[Analysis] Various fixes and optimizations

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]

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,

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}$.