From e579deab2be507f475f613f0083e28a52f8deb96 Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Tue, 6 Jan 2026 15:59:45 +0100 Subject: [PATCH] [Analysis] Int. 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a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex index ab5e850..90bda36 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex @@ -29,4 +29,121 @@ $$ \remark $f: X \to \R^n$ is called a \textit{Vector Field}. -\definition \textbf{Oriented Reparametrization} \ No newline at end of file +\definition \textbf{Oriented Reparametrization}\\ +\smalltext{For $\gamma: [a,b] \to \R^n$ (param. curve), $\phi:[c,d] \to [a,b]$ continuous} +$$ + \sigma: [c,d] \to \R^n \text{ s.t. } \sigma = \gamma \circ \phi +$$ +\subtext{diff.-able on $(c,d)$, strictly increasing and $\phi(c) = a, \phi(d) = b$} + +\newpage % to keep the elements below together +\lemma \textbf{Oriented Reparametrizations preserve Integrals} +$$ + \int_\gamma f(s)\cdot ds = \int_\sigma f(s)\cdot ds +$$ +\subtext{$\gamma: [a,b] \to \R^n$ param. curve$,\quad \sigma$ oriented reparam.$,\\ +\gamma([a,b]) \subset X,\quad f: X \to \R^n \text{ cont.}$} + +\remark Line Integrals of the form $\int_\gamma \nabla f(s) \cdot ds$ have: +$$ + \int_\gamma \nabla f(s) \cdot ds = \int_a^b \sum_{i=1}^{n}\frac{\partial g}{\partial x_i}\Bigl( \gamma(t) \Bigr) \gamma_i'(t) = f\Bigl( \gamma(b) \Bigr) - f\Bigl( \gamma(a) \Bigr) +$$ +\subtext{Follows from the Chain rule for $h(t) = g(\gamma(t))$}\\ +\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R,\quad f \in C^1,\quad \gamma: [a,b] \to X \text{ param. curve}$} + +\definition \textbf{Conservative Vector Field}\\ +\smalltext{$f: X\to\R$ conservative $\iffdef \forall \gamma_1,\gamma_2$ s.t. start \& end points match:} +$$ + \int_{\gamma_1} f(s)\cdot ds = \int_{\gamma_2} f(s)\cdot ds +$$ +\subtext{No matter which path, if start \& end match, the integral matches} + +\remark \textbf{Closed Curves in Conservative Vector Fields} +$$ + \forall\ \gamma: [a,a] \to \R:\quad \int_\gamma f(s) \cdot ds = 0 +$$ +\subtext{This is actually equivalent to $f$ being conservative.} + +\begin{subbox}{The Potential exists in Conservative Vector Fields} + \smalltext{$X \subset \R^n \text{ open},\quad f \text{ conservative}$} + \begin{align*} + \exists g \in C^1:\quad f = \nabla g + \end{align*} + \smalltext{If $x_1,x_2 \in X$ are joined by a $\gamma$, $g$ is unique up to $C \in \R$} + \begin{align*} + \nabla g_1 = f \implies g - g_1 \text{ is constant on } X + \end{align*} +\end{subbox} + +\definition \textbf{Path-Connected Set}\\ +$\forall x_1,x_2 \in X: \exists \gamma: [a,b] \to X$ s.t. $\gamma(a) = x_1, \gamma(b) = x_2$ + +\newpage + +\lemma \textbf{Property of Conservative Vector Fields}\\ +\smalltext{Easy way to e.g. disprove $f$ being conservative:} +$$ + \forall 1 \leq i \neq j \leq n:\quad \frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i} +$$ +\subtext{$X \subset \R^n \text{ open},\quad f: X\to\R^n,\quad f \in C^1,\quad f \text{ conserv.}$}\\ +\subtext{Only this was: This being true does not imply $f$ is conservative!} + +\definition \textbf{Star Shaped Set}\\ +$\exists x_0 \in X: \forall x \in X$ Line seg. $x_0 \to x$ is in $X$ + +\definition \textbf{Convex Set}\\ +$\forall x_1,x_2 \in X:$ Line seg. $x_1 \to x_2$ is in $X$\\ +\subtext{Convex implies star shaped.} + +\theorem \textbf{Some Star Shaped Sets are conservative}\\ +\smalltext{In open star-shaped sets $X \subset \R^n$:} \subtext{$f \in C^1$} +$$ + \forall 1 \leq i \neq j \leq n: \frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i} \implies f \text{ conservative} +$$ + +\definition $\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}$ \subtext{$f: X \to \R^3,\quad f \in C^1$} + +\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}$ + +\newpage +\subsection{The Riemann Integral in $\R^n$} + +\smalltext{For $f: X \to \R$ ($X \subset \R^n$ bounded \& closed), $\displaystyle\int_X f(x)\ dx$ fulfills:} + +\begin{enumerate} + \item \textbf{Composability}\\ + \smalltext{$\displaystyle\int_X f(x)\ dx = \int_a^b f(x)\ dx$} \subtext{$n=1, X=[a,b]$} + \item \textbf{Linearity}\\ + \smalltext{$\displaystyle\int_X \Bigl( af_1(x) + bf_2(x) \Bigr)\ dx = a \int_X f_1(x)\ dx + b \int_X f_2(x)\ dx$}\\ + \subtext{$f,g$ cont. on $X$, $a,b \in \R$} + \item \textbf{Positivity}\\ + \smalltext{$f \leq g \implies \displaystyle\int_X f(x)\ dx \leq \int_X g(x)\ dx$} + \item \textbf{Upper Bound}\\ + \smalltext{$\left\lvert \displaystyle\int_X f(x)\ dx \right\rvert \leq \int_X |f(x)|\ dx$} + \item \textbf{Triangle Inequality}\\ + \smalltext{$\left\lvert \displaystyle\int_X \Bigl( f(x) + g(x) \Bigr)\ dx \right\rvert \leq \displaystyle\int_X |f(x)|\ dx + \displaystyle\int_X |g(x)|\ dx$} + \item \textbf{Volume}\\ + \smalltext{$\displaystyle\int_X f(x)\ dx}$ is the volume of $\Bigl\{ (x,y) \in X \times \R \ \Big\vert\ 0 \leq y \leq f(x) \Bigr\}$\\ + \subtext{So the intuitive idea of $\int_a^b f(x)\ dx$ being the area carries over.} + \item \textbf{Domain Additivity}\\ + \smalltext{$\displaystyle\int_{X_1 \cup X_2} f(x)\ dx + \int_{X_1 \cap X_2} f(x)\ dx = \int_{X_1} f(x)\ dx + \int_{X_2} f(x)\ dx$}\\ + \subtext{If $X_1,X_2$ are compact, $f$ is cont. on $X_1 \cup X_2$} +\end{enumerate} + +\begin{subbox}{Fubini's Theorem: Multiple Integrals} + \smalltext{$f: X \to \R,\quad n = n_1 + n_2,\quad n_1,n_2 \geq 1$} + \begin{align*} + & X_{x_1} &:=\quad& \Bigl\{ x_2 \in \R^{n_2} \ \Big\vert\ (x_1,x_2) \in X \Bigr\} \subset \R^{n_2} \\ + & X_{1} &:=\quad& \Bigl\{ x_1 \in \R^{n_1} \ \Big\vert\ X_{x_1} \neq \emptyset \Bigr\} \subset \R^{n_1} + \end{align*} + If $g(x_1) := \displaystyle\int_{X_{x_1}}f\Bigl( (x_1,x_2) \Bigr)\ dx_2$ is continuous on $X_1$: + $$ + \int_X f(x)\ dx = \int_{X_1}\Biggl( \int_{X_{x_1}} f\Bigl( (x_1,x_2) \Bigr)\ dx_2 \Biggr)\ dx_1 + $$ + \smalltext{The role of $x_1,x_2$ can be swapped, if $f$ is continuous.} +\end{subbox} +\newpage \ No newline at end of file