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zp+(F?cIDPTtHzJ7+m-0jIii%S$#H^Z{N^(DpCA1S20mC184Hs?Rl;rRwBLeqMlbUev2R#TR))4tchw5`1~{Y~`sFO4nl3r;~YT z<%wAPiKv;0mYQ;j-;odMFQ#hsF<$^1o%N>f_N+}@|p^G F{{i&*SgZg5 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 90bda36..3ea5cbc 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/06_int.tex @@ -146,4 +146,136 @@ $$ $$ \smalltext{The role of $x_1,x_2$ can be swapped, if $f$ is continuous.} \end{subbox} -\newpage \ No newline at end of file +\newpage + +\definition \textbf{Parametrized $m$-Set in $\R^n$}\\ +$f: [a_1,b_1]\times\cdots\times[a_m,b_m] \to \R$\\ +s.t. $f \in C^1$ on $(a_1,b_1)\times\cdots\times(a_m,b_m)$\\ +\subtext{A param. $1$-set in $\R^n$ is just a param. curve} + +\definition \textbf{Negligible Subset}\\ +$B \subset \R^n$ s.t. $\exists k \geq 0$ param. $m_i$-sets: $f_i:X_i\to\R^n$ s.t. +$$ + B \subset f_1(X_1) \cup \cdots \cup f_k(X_k) +$$ +\subtext{$1 \leq i \leq k,\quad m_i < n$}\\ + +\begin{footnotesize} + \remark For an affine subspace $H \subset \R^n$ with $\dim(H) < n$, any $X \subset \R^n$ contained in $H$ is Negligible + + \remark The image of a param. curve $\gamma: [a,b] \to \R^n$ is negligible.\\ + \color{gray} $\gamma$ is a $1$-set in $\R^n$ \color{black} +\end{footnotesize} + +\lemma \textbf{Integral of Negligible Sets}\\ +\smalltext{For continuous $f: X \subset \R^n \to \R$}: +$$ + X \text{ negligible } \implies \int_X f(x)\ dx = 0 +$$ + +\subsection{Improper Integrals} +\smalltext{$I \subset \R$ bounded,$\quad J = [a,+\infty]$ for $a \in \R,\quad f$ cont. on $X = J \times I$} +$$ + \underset{x \to \infty}{\lim}\int_{[a,x]\times I}f(x,y)\ dxdy = \underbrace{\int_a^\infty \Biggl( \int_I f(x,y)\ dy \Biggr) dx}_\text{Order of Integration may change} +$$ +If this Limit is equal for both orders of Integration: + +\definition \textbf{Improper Integral in $\R^2$} +$$ + \int_{J \times I} f(x,y)\ dxdy := \underset{x \to \infty}{\lim}\int_{[a,x]\times I}f(x,y)\ dxdy +$$ + +\definition \textbf{Integral over $\R^2$} +$$ + \int_{\R^2} f(x,y)\ dxdy := \underset{R \to \infty}{\lim}\int_{[-R,R]^2} f(x,y)\ dxdy +$$ + +\begin{footnotesize} + \remark if $|f| \leq g$, and an impr. Integr. exists on g, it exists on $f$. +\end{footnotesize} + +\newpage +\subsection{Change of Variable} +\smalltext{This is to provide an Analogue of the Change of Variable in $\R$} +$$ + \int f\Bigl( g(x) \Bigr) g'(x)\ dx = \int f(y)\ dy +$$ + +\textbf{Prerequisites} + +\begin{footnotesize} + $\bar{X},\bar{Y} \subset \R^n$ compact, $\varphi: \bar{X} \to \bar{Y}$ cont.\\ + We have: $\bar{X} = X \cup B,\quad \bar{Y} = Y \cup C$ s.t. + \begin{enumerate} + \item $X,Y$ are open + \item $B,C$ are negligible + \item $\varphi$ on $X$ is a $C^1$ map $\varphi: X \to Y$ + \end{enumerate} +\end{footnotesize} + +\begin{subbox}{Change of Variable in $\R^n$} + \smalltext{$\bar{X},\bar{Y}$ as above, $\quad f$ cont. on $\bar{Y}$ arbitrary} + $$ + \int_{\bar{X}} f\Bigr( \varphi(x) \Bigl) \cdot \left\vert \det(\textbf{J}_\varphi(x)) \right\vert\ dx = \int_{\bar{Y}} f(y)\ dy + $$ +\end{subbox} + +\begin{footnotesize} + \remark Translations: $\varphi(x) = x + x_0$ have $\textbf{J}_\varphi(x) = \textbf{I}_n$\\ + so the volume is preserved: $\displaystyle\int_{\bar{X}}f(x+x_0)\ dx = \int_{x_0 + \bar{X}} f(x)\ dx$ + + \remark Linear maps: $\varphi(x) = \textbf{A}x$ have $\textbf{J}_\varphi(x) = \textbf{A}$\\ + The change of variable is: $\displaystyle\int_{\bar{X}} f\Bigl( \varphi(x) \Bigr)\ dx = \frac{1}{\left\vert \det(\textbf{A}) \right\vert} \int_{\bar{Y}}f(y)\ dy$ +\end{footnotesize} + +\remark \textbf{Commonly used Changes} +\begin{enumerate} + \item Polar Coordinates\\ + \smalltext{ + $\varphi(r, \theta) = \bigl(r\cos(\theta),\ r\sin(\theta)\bigr)$\\ + Where $dxdy = r\ dr\ d\theta$ + } + + \item Cylindrical Coordinates\\ + \smalltext{ + $\varphi(r, \theta, z) = \bigl( r\cos(\theta),\ rsin(\theta),\ z \bigr)$\\ + Where $dxdydz = r\ dr\ d\theta\ dz$ + } + + \item Spherical Coordinates\\ + \smalltext{ + $\varphi(r, \theta, \phi) = (r\sin(\phi)\cos(\theta),\ r\sin(\phi)\sin(\theta),\ r\cos(\phi))$\\ + Where $dxdydz = r^2\ \sin(\phi)\ dr\ d\theta\ d\phi$ + } +\end{enumerate} + +\newpage +\subsection{Green's Theorem} +\smalltext{An analogue of the Fundamental Theorem of Calculus in $\R^2$.} + +\definition \textbf{Simple Closed Parametrized Curve}\\ +$\gamma: [a,b] \to \R^2$ closed param. curve s.t. +\begin{enumerate} + \item $\gamma(t) \neq \gamma(s)\quad$ unless $s = t$, or $\{s,t\} = \{a,b\}$ + \item $\gamma'(t) \neq 0\quad \forall a < t < b$ +\end{enumerate} +\subtext{Example: $\varphi(t) = \Bigl( x_0 + r\cos(t),\ y_0 + r\sin(t) \Bigr)$\\ +(A circle, traversed \textit{once}, i.e. for $0 \leq t \leq 2\pi$)} + +\begin{subbox}{Green's Theorem} + \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 $\gamma_i(t)$} + $$ + \int_X \Biggl( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \Biggr)\ dxdy = \sum_{i=1}^{k} \int_{\gamma_i} f \cdot ds + $$ + \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.} + +\lemma \textbf{Volume using Green} +$$ + \text{Vol}(X) = \sum_{i=1}^{k} \int_{\gamma_i} x \cdot ds = \sum_{i=1}^{k} \int_{a_i}^{b_i} \gamma_{i,1}(t)\cdot \gamma_{i,2}'(t)\ dt +$$ +\subtext{Same assumptions as above.} + +\remark The \textit{Gauss-Ostrogradski} Formula exists for $\R^3$. \ No newline at end of file