From 09b635ebf0ebe6860bb865d4147fe76feb197bd8 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 1 Feb 2026 11:32:46 +0100 Subject: [PATCH] [Analysis] Various additions and fixes --- .../analysis-ii-cheat-sheet.pdf | Bin 352579 -> 353618 bytes .../cheat-sheet-jh/parts/00_intro.tex | 2 ++ .../01_partial_derivatives.tex | 3 ++- .../differentiation/06_critical_points.tex | 9 ++++++++- .../vectors/integration/00_line_integrals.tex | 6 +++--- .../vectors/integration/04_green_formula.tex | 3 +++ 6 files changed, 18 insertions(+), 5 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 ac6529a4f651537f056b1dcaea0b5f59d61a797b..fcd395d46686e59f120ad3316fe9394a31d93f91 100644 GIT binary patch delta 47032 zcmV)BK*PVo!WGiW6|i|00XCDt3oCzJTXP$?l782(VD^csV@?Klj8ms-eHL zd!Lf?K#?P9Y>Jdg%C`2`Z#OP8z!2b&678hCTe*+|F}O4u-CsAlG3DrLRE~Z;E8XWm 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The Analysis I cheat sheet has a derivatives and anti-derivatives table. + +Do note that a function like $e^{ax}$ is bounded as $x \rightarrow +\infty$ if $a \leq 0$ (exponent becomes smaller!) diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex index 19eb1a1..b8b5219 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex @@ -45,4 +45,5 @@ and the \drmvspace\rmvspace trace of the Jacobi Matrix, $\text{div}(f)(x_0) = \text{Tr}(J_f(x_0)) = \sum_{i = 1}^{n} \partial_{x_i} f_i(x_0)$ is called the \bi{divergence} of $f$ at $x_0$.\\ The gradient is simply the transpose of the Jacobian and it points in the direction of the \bi{steepest ascent}. -\rmvspace + +Do note that for functions $g : \R \rightarrow \R^n$, the derivative is taken component-wise! diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex index 22a81ab..24d8e87 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex @@ -49,5 +49,12 @@ For $2 \times 2$ matrices (i.e. 2D functions), we can use the following scheme ( (tr1) edge node [right] {$0$} (zero); \end{tikzpicture} \end{center} -As in Analysis I, it is important to also check the boundaries for maximums and minimums. +As in Analysis I, it is important to also check the boundaries for maximums and minimums (as it may also be possible that there are NO critical points in the set). For that, formulate formulas for the borders and check them for critical points. + +This is mostly intuition, but think of what segments the set consists of and note them down. +Then, for each of the sets of the segments, determine the critical points +(e.g. for set $A = \{ (x, y) \in \R^2 \divides x = 0, 0 \leq y \leq 3 \}$, we compute the critical points of $f(0, y)$) + +For cases where $x$ and $y$ are both not $0$, we have to parametrize the set +(e.g. for set $C = \{ (x, y) \in \R^2 \divides 3x + y = 3, 0 \leq x \leq 1 \}$, we have $\gamma(t) = (t, 3 - 3t)$ and compute the critical points of $f(\gamma(t))$) 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 706bf52..0662e28 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 @@ -13,9 +13,9 @@ \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 $x$\\ -Often, we use $V$ instead of $f$ to denote the vector field. -Ideally, to compute a line integral, we compute the derivative of $\gamma$ and $V(\gamma(t))$ separately, then simply do the integral after. +We usually call $f : X \rightarrow \R^n$ (or sometimes $V$ a \bi{vector field}, which maps each point $x \in X$ to a vector in $\R^n$, displayed as originating from $x$. +Ideally, to compute a line integral, we compute the derivative of $\gamma$ separately ($\gamma(t) = s$ usually, derive component-wise), +limits of integration are start and end of section. \hl{Be careful with hat functions} like $|x|$, we need two separate integrals for each side of the center! Alternatively, see section \ref{sec:green-formula} for a faster way. For calculating the area enclosed by the curve, see there too. 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 152c79e..4f167fc 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 @@ -58,3 +58,6 @@ To calculate the area enclosed by a curve using Green's formua, if not given a v \shade{gray}{Center of mass} The center of mass of an object $\cU$ is given by $\displaystyle \overline{x}_i = \frac{1}{\text{Vol}(\cU)} \int_{\cU} x_i \dx x$. + +\rmvspace +\shade{gray}{Dot product} For vectors $v, w \in \R^n$, we have $\displaystyle v \cdot w = \sum_{i = 1}^{n} v_i \cdot w_i$