diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 28f2ab8..9306199 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 2d40a89..49f4e7b 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/main.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/main.tex @@ -16,8 +16,12 @@ \section{Differential Equations} \input{parts/02_diffeq.tex} +\newpage +\section{Continuous functions in $\R^n$} +\input{parts/03_cont.tex} + \newpage \section{Differential Calculus in $\R^n$} -\input{parts/03_diff.tex} +\input{parts/04_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 index c745a65..966f0cb 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex @@ -1,5 +1,7 @@ Relevant definitions used throughout Analysis II. +\subtext{$\textbf{A} \in \R^{m \times n},\quad x,y \in \R^n,\quad \alpha \in \R$} + \definition \textbf{Euclidian Norm} $||x|| := \displaystyle\sqrt{\sum_{i=1}^{n} x_i^2}$\\ \subtext{Used to generalize $|x|$ in many Analysis I definitions} @@ -14,4 +16,5 @@ Relevant definitions used throughout Analysis II. \end{array} $ \end{center} -\subtext{$\forall x,y \in \R^n,\quad \alpha \in \R\\$} + +\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} \textbf{A}_{i, i}$ diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/03_cont.tex similarity index 84% rename from semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex rename to semester3/analysis-ii/cheat-sheet-rb/parts/03_cont.tex index 08a5573..9b53dee 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/03_diff.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/03_cont.tex @@ -58,6 +58,17 @@ $ \end{array} $ +\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. + +\subsection{Continuity in $\R^n$} + \definition \textbf{Continuity in $\R^n$}\\ $f \text{ continuous at } x_0 \in X \iffdef \forall \epsilon > 0, \exists \delta > 0:\\$ $$ @@ -73,15 +84,6 @@ $$ $$ \subtext{$X \subset \R^n,\quad f:X \to \R^m$} -\definition \textbf{Limits at points} -\begin{align*} - & \underset{x \neq x_0 \to x_0}{\lim} \Bigl( f(x) \Bigr) = y \iffdef \forall \epsilon > 0, \exists \delta > 0: \\ - & \forall x \neq x_0 \in X: \big\| x - x_0 \big\| < \delta \implies \big\| f(x) - y \big\| < \epsilon -\end{align*} -\subtext{$X \subset \R^n,\quad f:X \to \R^m,\quad x_0 \in X,\quad y \in \R^m$} - -The sequence test for Continuity works for point-limits too. - \lemma \textbf{Continuity of Compositions}\\ $f: X \to Y,\ g: Y \to \R^p \text{ continuous } \implies g \circ f \text{ continuous}$\\ \subtext{$X \subset \R^n,\quad Y \subset \R^m,\quad p \geq 1$} @@ -102,20 +104,31 @@ $X \subset \R^n$ closed $\iffdef \forall (x_k)_{k\geq 1} \in X:\quad \underset{x \definition \textbf{Compact} if closed and bounded.\\ \subtext{Example: The closed Disc $\Lambda = \{ x \in \R^n \sep \big\| x - x_0 \big\| \leq r \}$ is compact.} -\lemma The Cartesian Product preserves these properties. +\definition \textbf{Open}\\ +$X \subset \R^n$ open $\iffdef \forall x \in X,\ \exists \delta > 0:$ +$$\bigl\{ y \in \R^n \sep |x_i - y_i| < \delta,\ \ \forall i \leq n \bigr\} \subset X$$ +\subtext{In other words: Changing any coord. $x_i$ by $\delta$ keeps $x'$ in $X$}\\ +\subtext{Example: $\emptyset, \R^n$ are open (and closed)} -\lemma \textbf{Continous functions preserve closedness} + +\lemma The Cartesian Product preserves bounded/closed. + +\lemma \textbf{Continous functions preserve closed/open} + +\smalltext{$\forall \text{ closed/open } Y:$} $$ - \forall \text{ closed } Y:\quad f^{-1}(Y) = \bigl\{ x \in \R^n \sep f(x) \in Y \bigr\} \text{ is closed.} + f^{-1}(Y) = \bigl\{ x \in \R^n \sep f(x) \in Y \bigr\} \text{ is closed/open.} $$ \subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$} +\lemma \textbf{The complement of open sets is closed} +$$ + X \subset \R^n \text{ is open } \iff \underbrace{\bigl\{ x \in \R^n \sep x \notin X \bigr\}}_{\text{Complement}} \text{ is closed} +$$ + \begin{subbox}{Min-Max Theorem} \smalltext{For compact, non-empty $X \subset \R^n$, continuous $f: X \to \R$:} $$ \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) $$ -\end{subbox} - -\subsection{Partial Derivatives} - +\end{subbox} \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex new file mode 100644 index 0000000..5793715 --- /dev/null +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/04_diff.tex @@ -0,0 +1,62 @@ +\subsection{Partial Derivatives} + +\begin{subbox}{Partial Derivative} + \smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R,\quad 1 \leq i \leq n,\quad x_0 \in X$} + $$\dfd{i}(x_{0}) := g'(x_{0,i})$$ + \smalltext{for $g: \{ t \in \R \sep (x_{0, 1}, \ldots,\ t\ ,\ldots, x_{0, n}) \in X \} \to \R^n$} + $$ g(t) := \underbrace{f(x_{0,i}, \ldots, x_{0,t-1},\ t\ , x_{0, t+1},\ldots,x_{0, n})}_{ \text{ Freeze all }x_{0, k} \text{ except one } x_{0, i} \to t}$$ +\end{subbox} + +\notation $\dfd{i}(x_0) = \sdfd{i}(x_0) =\ssdfd{i}(x_0)$ + +\lemma \textbf{Properties of Partial Derivatives}\\ +\smalltext{Assuming $\sdfd{i} \text{ and } \partial_{x_i} g \text{ exist }$:} + +$ +\begin{array}{ll} + (i) & \partial x_i (f+g) = \partial x_i f + \partial x_i g \\ + (ii) & \partial x_i (fg) = \partial x_i (f)g + \partial x_i (g)f\quad \text{ if } m=1\\ + (iii) & \partial x_i \Bigr(\displaystyle\frac{f}{g}\Bigl) = \displaystyle\frac{\partial x_i(f)g - \partial x_i(g)f}{g^2}\quad \text{ if } g(x) \neq 0\ \forall x \in X\\ +\end{array} +$\\ +\subtext{$X \subset \R^n \text{ open},\quad f.g: X \to \R^n,\quad 1 \leq i \leq n$} + +\begin{subbox}{The Jacobian} + \smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R^n \text{ with partial derivatives existing}$} + $$ + \textbf{J}_f(x) := \begin{bmatrix} + \partial x_1 f_1(x) & \partial x_2 f_1(x) & \cdots & \partial x_n f_1(x) \\ + \partial x_1 f_2(x) & \partial x_2 f_2(x) & \ddots & \vdots \\ + \vdots & \vdots & \ddots & \vdots \\ + \partial x_1 f_n(x) & \partial x_2 f_n(x) & \cdots & \partial x_n f_m(x) + \end{bmatrix} + $$ +\end{subbox} +\subtext{Think of $f$ as a vector of $f_i$, then $\textbf{J}_f$ is that vector stretched for all $x_j$} + +\definition \textbf{Gradient} $\nabla f(x_0) := \begin{bmatrix} + \partial x_1 f(x_0) \\ + \vdots \\ + \partial x_n f(x_0) +\end{bmatrix} = \textbf{J}_f(x)^\top$\\ +\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R$} + +\definition \textbf{Divergence} $\text{div}(f)(x_0) := \text{Tr}\bigr(\textbf{J}_f(x_0)\bigl)$\\ +\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^n,\quad \textbf{J}_f \text{ exists}$} + +\subsection{The Differential} + +\smalltext{ + Partial derivatives don't provide a good approx. of $f$, unlike in the $1$-dimensional case. The \textit{differential} is a linear map which replicates this purpose in $\R^n$. +} + +\begin{subbox}{Differentiability in $\R^n$} + \smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R^n,\quad u: \R^n \to \R^m \text{ linear map}$} + $$ + df(x_0) := u + $$ + If $f$ is differentiable at $x_0 \in X$ with $u$ s.t. + $$ + \underset{x \neq x_0 \to x_0}{\lim} \frac{1}{\big\| x - x_0 \big\|}\Biggl( f(x) - f(x_0) - u(x - x_0) \Biggr) = 0 + $$ +\end{subbox} diff --git a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex index 2870a76..08e9cf8 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex @@ -55,6 +55,12 @@ \def \lemma{\colorbox{lightgray}{Lem.} } \def \method{\colorbox{lightgray}{Method} } +% partial derivatives +\def \dfd#1{\frac{\partial f}{\partial x_#1}} +\def \sdfd#1{\partial_{x_#1}f} +\def \ssdfd#1{\partial_#1f} +\def \dd#1#2{\frac{\partial_#1}{\partial x_#2}} % to replace f + % For intuiton and less important notes \def \subtext#1{ \color{gray}\footnotesize