From 629f97f4be59f2ce4686ca5cbf81433d0603faae Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 27 Oct 2025 08:16:11 +0100 Subject: [PATCH] [Analysis] Add tricks --- .../analysis-ii/analysis-ii-cheat-sheet.pdf | Bin 279224 -> 280596 bytes .../vectors/differentiation/00_continuity.tex | 9 +++++++++ 2 files changed, 9 insertions(+) diff --git a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf index 3d6938342252a3bd5dea3e30549dc3c405c6b1c9..a24de26f57af6ec76eb75c7ccc3db5512de17381 100644 GIT binary patch delta 87907 zcmV({K+?asg%OmH5wKhd0Wp&i0x6S*3BG?ay9!Q5rI036d7}lERp&x50PIRjbVPtnWNVURv-0|o9SI~KZ^crI=%mh2>aJuX?yA)1aM~0`5p?CFoge#beUW|oSFK)kJFoO9hGkA1cUIaE)sE*nG=qPW zR3`SqLLSLEN=GC&&`;?1Da^3xT%-Q4Kc+A1dV_5#MGLe*t4;g7J|sZH5S0FS2)GIY z#R!^9?y_F<@hkwYV#Q%Oet7RVI4|H97Xim+W#9QtQ&*nNn!5IJ59U=@FQeOT<3>#C zg+J`;7~HnL?zgqtE@Q%Mv0v?c}>`hO|)<=$>;U+tQ9&uib{?l$SKG@K%B zx>O5b3B$n%2`YMECuz69Cs4&u)nB^!RTO-;e7X?rpUFXvHvm+)2^p9DDBuDyEC+u8 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a/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex +++ b/semester3/analysis-ii/parts/vectors/differentiation/00_continuity.tex @@ -25,7 +25,16 @@ We write $\lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} f(x) = y$ iff $\forall (x_k)$ in $X$ s.t. $x_k \rightarrow x$ as $k \rightarrow +\infty$ and $x_k \neq x_0$ $(f(x_k))$ in $\R^m$ converges to $y$ \stepLabelNumber{all} \shortproposition Let $X \subseteq \R^n$, $y \subseteq \R^m$, $p \in \N$ and let $f: X \rightarrow Y$ and $g: Y \rightarrow \R^p$ be cont. Then $g \circ f$ is continuous +% ──────────────────────────────────────────────────────────────────── +\numberingOff +\inlineremark To find the limits, we have two tricks (for $\limit{(x, y)}{(a, b)}$): +\rmvspace +\begin{enumerate}[noitemsep] + \item \bi{(Substitution)} Substitute $y = x + (b - a)$, then limit is $\limit{x}{(a - b)}$ + \item \bi{(Polar coordinates)} Substitute $x = r \cos(\varphi)$ and $y = r \sin(\varphi)$ and the limit is $\limit{r}{0}$ +\end{enumerate} +\numberingOn\rmvspace \shortex \bi{(1)} $f_1 : \R^n \rightarrow \R^{m_1}$ and $f_2 : \R^n \rightarrow \R^{m_2}$ continuous $\Rightarrow f = (f_1, f_2): \R^n \rightarrow \R^{m_1 + m_2}$ is continuous (Cartesian product) \bi{(2)} Any linear map $f: \R^n \rightarrow \R^m$ is continuous. In particular, the identity map is continuous