diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 565d8f8..28f2ab8 100644 Binary files a/semester3/analysis-ii/cheat-sheet-rb/main.pdf and b/semester3/analysis-ii/cheat-sheet-rb/main.pdf differ diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.tex b/semester3/analysis-ii/cheat-sheet-rb/main.tex index 9571fdd..2d40a89 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/main.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/main.tex @@ -9,11 +9,15 @@ \begin{document} +\section{Linear Algebra} +\input{parts/01_linalg.tex} + +\newpage \section{Differential Equations} -\input{parts/01_diffeq.tex} +\input{parts/02_diffeq.tex} \newpage \section{Differential Calculus in $\R^n$} -\input{parts/02_diff.tex} +\input{parts/03_diff.tex} \end{document} diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex new file mode 100644 index 0000000..c745a65 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex @@ -0,0 +1,17 @@ +Relevant definitions used throughout Analysis II. + +\definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\ +\subtext{Used to generalize $|x|$ in many Analysis I definitions} + +\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} +\subtext{$\forall x,y \in \R^n,\quad \alpha \in \R\\$} diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/02_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diff.tex deleted file mode 100644 index e69de29..0000000 diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex similarity index 93% rename from semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex rename to semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex index 1370abe..2f4abbc 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diffeq.tex @@ -204,4 +204,18 @@ If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ \int \frac{1}{a(y)}\ \text{d}y = \int b(x) \dx + c $$ \end{subbox} -\subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} \ No newline at end of file +\subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} + +\subsection{Method Overview} + +\begin{center} + \begin{tabular}{l|l} + \textbf{Method} & \textbf{Use case} \\ + \hline + Variation of constants & LDE with $\ord(F)=1$ \\ + Characteristic Polynomial & Hom. LDE w/ const. coeff. \\ + Undetermined Coefficients & Inhom. LDE w/ const. coeff. \\ + Separation of Variables & ODE s.t. $y' = a(y)\cdot b(x)$ \\ + Change of Variables & e.g. $y' = f(ax + by + c)$ \\ + \end{tabular} +\end{center} \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex new file mode 100644 index 0000000..08a5573 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex @@ -0,0 +1,121 @@ +\subtext{Treating functions $f: X \subset \R^n \to \R / \C / \R^m,\quad m,n \geq 1$} + +\notation $f(x)$ for $f: I \subset \R^n \to \R^m$ means:\\ +$x = (x_1, \ldots, x_n),\quad f(x) = f\bigl( f_1(x), \ldots, f_m(x) \bigr)$ + +\subsection{Multivariate functions} + +\definition \textbf{Linear map} $f: \R^n \to \R^m$\\ +\subtext{In other words: $f(x) = \textbf{A}x,\quad \textbf{A} \in \C^{m \times n}$} + +Linear Maps are continuous + +\definition \textbf{Affine Linear map} $f(x) \mapsto \textbf{A}x + c$ + +\definition \textbf{Quadratic form} $Q: \R^n \to \R$\\ +\subtext{In other words: $Q(x) = \sum_{i=0}^{n}\sum_{j=0}^{m}\left( a_{i,j}x_i x_j \right)$} + +\definition \textbf{Monomials} $M(x): \R^n \to \R \mapsto \alpha x_1^{d_1}\cdots x_n^{d_n}$\\ +\subtext{For example: $f(x, y, z) = 16x^2yz^5$} + +\definition $\deg(M) := e = \sum_{i=1}^{n} d_i$\\ +\subtext{For example: $\deg(16x^2yz^5) = 8$} + +\definition \textbf{Polynomials} $P(x) := \sum_{i=0}^{n} M_i(x)$\\ +\subtext{For example: $P(x,y,z) = x^3 + 25x^2y^6z + xy$} + +Polynomials are continuous. + +\definition $\deg(P) := d \geq \max \{ \deg(M_i) \sep M_i \text{ in } P \}$\\ +\subtext{For example: $\deg(x^3 + 25x^2y^6z + xy) = 9$} + +Visualisations for some function types: + +\definition \textbf{Graph} $G_f := \{(x,y,z) \in \R^3 \sep z = f(x,y) \}$\\ +\subtext{Only for $f: \R^2 \to \R$. Visually, this is a surface in $\R^3$} + +\definition \textbf{Vector Plots} for $f: \R^2 \to \R^2$\\ +\subtext{Points in $(x,y) \in \R^2$ are displayed as vectors $f(x,y)$} + +\newpage +\subsection{Sequences in $\R^n$} + +\definition \textbf{Sequences in $\R^n$}\\ +$(x_k)_{k \geq 1}$ s.t. $x_k \in \R^n$ where $x_k = \bigl( x_{k,1},\ldots x_{k,n} \bigr)$ + +\definition \textbf{Convergence in $\R^n$}\\ +$$ + \lim_{k \to \infty} \Bigl( x_k \Bigr) = y \iff \forall \epsilon > 0, \exists N \geq 1: \forall k \geq N:\quad || x_k - y || < \epsilon +$$ + +Using this definition preserves many familiar results: + +\lemma \textbf{Equivalent conditions to Convergence}\\ +$ +\begin{array}{ll} + (i) & \forall i \text{ s.t. } 1 \leq i \leq n:\quad \underset{k \to \infty}{\lim} \Bigl(x_{k,i}\Bigr) = y_i \\ + (ii) & \underset{k \to \infty}{\lim} \Big\| x_k - y \Big\| = 0 +\end{array} +$ + +\definition \textbf{Continuity in $\R^n$}\\ +$f \text{ continuous at } x_0 \in X \iffdef \forall \epsilon > 0, \exists \delta > 0:\\$ +$$ + \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - f(x_0) \big\| < \epsilon\\ +$$ +$f$ continuous $\iffdef \forall x \in X: f$ continuous at $x$\\ +\subtext{$X \subset \R^n,\quad f:X \to \R^m$} + +\lemma \textbf{Continuitiy using Sequences}\\ +$f$ continuous at $x_0$ if and only if: +$$ + \forall (x_k)_{k \geq 1}:\quad \underset{k \to \infty}{\lim} \Bigl( x_k \Bigr) = x_0 \implies \underset{k \to \infty}{\lim}\Bigl(f(x_k)\Bigr) = f(x_0) +$$ +\subtext{$X \subset \R^n,\quad f:X \to \R^m$} + +\definition \textbf{Limits at points} +\begin{align*} + & \underset{x \neq x_0 \to x_0}{\lim} \Bigl( f(x) \Bigr) = y \iffdef \forall \epsilon > 0, \exists \delta > 0: \\ + & \forall x \neq x_0 \in X: \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - y \big\| < \epsilon +\end{align*} +\subtext{$X \subset \R^n,\quad f:X \to \R^m,\quad x_0 \in X,\quad y \in \R^m$} + +The sequence test for Continuity works for point-limits too. + +\lemma \textbf{Continuity of Compositions}\\ +$f: X \to Y,\ g: Y \to \R^p \text{ continuous } \implies g \circ f \text{ continuous}$\\ +\subtext{$X \subset \R^n,\quad Y \subset \R^m,\quad p \geq 1$} + +\lemma \textbf{Continuity using Coordinate Functions}\\ +$f: \R^n \to \R^m$ continuous $\iff \forall i \leq m: f_i$ continuous + +\subsection{Subsets of $\R^n$} + +\definition \textbf{Bounded}\\ +$X \subset \R^n$ bounded $\iffdef \Bigl\{ \big\| x \big\| \sep x \in X \Bigr\} \subset \R$ bounded.\\ +\subtext{Example: The open disc $D = \{ x \in \R^n \sep \big\| x - x_0 \big\| < r \}$ is bounded.} + +\definition \textbf{Closed}\\ +$X \subset \R^n$ closed $\iffdef \forall (x_k)_{k\geq 1} \in X:\quad \underset{x \to \infty}{\lim}\Bigl( x_k \Bigr) \in X$\\ +\subtext{Example: $\emptyset$, $\R^n$ are closed.} + +\definition \textbf{Compact} if closed and bounded.\\ +\subtext{Example: The closed Disc $\Lambda = \{ x \in \R^n \sep \big\| x - x_0 \big\| \leq r \}$ is compact.} + +\lemma The Cartesian Product preserves these properties. + +\lemma \textbf{Continous functions preserve closedness} +$$ + \forall \text{ closed } Y:\quad f^{-1}(Y) = \bigl\{ x \in \R^n \sep f(x) \in Y \bigr\} \text{ is closed.} +$$ +\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$} + +\begin{subbox}{Min-Max Theorem} + \smalltext{For compact, non-empty $X \subset \R^n$, continuous $f: X \to \R$:} + $$ + \exists x_1,x_2 \in X :\quad f(x_1) = \underset{x \in X}{\sup} f(x),\quad f(x_2) = \underset{x \in X}{\inf} f(x) + $$ +\end{subbox} + +\subsection{Partial Derivatives} + diff --git a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex index 02d2dbc..2870a76 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex @@ -52,6 +52,7 @@ \def \notation{\colorbox{lightgray}{Notation} } \def \remark{\colorbox{lightgray}{Remark} } \def \theorem{\colorbox{lightgray}{Th.} } +\def \lemma{\colorbox{lightgray}{Lem.} } \def \method{\colorbox{lightgray}{Method} } % For intuiton and less important notes