Clarify (#3786)

* Clarify
* Update chapters/functions.tex
* As suggested, and remove double empty lines.
Co-authored-by: Henrik Tidefelt <henrikt@wolfram.com>

Author: HansOlsson

chapters/functions.tex

@@ -1483,16 +1483,21 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati
 
 \subsection{Partial Derivatives of Functions}\label{partial-derivatives-of-functions}\index{partial derivative}\index{der@\robustinline{der}!partial derivative}
 
-A class defined as:
+A function class defined as follows is a partial derivative of another function:
 \begin{lstlisting}[language=grammar]
 IDENT "=" der "(" name "," IDENT { "," IDENT } ")" comment
 \end{lstlisting}
-is the partial derivative of a function, and may only be used as declarations of functions.
 
-The semantics is that a function (and only a function) can be specified in this form, defining that it is the partial derivative of the function to the right of the equal sign (looked up in the same way as a short class definition, and the looked up name must be a function), and partially differentiated with respect to each {\lstinline!IDENT!} in order (starting from the first one).
-Each {\lstinline!IDENT!} must be a scalar {\lstinline!Real!} input to the function.
+In \lstinline!$f$ = der($g$, $u_{1}, \ldots$)!, the function being defined is named $f$, and the function being differentiated is $g$.
+The name $g$ is looked up in the same way as a in short class definition, and the referenced class must be a function.
+Each $u_{i}$ must be a scalar {\lstinline!Real!} input to the function, and corresponds mathematically to prepending $\frac{\partial}{\partial u_{i}}$ to the function call.
+The $u_{i}$ are applied in increasing order of $i$ (although the partial derivatives commute for a broad class of functions).
 
-The comment allows a user to comment the function (in the info-layer and as one-line description, and as icon).
+\begin{nonnormative}
+In mathematical notation, the order of partial differentiation is reversed compared to the function definition; \lstinline!der($g$, $x$, $y$)! means $\frac{\partial}{\partial y}\frac{\partial}{\partial x}g$.
+\end{nonnormative}
+
+The \lstinline[language=grammar]!comment! has the same semantics as in a short class definition, for instance allowing the function to be given a description string, as well as \lstinline!Documentation! and \lstinline!Icon! annotations.
 
 \begin{example}
 The specific enthalpy can be computed from a Gibbs-function as follows:
@@ -1511,9 +1516,9 @@ \subsection{Partial Derivatives of Functions}\label{partial-derivatives-of-funct
   h := Gibbs(p, T) - T * Gibbs_T(p, T);
 end specificEnthalpy;
 \end{lstlisting}
+Thus \lstinline!der(Gibbs, T)! corresponds to $\frac{\partial \text{\lstinline!Gibbs(p, T)!}}{\partial \text{\lstinline!T!}} = \frac{\partial}{\partial \text{\lstinline!T!}}\text{\lstinline!Gibbs!}$.
 \end{example}
 
-
 \subsection{Using the Inverse Annotation}\label{using-the-inverse-annotation}
 
 If a function $f_1$ with one output formal parameter $y$ can be restricted to an informally defined domain and codomain, such that the mapping of the input formal parameter $u_{k}$ to $y$ is bijective for any fixed assignment to the other input formal parameters in the domain (see examples below), then it can be given an \fmtannotationindex{inverse} annotation to provide an explicit inverse $f_2$ to this mapping, provided that the function is only applied on this domain: