[Analysis] Update to new helper import scheme, continue differentiability summary

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

@@ -8,17 +8,36 @@ $\{ y = (y_1, \ldots, y_n) \in \R^n : |x_i - y_i| < \delta \smallhspace \forall
 \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
-\bi{(4)} $X \subseteq \R^n$ open $\Leftrightarrow$ $\forall x \in X \exists \delta > 0$ s.t. open ball of center $x$ and radius $\delta$ is contained in $X$
-
+\bi{(4)} $X \subseteq \R^n$ open $\Leftrightarrow$ $\forall x \in X \exists \delta > 0$ s.t. open ball of center $x$ and radius $\delta$ is contained in $X$\\
+% ────────────────────────────────────────────────────────────────────
 \compactdef{Partial derivative} Let $X \subseteq \R^n$ open, $f: X \rightarrow \R^m$ and $1 \leq i \leq n$.
 Then $f$ has partial derivative on $X$ with respect to the $i$-th variable (or coordinate),
 if $\forall x_0 = (x_{0, 1}, \ldots, x_{0, n}) \in X$, $g(t) = f(x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, x_{0, n})$ on set
 $I = \{ t \in \R : (x_{0, 1}, \ldots, x_{0, i - 1}, t, x_{0, i + 1}, \ldots, x_{0, n}) \in X \}$ is differentiable at $t = x_{0, i}$.
 The derivative $g'(x_{0, i})$ at $x_{0, i}$ is denoted:
 $\frac{\partial f}{\partial x_i}(x_0), \partial_{x_i} f(x_0) \text{ or } \partial_i f(x_0)$\\
-%
+% ────────────────────────────────────────────────────────────────────
+\stepLabelNumber{all}
 \shortproposition Let $X \subseteq \R^n$ open, $f, g : X \rightarrow \R^m$ and $1 \leq i \leq n$. Then:
 \bi{(1)} If $f$ \& $g$ have $\partial_i$ on $X$, then so does $f + g$ and $\partial_{x_i} (f + g) = \partial_{x_i}(f) + \partial_{x_i}(g)$
 \bi{(2)} If $m = 1$ (i.e. $\R^1$) and $f$ \& $g$ have $\partial_i$ on $X$, then so does $fg$ and $\partial_{x_i} (fg) = \partial_{x_i}(f)g + f \partial_{x_i}(g)$
 and if $g(x) \neq 0 \smallhspace \forall x \in X$, then if $f \div g$ has $\partial_i$ on $X$, then so does $f \div g$ and
-$\partial_{x_i}(f \div g) = (\partial_{x_i}(f) g - f \partial_{x_i}(g)) \div g^2$
+$\partial_{x_i}(f \div g) = (\partial_{x_i}(f) g - f \partial_{x_i}(g)) \div g^2$\\
+% ────────────────────────────────────────────────────────────────────
+\compactdef{Jacobi Matrix $J$} Element $J_ij = \partial_{x_j} f_i(x)$ for function $f: X \rightarrow \R^m$ with $X \subseteq \R^n$ open. $x_j$ is the $j$-th variable,
+$f_i$ is the $i$-th component of the equation (i.e. in the vector of the function). $J$ has $m$ rows and $n$ columns.\\
+% ────────────────────────────────────────────────────────────────────
+
+\drmvspace\drmvspace
+\stepLabelNumber{all}
+\compactdef{Gradient, Divergence} for $f : X \rightarrow \R$ with $X \in \R^n$ open, the \bi{gradient} is given by
+$\nabla f(x_0) = \begin{pmatrix}
+        \partial_{x_1} f(x_0) \\
+        \vdots                \\
+        \partial_{x_n} f(x_0)
+    \end{pmatrix}$
+and the
+
+\drmvspace\rmvspace
+trace of the Jacobi Matrix, $\text{div}(f)(x_0) = \text{Tr}(J_f(x_0)) = \sum_{i = 1}^{n} \partial_{x_i} f_i(x_0)$ is called the \bi{divergence} of $f$ at $x_0$.
+\rmvspace

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

@@ -0,0 +1,14 @@
+\newsectionNoPB
+\subsection{The differential}
+\setLabelNumber{all}{2}
+\compactdef{Differentiable function} We have function $f: X \rightarrow \R^m$, linear map $u : \R^n \rightarrow \R^m$ and $x_0 \in X$. $f$ is differentiable at $x_0$ with differential $u$ if
+$\displaystyle \lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} \frac{1}{||x - x_0||} (f(x) - f(x_0) - u(x - x_0) = 0$ where the limit is in $\R^m$.
+We denote $\dx f(x_0) = u$.
+If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$
+
+% ────────────────────────────────────────────────────────────────────
+\stepLabelNumber{all}
+\shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open
+\begin{itemize}[noitemsep]
+    \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$
+\end{itemize}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/03_higher_diff.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Higher derivatives}

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

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Change of variable}

semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/05_taylor_polynomials.tex

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Taylor polynomials}

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

@@ -0,0 +1,2 @@
+\newsectionNoPB
+\subsection{Critical points}