From 19ce8d3af8dc0811d3c2ab59f144215b3f1bef74 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sun, 1 Feb 2026 15:31:56 +0100 Subject: [PATCH] [Analysis] Various fixes and optimizations --- .../analysis-ii-cheat-sheet.pdf | Bin 353618 -> 354990 bytes .../linear-ode/02_constant-coefficient.tex | 13 ++++++++++--- .../vectors/integration/00_line_integrals.tex | 3 ++- .../vectors/integration/04_green_formula.tex | 4 ++++ 4 files changed, 16 insertions(+), 4 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 fcd395d46686e59f120ad3316fe9394a31d93f91..512f14b98949ce3bbb0a607101b17b63d38ec7b9 100644 GIT binary patch delta 68307 zcmV)CK*GP$$`!8F6|i^~1UE1>Hk0u%DSusCa~rpke)q3n_KB)vP6pS(Bz3CR$8oYg zl}+N<`;?rAkvNjZrbwBjZ0r2`?Z$0@A;2Le+DT=%av=p`aA`CeUpKmY%E{HFoc!>l zw4eWY^W^LAIiC<-FfQrj%_V*%6`Lrc3P#D~&3y9Pl$^~7pYPnC_nU2dvoU|HF3o5A{;t_}tEDaO 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b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex index 1a8aac6..a3990b6 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex @@ -16,11 +16,18 @@ The homogeneous equation will then be all the elements of the set summed up.\\ \shade{gray}{Inhomogeneous Equation}\rmvspace \begin{enumerate}[noitemsep] \item \bi{(Case 1)} $b(x) = c x^d e^{\alpha x}$, with special cases $x^d$ and $e^{\alpha x}$: - $f_p = Q(x) e^{\alpha x}$ with $Q$ a polynomial with $\deg(Q) \leq j + d$, where $j$ is multiplicity of root $\alpha$ (if $P(\alpha) \neq 0$, then $j = 0$) of characteristic polynomial + $f_p = Q(x) e^{\alpha x}$ with $Q$ a polynomial with $\deg(Q) \leq j + d$, + where $j$ is multiplicity of root $\alpha$ (if $P(\alpha) \neq 0$, then $j = 0$) of characteristic polynomial \item \bi{(Case 2)} $b(x) = c x^d \cos(\alpha x)$, or $b(x) = c x^d \sin(\alpha x)$: - $f_p = Q_1(x) \cdot \cos(\alpha x) + Q_2(x9 \cdot \sin(\alpha x))$, - where $Q_i(x)$ a polynomial with $\deg(Q_i) \leq d + j$, where $j$ is the multiplicity of root $\alpha i$ (if $P(\alpha i) \neq 0$, then $j = 0$) of characteristic polynomial + $f_p = Q_1(x) \cdot \cos(\alpha x) + Q_2(x) \cdot \sin(\alpha x))$, + where $Q_i(x)$ a polynomial with $\deg(Q_i) \leq d + j$, + where $j$ is the multiplicity of root $\alpha i$ (if $P(\alpha i) \neq 0$, then $j = 0$) of characteristic polynomial + \item \bi{(Case 3)} $b(x) = c e^{\alpha x} \cos(\beta x)$, or $b(x) = c e^{\alpha x} \sin(\beta x)$, use the Ansatz + $Q_1(x) e^{\alpha x} \sin(\beta x) + Q_2(x) e^{\alpha x} \cos(\beta x)$, agasin with the same polynomial. + Often, it is sufficent to have a polynomial of degree 0 (i.e. constant) \end{enumerate} +For inhomogeneous parts with addition or subtraction, the above cases can be combined. +For any cases not covered, start with the same form as the inhomogeneous part has (for trigonometric functions, duplicate it with both $\sin$ and $\cos$). \rmvspace\shade{gray}{Other methods}\rmvspace \begin{itemize}[noitemsep] 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 0662e28..584d61a 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 @@ -16,9 +16,10 @@ 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! +\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. +\bi{For computing}, we usually use the first integral in def \ref{all:4-1-1} (3). \setLabelNumber{all}{4} \compactdef{Oriented reparametrization} of $\gamma$ is parametrized curve $\sigma : [c, d] \rightarrow \R^n$ s.t $\sigma = \gamma \circ \varphi$, with $\varphi : [c, d] \rightarrow I$ cont. map, 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 4f167fc..7921801 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 @@ -61,3 +61,7 @@ The center of mass of an object $\cU$ is given by $\displaystyle \overline{x}_i \rmvspace \shade{gray}{Dot product} For vectors $v, w \in \R^n$, we have $\displaystyle v \cdot w = \sum_{i = 1}^{n} v_i \cdot w_i$ + +\rmvspace +\shade{gray}{Matrix-Vector product} Given vector $v\in \R^m$ and matrix $A \in \R^{n \times m}$, +we have $A \cdot v = u$ where $ \displaystyle u_j = \sum_{i = 1}^{m} v_i \cdot A_{j, i}$.