[Analysis] Various additions and fixes

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 $f(\gamma(t))$)