[Analysis] Various additions and fixes

semester3/analysis-ii/cheat-sheet-jh/parts/00_intro.tex

@@ -1,3 +1,5 @@
 \section{General tips}
 Use systems of equations if given some points, or other optimization techniques.
 The Analysis I cheat sheet has a derivatives and anti-derivatives table.
+
+Do note that a function like $e^{ax}$ is bounded as $x \rightarrow +\infty$ if $a \leq 0$ (exponent becomes smaller!)

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

@@ -45,4 +45,5 @@ 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 $x_0$.\\
 The gradient is simply the transpose of the Jacobian and it points in the direction of the \bi{steepest ascent}.
-\rmvspace
+
+Do note that for functions $g : \R \rightarrow \R^n$, the derivative is taken component-wise!

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

@@ -49,5 +49,12 @@ For $2 \times 2$ matrices (i.e. 2D functions), we can use the following scheme (
         (tr1) edge node [right] {$0$} (zero);
     \end{tikzpicture}
 \end{center}
-As in Analysis I, it is important to also check the boundaries for maximums and minimums.
+As in Analysis I, it is important to also check the boundaries for maximums and minimums (as it may also be possible that there are NO critical points in the set).
 For that, formulate formulas for the borders and check them for critical points.
+
+This is mostly intuition, but think of what segments the set consists of and note them down.
+Then, for each of the sets of the segments, determine the critical points
+(e.g. for set $A = \{ (x, y) \in \R^2 \divides x = 0, 0 \leq y \leq 3 \}$, we compute the critical points of $f(0, y)$)
+
+For cases where $x$ and $y$ are both not $0$, we have to parametrize the set
+(e.g. for set $C = \{ (x, y) \in \R^2 \divides 3x + y = 3, 0 \leq x \leq 1 \}$, we have $\gamma(t) = (t, 3 - 3t)$ and compute the critical points of $f(\gamma(t))$)

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 $x$\\
+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 $x$.
-Often, we use $V$ instead of $f$ to denote the vector field.
+Ideally, to compute a line integral, we compute the derivative of $\gamma$ separately ($\gamma(t) = s$ usually, derive component-wise),
-Ideally, to compute a line integral, we compute the derivative of $\gamma$ and $V(\gamma(t))$ separately, then simply do the integral after.
+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 $\displaystyle v \cdot w = \sum_{i = 1}^{n} v_i \cdot w_i$