[Analysis] Small fixes

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

@@ -42,6 +42,7 @@ we can compute the partial derivative using the chain rule as follows:
 $\frac{\partial g}{\partial \phi} = \frac{\partial g}{\partial x} \cdot \frac{\partial x}{\partial \phi}
 + \frac{\partial g}{\partial y} \cdot \frac{\partial y}{\partial \phi} + \frac{\partial g}{\partial z} \cdot \frac{\partial z}{\partial \phi}$
 where all $\frac{\partial g}{\partial x}$, etc are known from the gradient and the other elements can be computed quickly from the known equations.
+The chain rule for higher or lower dimensional functions is as one would expect from the above formula.
 
 Finally, evaluate $\frac{\partial g}{\partial \phi}$ at the required points and compute the result.
 

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

@@ -1,11 +1,12 @@
 \newsectionNoPB
 \subsection{Higher derivatives}
-\shortdef $f$ is in class $C^1$ if $f$ is differentiable and all its partial derivatives are continuous.
+\compactdef{Class} $f$ is in class $C^1$ if $f$ is differentiable and all its partial derivatives are continuous.
 $f$ is of class $C^k$ if it is differentiable and each of its partial derivatives are in $C^{k - 1}$.
 If $f \in C^k(X; \R^m)$ for all $k \geq 1$, then $f \in C^\infty(X; \R^m)$\\
 % ────────────────────────────────────────────────────────────────────
 \setLabelNumber{all}{4}
-\compactproposition{Mixed derivatives commute} $\partial_{x, y} f = \partial_{y, x}$, as well as $\partial_{x, y, z} = \partial_{x, z, y} = \ldots$, etc (all mixed derivatives commute)
+\compactproposition{Mixed derivatives commute} $\partial_{x, y} f = \partial_{y, x}$, as well as $\partial_{x, y, z} = \partial_{x, z, y} = \ldots$,
+etc (all mixed derivatives commute).
 Since we have symmetry, we can use the notation $\partial_{x_1^{m_1}, \ldots, x_n^{m_n}} f = \frac{\partial^k}{\partial x^m} f = D^m f = \partial^m f$,
 where $m = (m_1, \ldots, m_n)$ and $m_1 + \ldots + m_n = k$.
 There are ${n + k - 1 \choose k}$ possible values for $m$ and e.g. $(1, 1, 2)$ corresponds to the derivative $\frac{\partial^4 f}{\partial x \partial y \partial^2 z}$\\