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zFXD|ti7%2ZezSo8QV_t2z)ldf7X+^cAzT-F0E8_8;V8F6;8qe@4{UZ2g^Npc3y47( zWh7@uO$Vc^KrENU@dojafrRZ~3^yH%yh!3{kkkc|kxqXZ9{^HfKq|g7lE%gMZs4#q z&mKhoNjycI<(P2Z1Q~lk=8qto2j$EMxhFv0C!in>6mp{I7$`x)q!d+^GTxv9Z6THU zpb90EYA;aJ2x@nLI(!49zFQCXCO*Vhq~W4I^AlM(Mpn|3PfK_)+k6U zyG^5Du^L7PGzt-GfoK#eR%eDrVPYL#qfxk6rzUC?Ay)sN8by*Ml1x%aDoG=Dl1?1N tNis+#$s*ZAe|K}mx-^IezCF(T4;Wq}u9wLA0v-!AFgQ30B_%~qMhc!l&-efU diff --git a/ana13.tex b/ana13.tex index e9e5ff3..cffcdc6 100644 --- a/ana13.tex +++ b/ana13.tex @@ -11,7 +11,7 @@ \begin{salign*} f(x+h) - f(x) = \int_{0}^{1} \dv{}{s} f(x+sh) \d{s} &\stackrel{\text{Kettenregel}}{=} \int_{0}^{1} f'(x+sh) \cdot h \d{s} \\ &= \left( \int_{0}^{1} f'(x+sh) \d{s} \right) \cdot h. \end{salign*} -. \item Mittelwertsatz der Differentialrechnung: $\exists \tau \in (0,1)$ sodass: + \item Mittelwertsatz der Differentialrechnung: $\exists \tau \in (0,1)$ sodass: \begin{salign*} f(x+h) - f(x) = f'(x+ \tau h) \cdot h. \end{salign*} @@ -22,7 +22,7 @@ \end{enumerate} \end{bem} \begin{satz}[Mittelwertsatz] - Seien $D \in \R^{n}$ offen, $f: D \to \R$ stetig differenzierbar, sei $x \in D$ und $h \in \R^{n}$ sodass $x + sh \in D$, für $s \in [0,1]$. Dann gilt: + Seien $D \subset \R^{n}$ offen, $f: D \to \R$ stetig differenzierbar, sei $x \in D$ und $h \in \R^{n}$ sodass $x + sh \in D$, für $s \in [0,1]$. Dann gilt: \begin{salign*} f(x+h) - f(x) = \left( \int_{0}^{1} \nabla f(x+sh) \d{s}, h \right)_{2} = \left( \int_{0}^{1} \nabla f(x+sh) \d{s}\right)^{T} \cdot h. \end{salign*} @@ -32,9 +32,9 @@ \end{salign*} \end{satz} \begin{proof} - Sei $f: D \to \R^{m}$. Sei $g_{j}\colon [0,1] \to \R ,g_{j}(s) \coloneqq f_{j}(x+sh)$. Dann gilt: + Sei $f: D \to \R^{m}$. Sei $g_{j}\colon [0,1] \to \R, \ g_{j}(s) \coloneqq f_{j}(x+sh)$. Dann gilt: \begin{salign*} - f_{j}(x+h) - f_{j}(x) = g_{j}(1) - g_{j}(0) & \ \ \stackrel{\text{MWS}}{=} \ \ \int_{0}^{1} g_{j}'(s) \d{s} \ \ \ \ \stackrel{\text{Kettenregel}}{=} \ \ \ \ \int_{0}^{1} \sum_{i=1}^{n} \pdv{f_j}{x_{i}}(x+sh) \cdot h_{i} \d{s}. + f_{j}(x+h) - f_{j}(x) = g_{j}(1) - g_{j}(0) & \overset{\text{HDI}}{=} \int_{0}^{1} g_{j}'(s) \d{s} \overset{\text{Kettenregel}}{=} \int_{0}^{1} \sum_{i=1}^{n} \pdv{f_j}{x_{i}}(x+sh) \cdot h_{i} \d{s}. \end{salign*} Ist $m = 1$, so gilt: \begin{salign*} @@ -50,7 +50,7 @@ \begin{salign*} f(x+h) - f(x) = \int_{0}^{1} \left( \nabla^{T} f(x+sh) \cdot h \right)\d{s} = \nabla^{T} f(x+\tau h) \cdot h. \end{salign*} - Für $m \geq 2$ im Allgemeinen aber (da $\tau \in [0,1]$ nicht für alle Komponenten gleich gewählt werden kann): + Für $m \geq 2$ im Allgemeinen aber nicht (da $\tau \in [0,1]$ nicht für alle Komponenten gleich gewählt werden kann): \begin{salign*} f(x+h) - f(x) \neq J_{f}(x + \tau h) \cdot h. \end{salign*} @@ -75,10 +75,10 @@ \end{proof} \begin{definition} $D \subset \mathbb{K}^{n}$ heißt \underline{konvex}, genau dann wenn: für alle $x,x' \in D$ und für alle $\lambda \in [0,1]$ gilt $\lambda \cdot x + (1-\lambda)x' \in D$. \\ - Geometrisch: für zwei Punkte in $D$ liegt die Verbindungsstrecke der beiden Punkte stets ganz in $D$. + Geometrisch: Für zwei Punkte in $D$ liegt die Verbindungsstrecke der beiden Punkte stets ganz in $D$. \end{definition} \begin{korollar} - Seien $D \subset \R^{n}$ offen, $f: D \to \R^{n}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: + Seien $D \subset \R^{n}$ offen, $f: D \to \R^{m}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: \begin{salign*} \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall y \in K_{\varepsilon} \end{salign*} @@ -97,7 +97,7 @@ Sei $D$ nun konvex. Für $x,y \in D$ gilt dann: $$z = ty + (1-t)x = x + t(y-x) \in D, \ \ \ \ \ t \in [0,1].$$ Sei $g(t) \coloneqq f(x+ t(y-x))$ für $t \in [0,1]$. Dann gilt für $i \in \{1,\dotsc,m\}$: \begin{salign*} - f_{i}(y) - f_{i}(x) = g_{i}(1) - g_{i}(0) = \int_{0}^{1} g_{i}'(s) \d{s} = \int_{0}^{1} \sum_{j=1}^{n} \pdv{f_{i}(x+s(y-x))}{x_{j}}(y_{i}-x_{i}) \d{s}. + f_{i}(y) - f_{i}(x) = g_{i}(1) - g_{i}(0) = \int_{0}^{1} g_{i}'(s) \d{s} = \int_{0}^{1} \sum_{j=1}^{n} \pdv{f_{i}(x+s(y-x))}{x_{j}}(y_{j}-x_{j}) \d{s}. \end{salign*} Und damit in Vektorform: \begin{salign*} @@ -110,7 +110,7 @@ \end{salign*} \end{proof} \begin{bem} - Obige Lipschitz-Konstante liefert eine Abschätzung für die Ableitungen/ Jacobi-Matrix von $f$. + Obige Lipschitz-Konstante liefert eine Abschätzung für die Ableitungen/\allowbreak Jacobi-Matrix von $f$. \end{bem} \section{Taylor-Entwicklung} @@ -129,8 +129,8 @@ &\Longleftrightarrow \ \ \ f \ \text{ist total differenzierbar in} \ D \ \text{und} \ x \mapsto J_{f}(x) = (\partial_{i}f_{k})_{i,k} \ \text{stetig} \end{salign*} \item Ist $f \in C^{k}(D,\R^{m})$, dann sind die Ableitungen der $k-1$-ten Ordnung $\partial_{i_{k-1}}\dots\partial_{i_{1}}f: D \to \R^{m}$ total differenzierbar, weil stetig partiell differenzierbar. Also ist $\partial_{i_{k-1}}\dots\partial_{i_{1}}f$ stetig und damit sind alle Ableitungen $k-1$-ter Ordnung stetig. Ananolg folgt induktiv, dass alle Ableitungen $j$-ter Ordnung mit $j\leq k$ stetig auf $D$ sind. - \item Seien $D \subset \R^{m}$ offen, $f: D \to \R^{m}$. Existieren $\partial_{i}f, \partial_{j}f$ und $\partial_{j}\partial_{i}f$ auf $D$ \ ($i,j \in \{1,\dotsc,n\}$) \ und $\partial_{j}\partial_{i}f$ stetig in $a \in D$. Dann existiert $\partial_{i}\partial_{j}f$ und es gilt $$ \partial_{i}\partial_{j}f(a) = \partial_{j}\partial_{i}f(a).$$ - \item Seien $D \subset \R^{m}$ und $f \in C^{k}(D, \R^{m})$. Sei $\pi \in \mathcal{S}_{k}$ eine Permutation, dann gilt: $$ \partial_{i_{k}}\dots\partial_{i_{1}}f = \partial_{i_{\pi(k)}}\dots\partial_{i_{\pi(1)}}f, \ \ \ \ \ \ \ \forall i_{1},\dotsc,i_{k} \in \{1,\dotsc,n\}.$$ + \item Seien $D \subset \R^{n}$ offen, $f: D \to \R^{m}$. Existieren $\partial_{i}f, \partial_{j}f$ und $\partial_{j}\partial_{i}f$ auf $D$ \ ($i,j \in \{1,\dotsc,n\}$) \ und $\partial_{j}\partial_{i}f$ stetig in $a \in D$. Dann existiert $\partial_{i}\partial_{j}f$ und es gilt $$ \partial_{i}\partial_{j}f(a) = \partial_{j}\partial_{i}f(a).$$ + \item Seien $D \subset \R^{n}$ und $f \in C^{k}(D, \R^{m})$. Sei $\pi \in \mathcal{S}_{k}$ eine Permutation, dann gilt: $$ \partial_{i_{k}}\dots\partial_{i_{1}}f = \partial_{i_{\pi(k)}}\dots\partial_{i_{\pi(1)}}f, \ \ \ \ \ \ \ \forall i_{1},\dotsc,i_{k} \in \{1,\dotsc,n\}.$$ \end{enumerate} Reminder - Taylor-Entwicklung in $\R$: \begin{enumerate}[(1)] diff --git a/analysisII.pdf b/analysisII.pdf index 4d06f8a33518bb4f53c51d4efe6d7fde1b2baeaf..3406a51ccc781f1e707e081741539e86b61beb9c 100644 GIT binary patch delta 17286 zcmagCQ+Opzur(apwr$(CZ5tCiJGLjuj&0kv&B?^JHPQc`^PT(uV)e!9s_u(gPj&S= z{-g^3q6!KF4Fd}UZ-G!m1>@xWy+2C}(ARawpF|B@ug?e|PtMiajYI(746mul#gTCU zgC+Y2Zjxdnj24r@*CP;tCZ z57sw_4x@{sOXr;MWg1dMpwe2HrBZ4=T6pH=C{dtVWZBhHI~&hi)qitY#wmUS-Y>RS zVkWP?nD-BNmlK=0BdHBIZDgyWQ4u`hEI1zRt9qyBhtX@%q5Nd}cO_HOh2b^qhSbV#Y?x_m#8Zn?UXXoQl+h+;F#oQeLnS0q%T9@gwVC{YYo-nouL*^Pu=f}_yPh{v z)rX)@Dt2y4_6>1$20J>ue=Twxk0~jkilxI$O?y0IFQ?kF^Ik5QgAo~a-kFQaa%s^N zh0$SurZTIP$$}*qO+&N7LeW`bMn8^1(@C^Vr3+(iUO4pq!^I#SIr8rV7@Q~rW2+Uv zk0{@^?(La}$&ZpMQzcwby2zI*jWTHusdhd>}Fq?V8sb)S01Rtf&f)vcesax z6Hv$qs4Y}0e|!;3&A0x3SGev~#Modri)S?Lgr&{_u|Wq1baPPKPIw@@ z*TVI~nnP~}mKgm-VcEk0_G%+Sy?Zr34Ko-}7=O=KulD+(g=QAQD07C{OeE#yI9eym zFGH6TId~(PcPUa~tbaS3p0^_NrG7jbQr;qeCqoaPMC9O;4UTe*AQ<^R@RCWZ04p>a zk*wKF;_><##CFOfdRYI|#=FGpkb9R{G$sW24TyR-kLkK+>XLdn_r<7f%>~W$h0H>|#~ou|H`7HO>4mcAIfYVH&B?+=d+NPl@j}^~ z0_nqHVCB6h z2}+_V!rArR+EGdXQZpk$UhKFDAZ4YKI?G^5`trhHjr`K75bbf=FM7t1#zMr1`^_34 z{-r2#Y9PvEfAku9kp0r$lDFW5Pf#fI-=TSV=UAsw759ii*!E3%kFA>!segv& z<$thEj!<1yPB4OTwC`?{z@;kSxPHT6%6qqld1=BFM0QyL=n83&TB^*BB!MzFV56@! z3D~Sbhl8SW@ExEMCZP4T>ohdx80ql0+Brc^c}i6Na}5ZCmhQ1Slx@+P2j3nedRRrwJw5YS895LlD^r*NQE@hG%u{B!uNRnY z;N0qxPh${33hzY4v|MGQn{1WA$bZl*3M6hzXBEZnw?UnSB#lH)!_B`@wj&jqy22HW zvQ}yKuYWNeP2WI-T2aW<&%5>+IOX<+_!6){C?T>aD@dtcDgW7|E>{S*BN#%F5s$@w z39yC`8bp|~M)XPcH5W~e`sPtpZq5(S7z`#80!keKL^2sxm&(5y_)<*=XU4dr?KyLb zE&Kf-T*Tc{G-GpE`D}g-rp9}9YkwaaO3iU^eVQOvK1uB}&T5<~kBaL(gh+fIEjy6HaO_T>{zjU;$OU!HP zP*BkEKxBRiIKq~XKcqXIFfyF-{PMs$&dFB6U<8cmdg&pPm=H4?7Q1Xy?r%Bq6A&qG zpakOJJ4lB9Om(sBSY(eLbEK&iJKEC5&sXat4>LwB^X`sum1}?dZp6s1D5OTS9f?!$ahxw3cbG8^go~XP$QS2@Qu? zy_P^_>gz^h;bUWwhbAyj5Ne{m*I}ehE{7cS|KzK!=*PM1uRjwflS^Xo{aV@`+*|F{ zR6)!7RZdm%S=!oM0p8vVi3BeWfQf-Em^osih7{C3S zFlkyF^N01?G^~iPK0S?v#Q^k4f!csMCOM4zQJLF1_UYQcSV1A>wDgoQ!2 z&_F`*?}TPt^4HUVle!n%n+IJ8zz+{SA9ZZ3YUqyKQvgw(4fbkj0&1~HWlHdkJMY5_ zqkawa2l`MLTe?spLlEb&{%E2$!IH|j^4wRvm8BzK_2Xin72vbpUD zENdyxtv7#~@s;!lw%fpKo(C72B2b~eo!-_5G2rDXg9pk-RGKDlQd2guMh(2IFzuPA zKbe?yKK~u4Ehd=ay+oxc5uiS1Lfd9GZaJYRveRgo^FB{oeR3YJ@8*D1#Yk~9$BkIF z$WZ?Ck#jYX^oNq~ZLmTdP_{y5`%ee9L=Ew{5Es=tba%XyZKV;Va_g0XZXHx*+Z3%; z+K+>aTg0i)&7kk2L4xUVa7Mft#_l6z7eLxVLL)yPOPo%7Bh-!X$g;eIH)V5(>K$UmFh1~k?6y)`5&g9vjv=GK*B7XjqQN&-vQ+5 z#+;nNTbxj%rrKbf8LTvoKIzeGcA`T^O>3!y*wIWHS0~n6{2rWJeqHGfXLDF78s-{E z03lS(z>1Y`yiyn~l|H-r8BB5FY1``IowjWwBe5$H=0H*8n|d{ktAWV z)-{^Os)`CE!a-=&M=x@3JJU+(ALFNMVE(KAX3nhdruis}dH`-8Wymjm(kN8+8?mU4 zP7nqcP3ZZL{$x8+oF6Fd$L&TVX0$9c`qx8A%-Y>rY(bk)K+7u>?K
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