[Analysis] Diff. Calc.

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}

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} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} \textbf{A}_{i, i}$

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}

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}

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