBz&&h*v#tZ!#P7hW)|S75msr_Noh?TG4fN@()+Fud6z& zsM=Ii?JAp|!_O~I_KwKJttnaw@mzkuH~9gW`clW(9jxg%)^*Iro)>l93_oLxa_oH_6{ivxOO>HReF*H4YS-bS&MWLOH8b)Wa zB7WpfW0NW?@P_(bY1rnjbVK*so;Snyi~S94U6Ap&y&kc_Yf&5;VHO>sOXqbs`VfS>1m{m{k`6$}Ud=v9f_ZK0i47 zmDY8G($sU=t-7+h_VQ#FdyUgWO?pNDFNq*Ijs_LGyy6@QE|MmP0 zy8ijM^?ipD`k{If58ie7Cc|#X(9mYmSZK)CPfF$c>I)71S$v&gT}}to@p5lByIsaF zX3O}^pr3yd*uPKj;CN|EIHMo&(H`-7(i?JOQPdwyZ->2ai_&>|((4Z{d&7&tb#>cQ z)!%mhZT*R9@F#IAf0BMnu~>ihR<51y?S`}D>vwNo{`@O%Asfu& z^WJ2|rt_qp@9nf%KjWvLdn_#gUp7@%VRbg!EyAOdcfXvynZx;<@6W{5hf(oS52O6% z=L<^K?u>Q$klWI~h$EGG%7;;#)K|eS)}KYiZJtGK=ziOU=GL!_?Ocri17tJ`L<(hY zWOJ7+I{_#{HDY5mWn(!qIALZtHDNYoHZfv1H8(dkWo9-qV>CHtK0GyIV>M-CIWjn5 zW;ZoqHf1(3VmCE6H#KEuHZo&0IcArZI{`<3F+M&Db98cLVQmU{+BMYKQ;l&P$MNs? 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