[Analysis] Various additions and fixes

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.

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 $\displaystyle v \cdot w = \sum_{i = 1}^{n} v_i \cdot w_i$