From c5de4583fb440869abb01b33a424dfb233141d52 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 21 Oct 2025 16:25:19 +0200 Subject: [PATCH] [Analysis] Basically catch up --- .../analysis-ii/analysis-ii-cheat-sheet.pdf | Bin 274312 -> 279306 bytes .../linear-ode/02_constant-coefficient.tex | 2 +- .../01_partial_derivatives.tex | 22 ++++++++++++++++++ 3 files changed, 23 insertions(+), 1 deletion(-) diff --git a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf index 1542adeb94a3e8e768871ba7f85a7bbc06dada75..9c99f80b7c86c8fcb54638625e472b7ed97d5c5f 100644 GIT binary patch delta 123741 zcmV(!K;^%P-w=w15wJuG0TQ!C3QYk7mqPg@vz7}B2>~#Z!3!yq%M=fP!5bJmNu44q zIj4Mf9I5VHW#TCPb)w10{C3JfI zaD`Q14Ue9^ctqhnGNGY6TC++MGr78d^wR{o|F?-9!y{7h+Q`Yz6MT%nR0bda;nnTU ze_qXL1%-B#N3;@937v?FNm`r9efy}c1mRpvR_!B7s#+OSPgtdYj1qi8S;e@n>*oFD z!PVrEP?f3$tJ{ZAl`v4UHX?ept|T+HY-?|xgf3uZ;aO;D6+H{xX)6!IYP#SbU`?T` 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RoD%t}Yv*(8Jm3<)`WFjMkcd{!AHbcG@fo{54eK@L{Mst$zJ+$&VN$=f5{uGR$tQ1;U@9ZZ}&dQBXsP)GZvTcca>uXyDq>=Mq&mlgJGv(z{F4yA+T&6|dv^{!dDMTFq5-bpN>9 zmuTSA#*SwC4H(@bNcWGEJ&E+dtBo6#XoFkdB1-oUdwq$9VcL(CXTvaUZxN<@C$_#k z1HblpoW{dy-*11A=N55#c=GCtG^{;z$7(tcj>*F9>4^XJkiM91L;OCPoHYwNhFxBh zP~@i1rl06EC79#a735RWJpQ2*`4k`RowQzDFVCklaAT2+R3Ng%_y;`rK^D@hnbRd5 z^xD*Sw`dOsN4S4m&pMhnUkl%j_A*noGuv?7n|auL!ySL>aAs>pnrS;T@z0Q~7d5#R zS`kF0HbaU^cT|tRS@Ews{7tS?&vRsE)y}kj%S@Xo==6|{U4d%#7B+RpW|K5_g|LYm z*~rWd2%8ncW-G!vSEK{C&EZZuws4!?Vk@{!y?HU_-z_+C2SVQd(}T;?JIBk(eED-B z0lE0;$+CaB*sp)azsi#jR6paN*vUsgV>*)*Mr<~M0SR~?NSEowblF?9CHv49jK?*m z=PMWt-55)Q=Uan?Qg#G*OH_k9qf-QVpH zNn(FocZg(Zjlo}sCPK_^&CwYr7DG5&8TzCHVLR_>>6Og+IHA#cY&Nd3w68Xd6CXbi zslQTx%1SgT6I5DJrA?vIE>Pn3{Lg90r{5O+`bKJ*hcr3f1i4M(CB>Db?Xl&@m{2iB zw90Qd)ZL#fC+C-Eds03X?Im$r0cA;tO{gB$WzAW0K($!;deN-Qr59Jt>@EvX_*H)T z4|p=D#gowk69P9jmn%O6A%8bFT?#K#X?kTKGB-C0FIOO2F(5M_FJ zATS_O3NJ}+W^W)eH#HzIAW{l1Nkkx9JTXKyH#0*qGB`0qMKd%uL_#+>F+(vmI5{>k zF+wsnG(I3aF+?>tGea>lI59&-Gc-0tLN_=uLoqctIW{pdLNYcqJ`G(8FHB`_XLM*F zH!(Dm5dtZHCDX}E7jYQJ@$WM;j$3Y}mTBga{w%i^&GyZ*eX&hjY%jWW@ec?hI&_H$ zI+hT^6BJ#dQ_C$X4?z(@5OomR2+9r>9ea@fML9i-_n12{#ulPX{)u#|2iRYDb<;Mi8u zEU1Qm*-!(uFh@dek8iaO>bYVOEQbxS0=l6Kd^77|GgmZ12AW_l%!6iVfmUdP`OqLC zfAY(;v&uq}!+q1mus}lbss*Fz8d@``j_Dqr~dS-NTBP$ z7v{W7nBcVjQ?gZ*$4 z`XzMU^UWN9gK!8A!vGwCqi_t4OH$tF|DG5USO8g&LN??;F62QzP*$a)N=21oD#cXl zsMJv@p;AJndP?<_!YPH*3JJS<{pK?zTpRRMDdAb-sY=3l#?vg3vYVc&MS3@Snk_Pa z*zBoB9EMBUw<=16qrucroyE{=I>l<3w&PZ^0uCRb^a=+h-n zb0toOJ4wMcyByr)))hn{yXn4dcgs 0$ s.t. +$\{ y = (y_1, \ldots, y_n) \in \R^n : |x_i - y_i| < \delta \smallhspace \forall i \}$ is contained in $X$. +(= changing a coordinate of $x$ by $< \delta \rightarrow x' \in X$) +\shortproposition $X \subseteq \R^n$ open $\Leftrightarrow$ \bi{complement} $Y = \{ x \in \R^n : x \notin X \}$ is closed +\shortcorollary If $f: \R^n \rightarrow \R^m$ cont. and $Y \subseteq \R^m$ open, then $f^{-1}(Y)$ is open in $\R^n$ +\shortex \bi{(1)} $\emptyset$ and $\R^n$ are both open and closed. +\bi{(2)} Open ball $D = \{ x \in \R^n : ||x - x_0|| < r \}$ is open in $\R^n$ ($x_0$ the center and $r$ radius) +\bi{(3)} $I_1 \times \dots \times I_n$ is open in $\R^n$ for $I_i$ open +\bi{(4)} $X \subseteq \R^n$ open $\Leftrightarrow$ $\forall x \in X \exists \delta > 0$ s.t. open ball of center $x$ and radius $\delta$ is contained in $X$ + +\compactdef{Partial derivative} Let $X \subseteq \R^n$ open, $f: X \rightarrow \R^m$ and $1 \leq i \leq n$. +Then $f$ has partial derivative on $X$ with respect to the $i$-th variable (or coordinate), +if $\forall x_0 = (x_{0, 1}, \ldots, x_{0, n}) \in X$, $g(t) = f(x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, x_{0, n})$ on set +$I = \{ t \in \R : (x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, \ldots, x_{0, n}) \in X \}$ is differentiable at $t = x_{0, i}$. +The derivative $g'(x_{0, i})$ at $x_{0, i}$ is denoted: +$\frac{\partial f}{\partial x_i}(x_0), \partial_{x_i} f(x_0) \text{ or } \partial_i f(x_0)$\\ +% +\shortproposition Let $X \subseteq \R^n$ open, $f, g : X \rightarrow \R^m$ and $1 \leq i \leq n$. Then: +\bi{(1)} If $f$ \& $g$ have $\partial_i$ on $X$, then so does $f + g$ and $\partial_{x_i} (f + g) = \partial_{x_i}(f) + \partial_{x_i}(g)$ +\bi{(2)} If $m = 1$ (i.e. $\R^1$) and $f$ \& $g$ have $\partial_i$ on $X$, then so does $fg$ and $\partial_{x_i} (fg) = \partial_{x_i}(f)g + f \partial_{x_i}(g)$ +and if $g(x) \neq 0 \smallhspace \forall x \in X$, then if $f \div g$ has $\partial_i$ on $X$, then so does $f \div g$ and +$\partial_{x_i}(f \div g) = (\partial_{x_i}(f) g - f \partial_{x_i}(g)) \div g^2$