[Analysis] Various fixes and optimizations

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}$.