[Analysis] Update various sections

Improve legibility, add some more remarks

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/01_partial_derivatives.tex

@@ -2,9 +2,12 @@
 \subsection{Partial derivatives}
 \shortdef $X \subseteq \R^n$ \bi{open} if for any $x = (x_1, \ldots, x_n) \in X$ $\exists \delta > 0$ s.t.
 $\{ y = (y_1, \ldots, y_n) \in \R^n : |x_i - y_i| < \delta \smallhspace \forall i \}$ is contained in $X$.
-(= changing a coordinate of $x$ by $< \delta \rightarrow x' \in X$)
+(= changing a coordinate of $x$ by $< \delta \rightarrow x' \in X$)\\
-\shortproposition $X \subseteq \R^n$ open $\Leftrightarrow$ \bi{complement} $Y = \{ x \in \R^n : x \notin X \}$ is closed
+%
-\shortcorollary If $f: \R^n \rightarrow \R^m$ cont. and $Y \subseteq \R^m$ open, then $f^{-1}(Y)$ is open in $\R^n$
+\shortproposition $X \subseteq \R^n$ open $\Leftrightarrow$ \bi{complement} $Y = \{ x \in \R^n : x \notin X \}$ is closed\\
+%
+\shortcorollary If $f: \R^n \rightarrow \R^m$ cont. and $Y \subseteq \R^m$ open, then $f^{-1}(Y)$ is open in $\R^n$\\
+%
 \shortex \bi{(1)} $\emptyset$ and $\R^n$ are both open and closed.
 \bi{(2)} Open ball $D = \{ x \in \R^n : ||x - x_0|| < r \}$ is open in $\R^n$ ($x_0$ the center and $r$ radius)
 \bi{(3)} $I_1 \times \dots \times I_n$ is open in $\R^n$ for $I_i$ open

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex

@@ -47,12 +47,12 @@ The Jacobi matrix is $J_{g \circ f}(x_0) = J_g(f(x_0)) J_f(x_0)$ (RHS is a matri
 % ────────────────────────────────────────────────────────────────────
 \stepLabelNumber{all}
 \compactdef{Directional derivative} $f$ has a directional derivative $w \in \R^m$ in the direction of $v \in \R^n$,
-if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$
+if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$\\
 % ────────────────────────────────────────────────────────────────────
-\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.
+\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$.\\
 % ────────────────────────────────────────────────────────────────────
 \shortproposition Let $f$ as previously be differentiable. Then for any $x \in X$ and non-zero $v \in \R^n$,
-$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$
+$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$\\
 % ────────────────────────────────────────────────────────────────────
 \shortremark The values of the above directional derivative are linear with respect to the vector $v$.
 Suppose we know the dir. der. $w_1$ and $w_2$ in directions $v_1$ and $v_2$, then the directional derivative in direction $v_1 + v_2$ is $w_1 + w_2$

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/06_critical_points.tex

@@ -1,7 +1,7 @@
 \newsectionNoPB
 \subsection{Critical points}
 \stepLabelNumber{all}
-\compactdef{Critical Point} For $f: X \rightarrow \R^n$ differentiable, $x_0 \in X$ is called a \bi{critical point} of $f$ if $\nabla f(x_0) = 0$
+\compactdef{Critical Point} For $f: X \rightarrow \R^n$ differentiable, $x_0 \in X$ is called a \bi{critical point} of $f$ if $\nabla f(x_0) = 0$\\
 \shortremark As in 1 dimensional case, check edges of the interval for the critical point.\\
 %
 To determine the kind of critical point, we need to determine if $H_f(x_0)$ is definite:

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

@@ -16,7 +16,8 @@
 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.
-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 to using a line integral, see section \ref{sec:green-formula} for a faster way
 
 \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,
@@ -79,5 +80,3 @@ Below a chart to figure out some properties:
     \end{tikzpicture}
 \end{center}
 \dnrmvspace
-% TODO: Some tips and tricks
-% With the line integral, we can compute the length of the curve, as defined by the function.

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

@@ -1,5 +1,6 @@
 \newsection
 \subsection{The Green Formula}
+\label{sec:green-formula}
 \compactdef{Simple parametrized curve} $\gamma : [a, b] \rightarrow \R^2$ is a closed parametrized curve s.t.
 $\gamma(t) \neq \gamma(s)$ (if $s \neq t$ and $\{ s, t \} = \{ a, b \}$), s.t. $\gamma'(t) \neq 0$ for $a < t < b$.
 If $\gamma$ only piecewise in $C^1$ in $]a, b[$, then only apply when $\gamma'(t)$ exists.