From fe01067f512153f0432fcd879bf058dc743fc615 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 23 Jan 2026 13:07:46 +0100 Subject: [PATCH] [Analysis] Add notes --- .../analysis-ii-cheat-sheet.pdf | Bin 345535 -> 346660 bytes .../vectors/integration/01_int_in_rn.tex | 5 ++++- .../vectors/integration/04_green_formula.tex | 2 +- 3 files changed, 5 insertions(+), 2 deletions(-) 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 f760283d449fac0a5776dce77f2f724d563c5c2a..2a4034d45abb5e4c144c4b0916d0e2c2e17bcc31 100644 GIT binary patch delta 50593 zcmV(=K-s^)ixs4l6|h4J0ba923QGY5`)~O-vy%&*rvfxJmodTwDSzEPYjYd9k>C9* z=zPhjVovZLz{ytKuCtHqU8Qb)&Sev?uZof=iEE0~lGNJHufJ|QW`H3u98$8KN|h~= z00z(QM!$zlt|w&j?GxfZDf}f9I-#Pb4WDTER7_7UZlC-zfkxj>Y|AI4=0?lOy9pL! zO|2&QpH4PV`}@1=tAGEvn9&;AZYNJ@Eub$t5jB(4=*ewYWNJY;7n5aIL`iLw)@H(L zrIp|l%4)_NW7_B2f{V!$L2E;ll3gKmC3r0w#%T1Ys2f2!>w0fX!W6KxP!>j7MrC0- 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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 acf35c8..a69ec6e 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 @@ -67,6 +67,9 @@ Image of par. curve $\gamma: [a, b] \rightarrow \R^n$ is negligible, since $\gam \rmvspace \bi{How to compute the integral:} We compute each integral "inside out". For a definite integral, don't just find the anti-derivative, compute the actual integral! -For an integral as seen in the harder example, we compute it as we normally would, simply using the $\pm 2x$ as the $a$ and $b$ +For an integral as seen in the harder example, we compute it as we normally would, simply using the $\pm 2x$ as the $a$ and $b$. + +Using a change of variables into polar coordinates may come in handy, e.g. for a set like $\{ (x, y) \in \R^2 \divides 1 \leq x^2 + y^2 \leq 4 \}$, +we can use polar coordinates and the integral is then $\int_{0}^{2\pi} \int_{1}^{2} f(x, y) \dx r \dx \varphi$ (or flipped of course) \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 7e65f8c..01d839f 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 @@ -53,4 +53,4 @@ Be cognizant of what direction the integral goes, if the set is on the right han If the curve doesn't fully enclose the set, then we can simply compute the line integrals of the missing sections and subtract them from the final result. \shade{gray}{Center of mass} -% TODO: Finish the notes here +The center of mass of an object $\cU$ is given by $\displaystyle \overline{x}_i = \frac{1}{\text{Vol}(\cU)} \int_{\cU} x_i \dx x$.