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reP^7(r4FOv2mYx%@x||l@{*Y8?xB?!6MI(a z+%+9W(hU}B%_jtZ+JZ=THBH^c=>sc{XCaYA?oO>wVR4=G%a|+eL0Rds<9+yU0iM?I z{4zoJX4qS8!h?1?rxO3-N;clG9&?w$sM^tauIpXi`B2|2lwyqGZUsZVBgOorM1L3x zBcw8V-hy!7@T=O&p307GJ6mgrE!Svn(@EFgKTE!10P7UA?~2xmF%3;}VMqFiYiE3g z-n*XoOAIaryK!wcH>sxlb45jl;1{c-5WAL%41}1o4R15w^w!S-aCSov_rqX7vE$7g zq?ZYZ4a3e2ndb*mfZ2KUkyuo%z0E-!>>uB5O^_}dhy%p-5ma$>bo&T$f>=Q+NG!4r z7LFf^>;EbVkglLOySOMfFBco1B$qg^1SdNeuOyE+yQsJ%D=!<5IH#mA=>I$k*1X`N8V8e(oOQBFp^m+Ej5;2kdzb_zH#fJK zgvpN|F(5Wl3M~2BQd9~GT*TTqjEK;IzP<~g>WRdUig%B0pGl8RpLYzghm4S|>t2`| zR!V$El1wtC@kFDxZ6?Bp@A*+7u2_@bzccD7PtY?UjP(!OJ&OLaTFC*gWb+T zJ?X{6`%A%IY}xb+y)HWjLB)8SI5F-x*=_S8}Md8qR?UQm7aSVT8?_ARY1(g6U0jXZ(Wl*%7488Qz zP(-~p(}2Z*Mn(qa+>OrW}MEB1u1{T{2QU28P z3Q1YC0;*0@e}B&s)^YfK##1oxrCfl(Y?Mo=X%W!Q`%}$CWHWdJCUJR9{zj&`NT%!2 zQ4oWx?BWnKDMtbmVL!unq09&F8v#q;QN{?9IA|@ECUY`H`GT(+a|j0rJHO;oE@hx{ zmEzTx*_5JMNR!Jort5oBH_n`_xR!pi=!Pp?Wwd@_YFtS$O;$Ng(OdjHA;9Q$rC}3A zeJ=a~!b!h>uW4tj1be`qT)Xm~B2N^k#f@RV!Pn8-p%AP2qa4HV-$4 zSCBnzIMEB8XCALgl0R)a@z$<6k&+&$xZyWFxE7^moU0mkNg>yum-(V5in z?7SZSKv(=E{2Kir_<2X(aG+Tp^N*p#i_9l~G^v+1;>%~V^)YYCy6QlQq+rY(B(-2B zNh3Bb7#k5LEu 0\ \forall x \in \R^n_{\neq 0} \\ - \text{Negative Definite} &\iffdef& x^\top \textbf{A} x < 0\ \forall x \in \R^n_{\neq 0} - \end{array} - $ -\end{center} -\smalltext{If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite.} - -\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} (\textbf{A})_{i, i}$ - \begin{footnotesize} - \lemma \textbf{Determinant} of $\textbf{A} \in \R^{2\times2}$ - $$ - \det(\textbf{A}) - = - \det\left( - \begin{bmatrix} - a & b \\ - c & d - \end{bmatrix} - \right) = ad - bc - $$ + \definition \textbf{Scalar Product} $x \cdot y :=\sum_{i=0}^{n} (x_i \cdot y_i)$\\ + \definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\ + \lemma \textbf{Properties of} $||x||$ + \begin{center} + $ + \begin{array}{ll} + (i) & ||x|| \geq 0 \\ + (ii) & ||x|| \iff x = 0 \\ + (iii) & ||\alpha x|| = \alpha \cdot ||x|| \\ + (iv) & ||x + y|| \leq ||x|| + ||y||\quad \text{(Triangle Inequality)} + \end{array} + $ + \end{center} + \definition \textbf{Definiteness} + \begin{center} + $ + \begin{array}{lcl} + \text{Positive Definite} &\iffdef& x^\top \textbf{A} x > 0\ \forall x \in \R^n_{\neq 0} \\ + \text{Negative Definite} &\iffdef& x^\top \textbf{A} x < 0\ \forall x \in \R^n_{\neq 0} + \end{array} + $ + \end{center} + \color{gray}\scriptsize + If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite. + \color{black}\footnotesize + + \definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} (\textbf{A})_{i, i}$\\ \lemma \textbf{Inverse} of $\textbf{A} \in \R^{2\times2}$ $$ \textbf{A}^{-1} @@ -61,4 +46,61 @@ Relevant definitions used throughout Analysis II. -c & a \end{bmatrix} $$ -\end{footnotesize} + \color{gray}\scriptsize + For $A \in \R^{n \times n}$: Gauss Algorithm. + \color{black}\footnotesize + + \lemma \textbf{Determinant} of $\textbf{A} \in \R^{2\times2}$ + $$ + \det(\textbf{A}) + = + \det\left( + \begin{bmatrix} + a & b \\ + c & d + \end{bmatrix} + \right) = ad - bc + $$ + \color{gray}\scriptsize + For $A \in \R^{n \times n}$: Cofactor method. + \color{black}\footnotesize + + \lemma \textbf{Determinant} of $A \in \R^{3\times3}$ (Sarrus) + % https://tex.stackexchange.com/a/32981/184539 + $$\begin{tikzpicture}[>=stealth] + \matrix [% + matrix of math nodes, + column sep=1em, + row sep=1em + ] (sarrus) {% + a_{11} & a_{12} & a_{13} & a_{11} & a_{12} \\ + a_{21} & a_{22} & a_{23} & a_{21} & a_{22} \\ + a_{31} & a_{32} & a_{33} & a_{31} & a_{32} \\ + }; + + \path ($(sarrus-1-1.north west)-(0.5em,0)$) edge ($(sarrus-3-1.south west)-(0.5em,0)$) + ($(sarrus-1-3.north east)+(0.5em,0)$) edge ($(sarrus-3-3.south east)+(0.5em,0)$) + (sarrus-1-1) edge (sarrus-2-2) + (sarrus-2-2) edge[->] (sarrus-3-3) + (sarrus-1-2) edge (sarrus-2-3) + (sarrus-2-3) edge[->] (sarrus-3-4) + (sarrus-1-3) edge (sarrus-2-4) + (sarrus-2-4) edge[->] (sarrus-3-5) + (sarrus-3-1) edge[dashed] (sarrus-2-2) + (sarrus-2-2) edge[->,dashed] (sarrus-1-3) + (sarrus-3-2) edge[dashed] (sarrus-2-3) + (sarrus-2-3) edge[->,dashed] (sarrus-1-4) + (sarrus-3-3) edge[dashed] (sarrus-2-4) + (sarrus-2-4) edge[->,dashed] (sarrus-1-5); + + \foreach \c in {1,2,3} {\node[anchor=south] at (sarrus-1-\c.north) {$+$};}; + \foreach \c in {1,2,3} {\node[anchor=north] at (sarrus-3-\c.south) {$-$};}; + \end{tikzpicture}$$ + \lemma \textbf{Properties of Eigenvalues} + $$ + \text{Tr}(\textbf{A}) = \sum_{i=0}^{n} \lambda_i \qquad\qquad \text{det}(\textbf{A}) = \prod_{i=0}^{n} \lambda_i + $$ + \color{gray}\scriptsize + To find $\lambda_i$ solve $\det(\textbf{A} - \lambda \textbf{I}) = 0$. + \color{black}\footnotesize +\end{footnotesize} \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex index 88b5259..6bb033a 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diffeq_sol.tex @@ -118,11 +118,7 @@ If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ \end{footnotesize} \begin{subbox}{Separation of Variables} - Form: - $$ - y' = a(y)\cdot b(x) - $$ - Solve using: + \smalltext{Form: $ y' = a(y)\cdot b(x) $} $$ \int \frac{1}{a(y)}\ \text{d}y = \int b(x) \dx + c $$ diff --git a/semester3/analysis-ii/cheat-sheet-rb/util/setup.tex b/semester3/analysis-ii/cheat-sheet-rb/util/setup.tex index 5cf070a..290f538 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/util/setup.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/util/setup.tex @@ -38,4 +38,4 @@ % Flexible graphs / visualisations inside latex \usepackage{tikz} -\usetikzlibrary{positioning, arrows.meta} \ No newline at end of file +\usetikzlibrary{positioning, arrows.meta, calc, matrix} \ No newline at end of file