[Analysis] Final tweaks

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

@@ -67,3 +67,5 @@ where the lowest value is the global minimum and the highest value the global ma
 Always consider the corners as possible maxima or minima (if some corners are critical points, all are highly likely to be).
 
 The tangent plane at a critical point of a function $f : \R^n \rightarrow \R$, is of the form $\{ (x, y, z) \divides z = \text{const} \}$, with $z = f(x_0)$.
+
+Note that a global minimum or maximum is the maximal / minimal element of the range of the function.

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

@@ -43,6 +43,7 @@ If any two points of $X$ can be joined by a parametrized curve, then $g$ is uniq
 \shortremark Two points $x, y \in X$ can be joined by parametrized curve $\gamma$ if $\gamma(a) = x$ and $\gamma(b) = y$. In that case, $X$ is called \bi{path-connected}.
 It is true when $X$ is \textit{convex} (e.g. when $X$ is a disc or a product of intervals).
 If $f$ is a vector field on $X$, then $g$ is called a \bi{potential} for $f$ and it is not unique, since we can add a constant to $g$ without changing the gradient.\\
+\shade{gray}{Finding a potential} We want a $g$ s.t. $\nabla g = f$ for some conservative $f$, so we find an anti-derivative $g$ for which $\nabla g = f$\\
 %
 \stepLabelNumber{all}
 \shortproposition For a vectorfield to be conservative, a \textit{necessary condition} is that $\displaystyle\frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{x_i}$