[Analysis] Clean up, add some notes

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex

@@ -26,7 +26,7 @@ The homogeneous equation will then be all the elements of the set summed up.\\
 \begin{itemize}[noitemsep]
     \item \bi{Change of variable} Apply substitution method here, substituting for example for $y' = f(ax + by + c)$ $u = ax + by$ to make the integral simpler.
           Mostly intuition-based (as is the case with integration by substitution)
-    \item \bi{Separation of variables} For equations of form $y' = a(y) \cdot b(x)$ (NOTE: Not linear),
+    \item \bi{Separation of variables} For equations of form $y' = a(y) \cdot b(x)$ (Note: Not linear),
           we transform into $\frac{y'}{a(y)} = b(x)$ and then integrate by substituting $y'(x) dx = dy$, changing the variable of integration.
           Solution: $A(y) = B(x) + c$, with $A = \int \frac{1}{a}$ and $B(x) = \int b(x)$.
           To get final solution, solve for the above equation for $y$.

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex

@@ -11,7 +11,6 @@ $(x_k)$ converges to $y$ as $k \rightarrow +\infty$ if $\forall \varepsilon > 0
 \bi{(1)} Let $x_0 \in X$. $f$ continuous in $\R^n$ if $\forall \varepsilon > 0 \smallhspace \exists \delta > 0$ s.t. if $x \in X$ satisfies $||x - x_0|| < \delta$,
 then $||f(x) - f(x_0)|| < \varepsilon$
 \bi{(2)} $f$ continuous \textit{on} $X$ if continuous at $x_0 \smallhspace \forall x_0 \in X$
-% TODO: Add tricks from TA slides here (week 05 / 04)
 % ────────────────────────────────────────────────────────────────────
 \shortproposition Let $X$ and $f$ as prev. Let $x_0 \in X$. $f$ continuous at $x_0$ iff $\forall (x_k)_{k \geq 1}$ in $X$ s.t.
 $x_k \rightarrow x_0$ as $k \rightarrow +\infty$, $(f(x_k))_{k \geq 1}$ in $\R^m$ converges to $f(x)$\\

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/04_change_of_variable.tex

@@ -2,4 +2,3 @@
 \subsection{Change of variable}
 The idea is to substitute variables for others that make the equation easier to solve.
 A common example is to switch to polar coordinates from cartesian coordinates, as already demonstrated with continuity checks
-% TODO: Add notes from TA notes

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/00_line_integrals.tex

@@ -17,7 +17,8 @@ We usually call $f : X \rightarrow \R^n$ a \bi{vector field}, which maps each po
 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.
 \hl{Be careful with hat functions} like $|x|$, we need two separate integrals for each side of the center!
-Alternatively to using a line integral, see section \ref{sec:green-formula} for a faster way
+Alternatively, see section \ref{sec:green-formula} for a faster way.
+For calculating the area enclosed by the curve, see there too.
 
 \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,
@@ -63,6 +64,7 @@ $\text{curl}(f) = \begin{bmatrix}
     \end{bmatrix}$
 \dnrmvspace
 
+If $\text{curl}(f) = 0$, then $f$ is irrational.
 Below a chart to figure out some properties:
 \begin{center}
     \begin{tikzpicture}[node distance = 0.5cm and 0.5cm, >={Classical TikZ Rightarrow[width=7pt]}]

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex

@@ -53,5 +53,9 @@ That set is derived from the image that is given for the line.
 Be cognizant of what direction the integral goes, if the set is on the right hand side of the curve, the final result has to be negated to change the direction of the integral.
 If the curve doesn't fully enclose the set, then we can simply compute the line integrals of the missing sections and subtract them from the final result.
 
+We can also use known formulas to compute the area of discs, etc (like $r^2 * \pi$ for a circle).
+To calculate the area enclosed by a curve using Green's formua, we can use the vector field
+% TODO: Finish
+
 \shade{gray}{Center of mass}
 The center of mass of an object $\cU$ is given by $\displaystyle \overline{x}_i = \frac{1}{\text{Vol}(\cU)} \int_{\cU} x_i \dx x$.