diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 8f1fed0..f92bcaa 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/parts/01_linalg.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex index e74de77..8f95c81 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/01_linalg.tex +++ b/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} diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex index 9b53dee..296e178 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/04_cont.tex +++ b/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.} $$ -\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$} +\subtext{$f: \R^n \to \R^m$ is continuous,$\quad Y \subset \R^m$}\\ +\subtext{Note: $X$ open/closed, does \textit{not} imply $f(X)$ open/closed} \lemma \textbf{The complement of open sets is closed} $$ diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex index c79afae..43222e3 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex @@ -295,12 +295,16 @@ $$ Don't \textit{have to} exist if $X$ is open, only if $X$ is compact.\\ \subtext{However, for compact sets, the lemma above no longer applies.} -\method \textbf{Critical points on Compact Sets}\\ -Decompose $X = X' \cup B$, s.t. $X'$ is open, $B$ is a \textit{boundary}. +\method \textbf{Critical points on Compact Sets} + +\footnotesize +Decompose $X = X' \cup B$, s.t. $X'$ (Interior) is open, $B$ is a \textit{boundary}. \begin{enumerate} - \item Find critical points in $X'$ - \item Check if any $x \in B$ is a maximum/minimum + \item Find critical points in $X'$: via $\nabla f$, check state using $\textbf{H}_f$ + \item Check if any $x \in B$ is a maximum/minimum\\ + For this: try to parametrize (sections of) $B$, check corners. \end{enumerate} +\normalsize \definition \textbf{Non-degenerate Critical Point} $$ diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex index 3ea5cbc..b9d51e9 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex +++ b/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 $$ \smalltext{For a $C^1$ Vector field $f=(f_1,f_2)$ containing $X$} \end{subbox} -\subtext{So, some Riemann Integrals can be converted to a sum of line integrals.} +\subtext{So, a sum of line integrals can be written as the Integral of the curl.\\ This is very useful for computing complex line integrals.} \lemma \textbf{Volume using Green} $$