diff --git a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf index cc10e69..c0d51ab 100644 Binary files a/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf and b/semester3/analysis-ii/cheat-sheet-jh/analysis-ii-cheat-sheet.pdf differ diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex index 2b76926..ff5d1ce 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex +++ b/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 diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex index 8c1131e..a3ba31f 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/01_int_in_rn.tex +++ b/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 $X$, $\displaystyle\int_{X} f(x) \dx x = 0$ diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex index ad540dd..bd4c7fe 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/02_improper_int.tex +++ b/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 diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex index 59d1059..8e3f59d 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/03_change_of_variable_formula.tex +++ b/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 diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex index 39afce6..4a338e1 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex +++ b/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*} diff --git a/semester3/numcs/numcs-summary.pdf b/semester3/numcs/numcs-summary.pdf index 24abcbb..adfdf8c 100644 Binary files a/semester3/numcs/numcs-summary.pdf and b/semester3/numcs/numcs-summary.pdf differ