Merge branch 'main' of https://github.com/janishutz/eth-summaries

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

@@ -1,2 +1,60 @@
 \newsectionNoPB
 \subsection{Line integrals}
+\shortdef Let $I = [a, b]$ be a closed and bounded interval in $\R$. $f : I \rightarrow \R$ with $f(t) = (f_1(t), \ldots, f_n(t))$ continuous (also $f_i$ cont.).
+\mrmvspace
+\begin{enumerate}[noitemsep, label=(\arabic*)]
+    \item Then $\displaystyle \int_{a}^{b} f(t) \dx t = \left( \int_{a}^{b} f_1(t), \ldots, \int_{a}^{b} f_n(t) \right)$
+    \item \bi{Parametrized Curve} in $\R^n$ is a continuous map $\gamma: I \rightarrow \R^n$, piecewise in $C^1$, i.e. for $k \geq 1$, we have partition $a = t_0 < t_1 < \ldots < t_k = b$,
+          such that if $f$ is restricted to interval $]t_{j - 1}, t_j$, restriction is $C^1$. $\gamma$ is a \textit{path} between $\gamma(a)$ and $\gamma(b)$
+    \item \bi{Line integral} $X \subseteq \R^n$ is the image of $\gamma$ as above and $f : X \rightarrow \R^n$ cont.\\
+          Integral $\displaystyle\int_{a}^{b} f(\gamma(t)) \cdot \gamma'(t) \dx t \in \R$
+          is line integral of $f$ along $\gamma$, denoted $\displaystyle\int_{\gamma} f(s) \dx s$ or $\displaystyle\int_{\gamma} f(s) \dx \vec{s}$ or $\displaystyle\int_{\gamma} \omega$,\\
+          with $\omega = f_1(x) \dx x_1 + \ldots f_n(x) \dx x_n$
+\end{enumerate}
+
+\rmvspace
+We usually call $f : X \rightarrow \R^n$ a \bi{vector field}, which maps each point $x \in X$ to a vector in $\R^n$, displayed as originating from $x$
+
+\setLabelNumber{all}{4}
+\compactdef{Oriented reparametrization} of $\gamma$ is parametrized curve $\sigma : [c, d] \rightarrow \R^n$ s.t $\sigma = \gamma \circ \varphi$, with $\varphi : [c, d] \rightarrow I$ cont. map,
+differentiable on $]a, b[$ and for which $\varphi(a) = c$ and $\varphi(b) = d$.
+Conversely, $\gamma = \sigma \circ \varphi^{-1}$
+
+\rmvspace
+\shortproposition For $f : X \rightarrow \R^n$ with $X$ containing the image of of $\gamma$ and equivalently $\sigma$,
+we have $\displaystyle\int_{\gamma} f(s) \cdot \dx \vec{s} = \int_{\sigma} f(s) \cdot \dx \vec{s}$
+
+\mrmvspace
+\setLabelNumber{all}{8}
+\compactdef{Conservative Vector Field} If for any $x_1, x_2 \in X$ the line integral $\displaystyle\int_{\gamma} f(s) \dx \vec{s}$ is independent choice of $\gamma$ in $X$
+
+\mrmvspace
+\shortremark $f$ conservative iff $\int_{\gamma} f(s) \dx \vec{s} = 0$ for a \textit{closed} ($\gamma(a) = \gamma(b)$) parametrized curve\\
+%
+\shorttheorem Let $X$ be open set, $f$ conservative vector field. Then $\exists C^1$ function $g$ s.t. $f = \nabla g$.
+If any two points of $X$ can be joined by a parametrized curve, then $g$ is unique up to a constant: if $\nabla g_1 = f$, then $g - g_1$ is constant on $X$
+
+\shortremark Two points $x, y \in X$ can be joined by parametrized curve $\gamma$ if $\gamma(a) = x$ and $\gamma(b) = y$. In that case, $X$ is called \bi{path-connected}.
+It is true when $X$ is \textit{convex} (e.g. when $X$ is a disc or a product of intervals).
+If $f$ is a vector field on $X$, then $g$ is called a \bi{potential} for $f$ and it is not unique, since we can add a constant to $g$ without changing the gradient.\\
+%
+\stepLabelNumber{all}
+\shortproposition For a vectorfield to be conservative, a \textit{necessary condition} is that $\displaystyle\frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{x_i}$
+for any $1 \leq i \neq j \leq n \in \N$\\
+%
+\stepLabelNumber{all}
+\compactdef{Start Shaped Set} $X \subseteq \R^n$ is star shaped if $\exists x_0 \in X$ s.t. $\forall x \in X$, the line segment from $x$ to $x_0$ is contained in $X$,
+and we also say that $X$ is \textit{star shaped around} $x_0$\\
+%
+\stepLabelNumber{all}
+\shorttheorem Let $X$ start shaped and open, $f$ a $C^1$ vector field fulfilling Proposition \ref{all:4-1-13}. Then $f$ is conservative.
+
+\drmvspace
+\setLabelNumber{all}{20}
+\shortdef Let $X \subseteq \R^3$ open and $f$ a $C^1$ vector field. Then the \bi{curl} of $f$ is the conservative vector field
+$\text{curl}(f) = \begin{bmatrix}
+        \partial_y f_3 - \partial_z f_2 \\
+        \partial_z f_1 - \partial_x f_3 \\
+        \partial_x f_2 - \partial_y f_1
+    \end{bmatrix}$
+\ddrmvspace

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

@@ -1,2 +1,48 @@
 \newsectionNoPB
 \subsection{Riemann integral in Vector Space}
+The integral of a continuous function $f: X \rightarrow \R$ with $X \subseteq \R^n$ bounded and closed, is denoted $\int_X f(x) \dx x$ with properties:
+\rmvspace
+\begin{enumerate}[label=(\arabic*), noitemsep]
+    \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{(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 $\text{Vol}_n(X)$
+    \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*}
+
+          \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*}