[Analysis] Small notes

semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex

@@ -30,4 +30,35 @@ Relevant definitions used throughout Analysis II.
 \end{center}
 \smalltext{If $0$ is allowed, $\textbf{A}$ is called positive/negative semi-definite.}
 
-\definition \textbf{Trace} $\text{Tr}(\textbf{A}) := \displaystyle\sum_{i=0}^{\text{min}(m,n)} \textbf{A}_{i, i}$
+\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 $\textbf{A} \in \R^{2\times2}$
+    $$
+        \det(\textbf{A})
+        =
+        \det\left(
+            \begin{bmatrix}
+                a & b \\
+                c & d
+            \end{bmatrix}
+        \right) = ad - bc
+    $$
+
+    \lemma \textbf{Inverse} of $\textbf{A} \in \R^{2\times2}$
+    $$
+    \textbf{A}^{-1}
+    =
+    \begin{bmatrix}
+        a & b \\
+        c & d
+    \end{bmatrix}^{-1}
+    =
+    \frac{1}{\det(\textbf{A})}
+    \begin{bmatrix}
+        d & -b \\
+        -c & a
+    \end{bmatrix}
+    $$
+\end{footnotesize}

semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex

@@ -119,7 +119,8 @@ $$\bigl\{ y \in \R^n \sep  |x_i - y_i| < \delta,\ \ \forall i \leq n \bigr\} \su
 $$
     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,$\quad Y \subset \R^m$}
+\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$}\\
+\subtext{Note: $X$ open/closed, does \textit{not} imply $f(X)$ open/closed}
 
 \lemma \textbf{The complement of open sets is closed}
 $$

semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex

@@ -295,12 +295,16 @@ $$
 Don't \textit{have to} exist if $X$ is open, only if $X$ is compact.\\
 \subtext{However, for compact sets, the lemma above no longer applies.}
 
-\method \textbf{Critical points on Compact Sets}\\
-Decompose $X = X' \cup B$, s.t. $X'$ is open, $B$ is a \textit{boundary}. 
+\method \textbf{Critical points on Compact Sets}
+
+\footnotesize
+Decompose $X = X' \cup B$, s.t. $X'$ (Interior) is open, $B$ is a \textit{boundary}. 
 \begin{enumerate}
-    \item Find critical points in $X'$
-    \item Check if any $x \in B$ is a maximum/minimum 
+    \item Find critical points in $X'$: via $\nabla f$, check state using $\textbf{H}_f$
+    \item Check if any $x \in B$ is a maximum/minimum\\
+    For this: try to parametrize (sections of) $B$, check corners.
 \end{enumerate}
+\normalsize
 
 \definition \textbf{Non-degenerate Critical Point}
 $$

semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex

@@ -85,8 +85,13 @@ $\forall x_1,x_2 \in X: \exists \gamma: [a,b] \to X$ s.t. $\gamma(a) = x_1, \gam
 $$
     \forall 1 \leq i \neq j \leq n:\quad \frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i}
 $$
+\smalltext{Equivalently:}
+$$
+    \textbf{J}_f(x) = \textbf{J}_f(x)^\top
+$$
+
 \subtext{$X \subset \R^n \text{ open},\quad f: X\to\R^n,\quad f \in C^1,\quad f \text{ conserv.}$}\\
-\subtext{Only this was: This being true does not imply $f$ is conservative!}
+\subtext{Only this way: This being true (alone) does not imply $f$ is conservative!}
 
 \definition \textbf{Star Shaped Set}\\
 $\exists x_0 \in X: \forall x \in X$ Line seg. $x_0 \to x$ is in $X$
@@ -109,6 +114,21 @@ $$
 
 \remark $\text{curl}(f) = 0 \iff \forall 1 \leq i \neq j \leq 3: \displaystyle\frac{\partial f_i}{\partial x_j} = \displaystyle\frac{\partial f_j}{\partial x_i}$
 
+\remark $\text{curl}(\nabla f) = 0$ if $f: \R^3 \to \R$ is in $C^2$.
+
+\method \textbf{Finding the Potential}
+
+\smalltext{
+    We want $g$ s.t. $\nabla g = f$ for some conservative $f$
+    \begin{enumerate}
+        \item Find $\displaystyle\int f_i\ dx_i$ for all $i \leq n$
+        \item Define $g$ as the union of terms in $\displaystyle\int f_i\ dx_i$
+        \item $g$ now has $\partial x_i g_i = f_i$ thus $\nabla g = f$ 
+    \end{enumerate} 
+    Note that the union step \textit{only works} if $f$ is conservative.
+}
+
+
 \newpage
 \subsection{The Riemann Integral in $\R^n$}
 
@@ -266,11 +286,11 @@ $\gamma: [a,b] \to \R^2$ closed param. curve s.t.
     \smalltext{$X \subset \R^2 \text{ compact with Boundary } \partial X = \displaystyle\underset{1 \leq i \leq n}{\bigcup} \gamma_i$ as above}\\
     \smalltext{Assume: $\gamma_i: [a_i,b_i] \to \R^2$ s.t. $X$ is always \textit{left} of $\gamma_i'(t)$ at $\gamma_i(t)$}
     $$
-    \int_X \Biggl( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \Biggr)\ dxdy = \sum_{i=1}^{k} \int_{\gamma_i} f \cdot ds
+    \int_X \Biggl( \underbrace{\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y}}_{\text{curl}(f)} \Biggr)\ dxdy = \sum_{i=1}^{k} \int_{\gamma_i} f \cdot ds
     $$
     \smalltext{For a $C^1$ Vector field $f=(f_1,f_2)$ containing $X$}
 \end{subbox}
-\subtext{So, some Riemann Integrals can be converted to a sum of line integrals.}
+\subtext{So, a sum of line integrals can be written as the Integral of the curl.\\ This is very useful for computing complex line integrals.}
 
 \lemma \textbf{Volume using Green}
 $$