[Analysis] Theory summary complete

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex

@@ -57,3 +57,4 @@ $\text{curl}(f) = \begin{bmatrix}
         \partial_z f_1 - \partial_x f_3 \\
         \partial_x f_2 - \partial_y f_1
     \end{bmatrix}$
+\ddrmvspace

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex

@@ -6,33 +6,43 @@ The integral of a continuous function $f: X \rightarrow \R$ with $X \subseteq \R
     \item \bi{(Compatibility)} If $n = 1$ and $X = [a, b]$, integral is the indefinite integral as per Analysis I
     \item \bi{(Linearity)} If $f$, $g$ are continuous on $X$ and $a, b \in \R$, then $\displaystyle \int_X (a f(x) + b g(x)) \dx x = a \int_X f(x) \dx x + b \int_X g(x) \dx x$
     \item \bi{(Positivity)} If $f \leq g$, then so is the integral and if $f \geq 0$, so is the integral and if $Y \subseteq X$, then int. over $Y$ is $\leq$ over $X$
-    \item \bi{(Upper bound \& Triangle Inequality)} $\displaystyle \left| \int_{X} f(x) \dx x \right| \leq \int_{X} |f(x)|\dx x$ and 
-        $\displaystyle \left| \int_{X} (f(x) + g(x)) \dx x \right| \leq \int_{X} |f(x)| \dx x \int_X |g(x)|$
+    \item \bi{(Upper bound \& Triangle Inequality)} $\displaystyle \left| \int_{X} f(x) \dx x \right| \leq \int_{X} |f(x)|\dx x$ and
+          $\displaystyle \left| \int_{X} (f(x) + g(x)) \dx x \right| \leq \int_{X} |f(x)| \dx x \int_X |g(x)|$
     \item \bi{(Volume)} The integral of $f$ is the volume of $\{ (x, y) \in X \times \R : 0 \leq y \leq f(x) \} \subseteq \R^{n + 1}$.
-        If $X$ is a bounded rectangle, e.g. $X = [a_1, b_1] \times \ldots \times [a_n, b_n] \subseteq \R^n$ and $f = 1$, then $\int_{X} \dx x = (b_n - a_n) \dots (b_1 - a_1)$.
-        We write $\text{Vol}(X)$ or $\text{Vol}_n(X)$
+          If $X$ is a bounded rectangle, e.g. $X = [a_1, b_1] \times \ldots \times [a_n, b_n] \subseteq \R^n$ and $f = 1$, then $\int_{X} \dx x = (b_n - a_n) \dots (b_1 - a_1)$.
+          We write $\text{Vol}(X)$ or $\text{Vol}_n(X)$
     \item \bi{(Multiple integral)} \textit{(Fubini)} If $n_1, n_2 \in \Z$ s.t. $n = n_1 + n_2$,
-        then for $x_1 \in \R^{n_1}$, let $Y_{x_1} = \{ x_2 \in \R^{n_2} : (x_1, x_2) \in X \} \subseteq \R^{n_2}$.
-        Let $X_1$ be the set of $x_1 \in \R^n$ such that $Y_{x_1}$ is not empty. Then $X_1$ and $Y_{x_1}$ are compact.\\
-        If $\displaystyle g(x_1) = \int_{Y_{x_1}} f(x_1, x_2) \dx x_2$ is continuous on $X_1$, then
-        \dnrmvspace
-        \begin{align*}
-            \int_{X} f(x_1, x_2) \dx x = \int_{X_1} g(x_1) \dx x = \int_{X_1} g(x_1) \dx x_1 = \int_{X_1} \left( \int_{Y_{x_1}} f(x_1, x_2) \dx x_2 \right) \dx x_1
-        \end{align*}
-        
-        \rmvspace
-        Exchanging the role of $x_1$ and $x_2$ we have (with $Z_{x_2} = \{ x_1 : (x_1, x_2) \in X \}$) if integral over $x_1$ is continuous.
-        \rmvspace
-        \begin{align*}
-            \int_{X} f(x_1, x_2) \dx x = \int_{X_2} \left( \int_{Z_{x_2}} f(x_1, x_2) \dx x_1 \right) \dx x_2
-        \end{align*}
-        \drmvspace
-    \item \bi{(Domain additivity)} If $X_1$ and $X_2$ are compact and $f$ continuous on $X = X_1 \cup X_2$, then (for $Y = X_1 \cap X_2$)
-        \rmvspace
-        \begin{align*}
-            \int_X f(x) \dx x + \int_Y f(x) \dx x = \int_{X_1} f(x) \dx x + \int_{X_2} f(x) \dx x
-        \end{align*}
+          then for $x_1 \in \R^{n_1}$, let $Y_{x_1} = \{ x_2 \in \R^{n_2} : (x_1, x_2) \in X \} \subseteq \R^{n_2}$.
+          Let $X_1$ be the set of $x_1 \in \R^n$ such that $Y_{x_1}$ is not empty. Then $X_1$ and $Y_{x_1}$ are compact.\\
+          If $\displaystyle g(x_1) = \int_{Y_{x_1}} f(x_1, x_2) \dx x_2$ is continuous on $X_1$, then
+          \dnrmvspace
+          \begin{align*}
+              \int_{X} f(x_1, x_2) \dx x = \int_{X_1} g(x_1) \dx x = \int_{X_1} g(x_1) \dx x_1 = \int_{X_1} \left( \int_{Y_{x_1}} f(x_1, x_2) \dx x_2 \right) \dx x_1
+          \end{align*}
 
-        \rmvspace
-        In particular, if $Y$ empty (or size is ``negligible''), then $\int_{X} f(x) \dx x = \int_{X_1} f(x) \dx x + \int_{X_2} f(x) \dx x$
+          \rmvspace
+          Exchanging the role of $x_1$ and $x_2$ we have (with $Z_{x_2} = \{ x_1 : (x_1, x_2) \in X \}$) if integral over $x_1$ is continuous.
+          \rmvspace
+          \begin{align*}
+              \int_{X} f(x_1, x_2) \dx x = \int_{X_2} \left( \int_{Z_{x_2}} f(x_1, x_2) \dx x_1 \right) \dx x_2
+          \end{align*}
+          \drmvspace
+    \item \bi{(Domain additivity)} If $X_1$ and $X_2$ are compact and $f$ continuous on $X = X_1 \cup X_2$, then (for $Y = X_1 \cap X_2$)
+          \rmvspace
+          \begin{align*}
+              \int_X f(x) \dx x + \int_Y f(x) \dx x = \int_{X_1} f(x) \dx x + \int_{X_2} f(x) \dx x
+          \end{align*}
+
+          \rmvspace
+          In particular, if $Y$ empty (or size is ``negligible''), then $\int_{X} f(x) \dx x = \int_{X_1} f(x) \dx x + \int_{X_2} f(x) \dx x$
 \end{enumerate}
+\setLabelNumber{all}{3}
+\shortdef For $m \leq n \in \N$, a \bi{parametrized $m$-set} in $\R^n$ is a continuous map $f: [a_1, b_1] \times \ldots \times [a_m, b_m] \rightarrow \R^n$,
+which is $C^1$ on $]a_1, b_1[ \times \ldots \times ]a_m, b_m[$.
+$B \subseteq \R^n$ is \bi{negligible} if $\exists k \geq 0 \in \Z$ and parametrized $m_i$-sets $f_i: X_i \rightarrow \R^n$ with $1 \leq i \leq k$ and $m_i < n$ s.t.
+$X \subseteq f_1(x_1) \cup \ldots \cup f_k(X_k)$. A parametrized $1$-set in $\R^n$ is a parametrized curve.
+\shortex Any $\R \times \{ 0 \} \subseteq \R^2$ is negligible in $\R^2$, or more generally,
+if $H \subseteq \R^n$ is an affine subspsace of dimension $m < n$, then any subset of $\R^n$ that is contained in $H$ is negligible.
+Image of par. curve $\gamma: [a, b] \rightarrow \R^n$ is negligible, since $\gamma$ is a $1$-set in $\R^n$
+
+\shortproposition $X$ compact set, negligible. Then for any cont. function on $X$, $\displaystyle\int_{X} f(x) \dx x = 0$

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex

@@ -1,2 +1,11 @@
 \newsectionNoPB
 \subsection{Improper integrals}
+As in the one-dimensional case, we are looking at integrals that are undefined at the edge of the interval and thus, we apply a limit to them,
+thus approaching said edge of the interval.
+
+For example, in the two-dimensional case, disc $D_R = [-R, R]^2$ with radius $R$
+\rmvspace
+\begin{align*}
+    \limit{R}{\infty} \int_{D_R} f(d, y) \dx x \dx y
+\end{align*}
+\ddrmvspace

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex

@@ -1,2 +1,9 @@
 \newsectionNoPB
 \subsection{Change of Variable Formula}
+\compacttheorem{Change of variable formula} $\overline{X}, \overline{Y} \subseteq \R^n$ compact, $\varphi : \overline{X} \rightarrow \overline{Y}$ continuous.
+For the open sets $X, Y$, negligible sets $B, C$ and restriction of $\varphi : X \rightarrow Y$ to open set $X$ is a $C^1$ bijection,
+we can write $\overline{X} = X \cup B$ and $\overline{Y} = Y \cup C$.
+The Jacobian $J_\varphi(x)$ is invertible at all $x \in X$.
+For any cont. func. $f$ on $\overline{Y}$ we have $\displaystyle \int_{\overline{X}} f(\varphi(x)) |\det(J_\varphi(x))| \dx x = \int_{\overline{Y}} f(y) \dx y$
+% TODO: Add notes from TA's notes for how to apply it
+\rmvspace

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex

@@ -1,2 +1,22 @@
 \newsectionNoPB
 \subsection{The Green Formula}
+\compactdef{Simple parametrized curve} $\gamma : [a, b] \rightarrow \R^2$ is a closed parametrized curve s.t.
+$\gamma(t) \neq \gamma(s)$ (if $s \neq t$ and $\{ s, t \} = \{ a, b \}$), s.t. $\gamma'(t) \neq 0$ for $a < t < b$.
+If $\gamma$ only piecewise in $C^1$ in $]a, b[$, then only apply when $\gamma'(t)$ exists.
+
+\stepLabelNumber{all}
+\compacttheorem{Green's Formula} $X \subseteq \R^2$ compact set with boundary $\partial X = \gamma_1 \cup \ldots \cup y_k$
+with $\gamma_i = (\gamma_{i, 1}, \gamma_{i, 2}) : [a_i, b_i] \rightarrow \R^2$ a simple closed parametrized curve, with property that $X$ lies ``to the left'' of tangent vector $\gamma_i'(t)$ based at $\gamma_i(t)$.
+$f = (f_1, f_2)$ is a vector field of class $C^1$ on open set containing $X$. Then:
+\drmvspace
+\begin{align*}
+    \int_{X} \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial_y} \right) \dx x \dx y = \sum_{i = 1}^{k} \int_{\gamma_i} f \cdot \dx \vec{s}
+\end{align*}
+
+\stepLabelNumber{all}\dhrmvspace
+\inlinecorollary $X \subseteq \R^2$ compact with boundary $\partial X$ as before.
+$\gamma_i$ as above, then
+\drmvspace
+\begin{align*}
+    \text{Vol}(X) = \sum_{i = 1}^{k} \int_{\gamma_i} x \dx \vec{s} = \sum_{i = 1}^{k} \int_{a_i}^{b_i} \gamma_{i, 1}(t) \gamma_{i, 2}'(t) \dx t
+\end{align*}