diff options
Diffstat (limited to 'Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex')
| -rw-r--r-- | Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex | 1188 |
1 files changed, 1188 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex new file mode 100644 index 00000000000..1274318f916 --- /dev/null +++ b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex @@ -0,0 +1,1188 @@ +%% LyX 2.2.0 created this file. For more info, see http://www.lyx.org/. +%% Do not edit unless you really know what you are doing. +\documentclass[english,extend]{article} +\usepackage{lmodern} +\renewcommand{\sfdefault}{lmss} +\renewcommand{\ttdefault}{lmtt} +\usepackage[T1]{fontenc} +\usepackage[latin9]{inputenc} +\setcounter{secnumdepth}{2} +\setcounter{tocdepth}{2} +\usepackage{booktabs} +\usepackage{amstext} + +\makeatletter + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. +%% Because html converters don't know tabularnewline +\providecommand{\tabularnewline}{\\} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. + \newenvironment{example}{\begin{center}\ttfamily}{\end{center}} +\newenvironment{lyxcode} +{\par\begin{list}{}{ +\setlength{\rightmargin}{\leftmargin} +\setlength{\listparindent}{0pt}% needed for AMS classes +\raggedright +\setlength{\itemsep}{0pt} +\setlength{\parsep}{0pt} +\normalfont\ttfamily}% + \item[]} +{\end{list}} +\newcommand{\strong}[1]{\textbf{#1}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. +\usepackage{diffcoeff} + +\@ifundefined{showcaptionsetup}{}{% + \PassOptionsToPackage{caption=false}{subfig}} +\usepackage{subfig} +\makeatother + +\usepackage{babel} +\begin{document} + +\title{\texttt{diffcoeff}~\\ +a \LaTeX{} package for writing\texttt{}~\\ +differential coefficients easily} + +\author{Andrew Parsloe\\ +{\small{}(aparsloe@clear.net.nz)}} +\maketitle +\begin{abstract} +\noindent \texttt{diffcoeff.sty} allows the easy writing of ordinary and +partial differential coefficients of arbitrary (algebraic or numeric) order. +For mixed partial derivatives, the overall order (the superscript on $\partial$ +in the numerator) is calculated by the package. Optional arguments allow +the easy specification of a point of evaluation for ordinary derivatives, +or variables held constant for partial derivatives, and the placement of +the differentiand (in the numerator or appended). Some tweaking of the +display is possible through key = value settings. Secondary commands provide +analogous coefficients constructed from $D,\thinspace\Delta,$ and $\delta$, +and a command for writing Jacobians. The package uses \texttt{expl3} and +\texttt{xparse} from the \LaTeX{}3 bundles, \texttt{l3kernel} and \texttt{l3packages}. +\end{abstract} + +\section{Requirements} + +The \LaTeX{} package \texttt{diffcoeff.sty} is written in the expl3 language +of \LaTeX{}3\texttt{ }and requires the bundles \texttt{l3kernel} and \texttt{l3packages} +(the latter for the \texttt{xparse} module). However, granted the presence +of these bundles in your \TeX{} distribution, the \LaTeX{}3 element should +be invisible to the user. + +The package is invoked in the usual way by entering +\begin{lyxcode} +\textbackslash{}usepackage\{diffcoeff\} +\end{lyxcode} +in the preamble of your document. + +\paragraph{Note on terminology} + +I refer throughout to the quantity or function being differentiated as +the \emph{differentiand} (in line with \emph{integrand}, \emph{operand}, +etc.). + +\section{Ordinary differential coefficients \label{sec:Ordinary-differential-coefficien}} + +Writing\textbf{ }\texttt{\textbackslash{}diff\{y\}\{x\}} will produce $\diff{y}{x}$ +in text style (i.e. placed between \texttt{\$ \$}) or +\[ +\diff{y}{x} +\] +in display style (i.e. placed between \texttt{\textbackslash{}{[} \textbackslash{}{]}} +). In fact \texttt{\textbackslash{}diff yx} (omitting the braces) will +produce these results, with a saving on keystrokes. The braces are needed +only when differentiand or variable of differentiation is more than a single +token. + +There is one other form: we can insert a slash, `/', between numerator +and denominator: \texttt{\textbackslash{}diff f/x} produces $\diff f/x$ +which may be preferred for textstyle differential coefficients on occasion. +Nothing is gained in this particular instance. It is quicker to type the +five keystrokes d, f, /, d, x than it is to type the nine of \textbackslash{}, +d, i, f, f, , f, /, x but there are occasions when this is not always the +case. + +An optional first argument allows the order of differentiation to be specified. +The order need not be a number; an algebraic order of differentiation is +perfectly acceptable or, indeed, a mix: +\begin{example} +\textbackslash{}diff{[}2{]}\{y\}\{x\} $\Longrightarrow\quad{\displaystyle \diff[2]{y}{x},}$\medskip{} + +\textbackslash{}diff{[}n+1{]}\{y\}\{x\} $\Longrightarrow\quad{\displaystyle \diff[n+1]{y}{x}}.$ +\end{example} +(And again the braces can be omitted for single letters like \textbf{x} +and \textbf{y}.) + +In slash style, \texttt{\textbackslash{}diff{[}2{]}f/x} (11 keystrokes) +produces $\diff[2]f/x$, not significantly more typing than \texttt{d\textasciicircum{}2f/dx\textasciicircum{}2} +(9 keystrokes). + +If you want to specify a point at which the derivative is evaluated, append +a final optional argument, but note that it is given in \emph{braces} rather +than square brackets: +\begin{example} +\textbackslash{}diff{[}2{]}\{y\}\{x\}\{0\} $\Longrightarrow\quad{\displaystyle \diff[2]{y}{x}{0}}$ +\end{example} +\noindent (In this example it seems neater \emph{not} to finish with a +full stop or other punctuation.) The use of braces means that the differential +coefficient can be followed immediately by a mathematical expression wrapped +in \texttt{\textbackslash{}\{ \textbackslash{}\}}, or \texttt{{[} {]}}, +without the expression being confused with the (final) optional argument. +Note also that there must be \emph{no space} before the argument: it follows +\emph{immediately} on the second mandatory argument (if it follows at all). + +We could save a few keystrokes by writing this last example as \texttt{\textbackslash{}diff{[}2{]}yx\{0\}}. +The braces around the final optional argument can \emph{not} be omitted +\textendash{} otherwise there is no way of knowing that it \emph{is }the +final optional argument and not part of a following expression. + +In slash style, the trailing optional argument can be used, but perhaps +should not be. It looks ugly: +\begin{example} +\textbackslash{}diff{[}2{]}y/x\{0\} $\Longrightarrow\quad{\displaystyle \diff[2]{y}/{x}{0}}$ +\end{example} +Slash style is a more casual rendering of the derivative, intended for +inline use within text and it would be better to use a phrase like `evaluated +at zero'. + +\subsection{\textbackslash{}diffset: formatting tweaks} + +There are a number of tweaks one can make to the display of a derivative. +Many people now use upright (roman) forms for the `d's of a differential +coefficient, rather than math italic. To do this, put the command +\begin{example} +{\footnotesize{}\textbackslash{}}diffset{[}roman = true{]} +\end{example} +\noindent in the preamble of your document (following the \texttt{\textbackslash{}usepackage\{diffcoeff\}} +of course). The default is math italic. + +It is possible that you may want more space between the `d' in the numerator +of a differential coefficient and the superscripted order of the derivative. +Using an upright `d' alleviates this problem, but if using the default +math italic for the `d's, the separation can be altered by using the +\begin{example} +\textbackslash{}diffset{[}d-sep = $n${]} +\end{example} +\noindent command which adds an extra $n$~mu to \TeX{}'s spacing. The +default value for $n$ is 1 (i.e. 1~mu). The new separation will affect +all derivatives following the new setting. Put in the preamble, the new +separation will be document-wide. + +A third tweak changes the delimiters used to indicate the point of evaluation. +By default there is nothing on the left side and a vertical rule with the +point of evaluation subscripted to it on the right. You may prefer subscripted +parentheses. In that case write +\begin{example} +\textbackslash{}diffset{[}d-delims~=~(){]}\textmd{.} +\end{example} +Whatever delimiters you choose need to work with \LaTeX{}'s \texttt{\textbackslash{}left} +and \texttt{\textbackslash{}right} commands and consist of exactly two +tokens. \texttt{{[}} and \texttt{{]}}\textbf{ }are acceptable as also are +pairs like \texttt{\textbackslash{}lceil \textbackslash{}rceil}, \texttt{\textbackslash{}lfloor +\textbackslash{}rfloor} but if you want to use \texttt{\textbackslash{}\{}\textbf{ +}and\textbf{ }\texttt{\textbackslash{}\}} you need to place the \texttt{\textbackslash{}diffset} +command between maths delimiters. The default pair, as indicated, is \texttt{. |}, +t or full stop being \LaTeX{}'s way of suppressing (in this case) the +left delimiter. + +If you change the delimiters, say to \textbf{( )}, then the position of +the subscript may need adjusting. To do this, use the command +\begin{example} +\textbackslash{}diffset{[}d-nudge = $n${]} +\end{example} +A suggested setting for parentheses \textbf{( )} is $-6$ (in fact $-6$~mu +but the `mu' is supplied by \texttt{diffcoeff}). Thus the total change +would be +\begin{example} +\textbackslash{}diffset{[}d-delims = ( ), d-nudge = -6{]} +\end{example} +producing, for example, +\begin{example} +\textbackslash{}diff{[}n{]}\{y\}\{x\}\{0\} $\diffset[d-delims=(),d-nudge=-6]\Longrightarrow\quad{\displaystyle \diff[n]{y}{x}{0}}.$ +\end{example} +The default setting for \textbf{. |} is 0. Simply writing +\begin{example} +\textbackslash{}diffset $\diffset$ +\end{example} +will return all settings to their defaults. + +\subsection{Variations} + +\subsubsection{Appending the differentiand: \textbackslash{}diff{*}} + +If you want the differentiand to follow the differential coefficient rather +than sit in the numerator, perhaps because it is a fraction itself or because +it is long, like a polynomial ($ax^{2}+bx+c$), then one way to achieve +that is to leave the first mandatory argument in the \texttt{\textbackslash{}diff} +command empty and immediately follow the differential operator with the +differentiand: +\begin{example} +\textbackslash{}diff\{\}\{x\}(ax\textasciicircum{}2+bx+c) $\Longrightarrow\quad{\displaystyle \diff{}{x}(ax^{2}+bx+c)}.$ +\end{example} +Another is to use the star form of the \texttt{\textbackslash{}diff }command, +\begin{example} +\textbackslash{}diff{*}{[}2{]}\{\textbackslash{}frac\{F(x)\}\{G(x)\}\}\{x\} +$\Longrightarrow\quad{\displaystyle \diff*[2]{\frac{F(x)}{G(x)}}{x}.}$ +\end{example} +\noindent The LaTeX expression can be harder to read if, as here, one is +using a command like \texttt{\textbackslash{}frac} with its own pairs of +braces, but it is much easier, if one isn't sure whether the differentiand +should be appended or in the numerator, simply to insert or delete an asterisk +than move the differentiand from one place to the other. The star form +becomes especially useful if you want to both append the differentiand +\emph{and }indicate the point of evaluation, since it saves having to set +up the \texttt{\textbackslash{}left.} and \texttt{\textbackslash{}right|}\textbf{ +}delimiters and the subscript: +\begin{example} +\textbackslash{}diff{*}\{\textbackslash{}frac\{F(x)\}\{G(x)\}\}\{x\}\{0\} +$\Longrightarrow\quad{\displaystyle \diffset[d-delims=.|,d-nudge=0]\diff*{\frac{F(x)}{G(x)}}{x}{0}}$ +\end{example} +In slash style with the star option, an example above becomes +\begin{example} +\textbackslash{}diff{*}\{(ax\textasciicircum{}2+bx+c)\}/\{x\} $\Longrightarrow\quad\text{\ensuremath{{\displaystyle \diff*{(ax^{2}+bx+c)}/{x}}}}$, +\end{example} +where the derivative is automatically enclosed in parentheses by \texttt{diffcoeff}. + +\subsubsection{Multi-character variables of differentiation} + +Derivatives of a function-of-a-function may require forming a differential +coefficient in which the variable of differentiation is more complicated +than a single symbol like \texttt{x} or \texttt{\textbackslash{}alpha}. +For instance, to differentiate $\ln x^{2}$ (the logarithm of $x^{2}$) +one first differentiates in $x^{2}$ then in $x$. The initial differentiation +can be rendered +\begin{example} +\textbackslash{}diff\{\textbackslash{}ln x\textasciicircum{}2\}\{x\textasciicircum{}2\} +$\Longrightarrow\quad{\displaystyle \diff{\ln x^{2}}{x^{2}}}$; \medskip{} + +diff\{\textbackslash{}ln x\textasciicircum{}2\}/\{x\textasciicircum{}2\} +$\Longrightarrow\quad{\displaystyle \diff{\ln x^{2}}/{x^{2}}}.$ +\end{example} +\noindent Because of the superscript in the variable of differentiation +$x^{2}$, parentheses have been automatically inserted in the denominator. +This does not happen in a first-order derivative unless there is a superscript +present. For instance, +\begin{example} +\textbackslash{}diff\{\textbackslash{}ln\textbackslash{}sin x\}\{\textbackslash{}sin +x\} $\Longrightarrow\quad{\displaystyle \diff{\ln\sin x}{\sin x}.}$ +\end{example} +\noindent displays without parentheses. However, for higher order derivatives +parentheses are \emph{always} inserted to avoid confusion: +\begin{example} +\textbackslash{}diff{[}2{]}\{\textbackslash{}ln\textbackslash{}sin x\}\{\textbackslash{}sin +x\} $\Longrightarrow\quad{\displaystyle \diff[2]{\ln\sin x}{\sin x}.}$ +\end{example} + +\paragraph{Positioning the d in the numerator} + +When appending a differentiand, you may want to change the position of +the `d' in the numerator, particularly if the variable of differentiation +is a multi-character symbol or the order of differentiation is a multi-character +value like $n+1$. + +If you `manually' append the differentiand, then there are various ways +of altering the placement of the `d' from the default midpoint: use \texttt{\textbackslash{}hfill} +to push it hard to the left; use \texttt{\textbackslash{}hfil} to\textbf{ +}push it to the left an intermediate amount; use \texttt{\textbackslash{}hphantom} +or \texttt{\textbackslash{}hspace}, both with a braced argument, to push +it to the left some custom amount; use \texttt{\textbackslash{}hspace}\textbf{ +}with a \emph{negative} braced argument to push it to the right.\emph{ +}These same means can be used to shift the `d' when using the starred +form of \texttt{\textbackslash{}diff}.\textbf{ }The effect is exactly the +same, too: +\begin{example} +\textbackslash{}diff{[}n+1{]}\{\textbackslash{}hphantom\{\textbackslash{}sin +x\}\}\{\textbackslash{}sin x\}\textbackslash{}ln\textbackslash{}sin x $\Longrightarrow\quad{\displaystyle \diff[n+1]{\hphantom{\sin x}}{\sin x}\ln\sin x},$\medskip{} + +\textbackslash{}diff{*}{[}n+1{]}\{\textbackslash{}hphantom\{\textbackslash{}sin +x\}\textbackslash{}ln\textbackslash{}sin x\}\{\textbackslash{}sin x\} $\Longrightarrow\quad{\displaystyle \diff*[n+1]{\hphantom{\sin x}\ln\sin x}{\sin x}}.$ +\end{example} +\noindent In the starred form \texttt{diffcoeff} understands that the formatting +is not appended with the differentiand but stays in the numerator. (But +a \emph{second} \texttt{\textbackslash{}hphantom} or \texttt{\textbackslash{}hfil} +etc. would be appended.) These are to be compared with +\begin{example} +\textbackslash{}diff{*}{[}n+1{]}\{\textbackslash{}ln\textbackslash{}sin +x\}\{\textbackslash{}sin x\} $\Longrightarrow\quad{\displaystyle \diff*[n+1]{\ln\sin x}{\sin x},}$ +\end{example} +where no phantom has been used. Which is better? Deleting the asterisk +gives +\begin{example} +\textbackslash{}diff{[}n+1{]}\{\textbackslash{}ln\textbackslash{}sin x\}\{\textbackslash{}sin +x\} $\Longrightarrow\quad{\displaystyle \diff[n+1]{\ln\sin x}{\sin x},}$ +\end{example} +In slash style, the phantom (or \texttt{\textbackslash{}hfil} etc.) is +ignored: +\begin{example} +\textbackslash{}diff{*}{[}n+1{]}\{\textbackslash{}hphantom\{\textbackslash{}sin +x\textbackslash{}sin x\textbackslash{}sin x\}\textbackslash{}ln\textbackslash{}sin +x\}/\{\textbackslash{}sin x\} $\Longrightarrow\quad{\displaystyle \diff*[n+1]{\hphantom{\sin x\sin x\sin x}\ln\sin x}/{\sin x}}.$ +\end{example} + +\subsubsection{Iterated derivatives} + +A second derivative is an iterated derivative, i.e., one in which a differential +coefficient forms the differentiand of another differential coefficient: +\begin{example} +\textbackslash{}diff{[}2{]}yx = \textbackslash{}diff{*}\{\textbackslash{}diff +yx\}x $\Longrightarrow{\displaystyle \diff[2]yx=\diff*{\diff yx}x},$ +\end{example} +or even +\begin{example} +\textbackslash{}diff{[}2{]}yx = \textbackslash{}diff\{\textbackslash{}diff +yx\}x $\Longrightarrow{\displaystyle \diff[2]yx=\diff{\diff yx}x},$ +\end{example} +where omission of unnecessary braces has aided readability. Note how easy +it is to switch between the different forms on the right, simply by inserting +or removing an asterisk. + +\subsection{Forming `derivatives' with D, \textbackslash{}Delta, \textbackslash{}delta} + +Often one wants to construct analogues of a differential coefficient but +with symbols other than $d$ or $\partial$. The \texttt{diffcoeff} package +offers three alternatives, all with the same pattern of optional and mandatory +arguments as for \texttt{\textbackslash{}diff}, except for the slash form. +There is \emph{no} slash option. + +An uppercase $D$ is used in place of $d$ for the \emph{material} or \emph{substantive} +derivative of a quantity in (for example) fluid dynamics. Write \texttt{\textbackslash{}Diff} +to invoke this command:\footnote{The \texttt{\textbackslash{}diffp} command, the partial derivative, in +the example is discussed in the next section.} +\begin{example} +\textbackslash{}Diff\{\textbackslash{}rho\}\{t\}=\textbackslash{}diffp\textbackslash{}rho +t + \textbackslash{}mathbf\{u\textbackslash{}cdot\}\textbackslash{}nabla\textbackslash{}rho +$\Longrightarrow{\displaystyle \Diff{\rho}{t}=\diffp\rho t+\mathbf{u\cdot}\nabla\rho.}$ +\end{example} +(The braces could also be removed from the arguments of \texttt{\textbackslash{}Diff} +as they have been from the arguments of \texttt{\textbackslash{}diffp}.)\texttt{ } + +The `D's are romanised (along with the `d's of ordinary derivatives) +with the +\begin{example} +\textbackslash{}diffset{[}roman = true{]} +\end{example} +command. The default is math italic. + +The command \texttt{\textbackslash{}diffd} will form a fraction often used +in introductory calculus texts (and other places):\footnote{I considered using \texttt{\textbackslash{}diffg} for this command as in +`diff greek' but decided that the more likely mind-phrase is `diff delta', +leading to the use of `d' rather than `g'.} +\begin{example} +\textbackslash{}diffd\{y\}\{x\} $\Longrightarrow{\displaystyle \diffd yx.}$ +\end{example} +Similarly, \texttt{\textbackslash{}Diffd} forms a fraction with $\Delta$: +\begin{example} +\textbackslash{}Diffd\{y\}\{x\} $\Longrightarrow{\displaystyle \Diffd{y}{x}.}$ +\end{example} +Higher order forms of these derivatives are produced in the same way as +with \texttt{\textbackslash{}diff}, using an optional argument to specify +the order: +\begin{example} +\textbackslash{}diffd{[}2{]}\{y\}\{x\} $\Longrightarrow{\displaystyle \diffd[2]yx.}$ +\end{example} +A final optional argument, enclosed in braces, specifies a point of evaluation, +care being taken, as ever, to ensure that there is no space between it +and the second mandatory argument: +\begin{example} +\textbackslash{}Diffd\{y\}\{x\}\{x=0\} $\Longrightarrow{\displaystyle \Diffd yx{x=0}.}$ +\end{example} + +\section{Partial differential coefficients\label{sec:Partial-differential-coefficient}} +\noindent \begin{flushleft} +Partial differential coefficients follow the same pattern as for ordinary +derivatives, with some generalisations arising from the greater possibilities. +The command this time is \texttt{\textbackslash{}diffp}. Thus \textbf{\textbackslash{}diffp\{F\}\{x\}} +produces $\diffp{F}{x}$ in text style and +\[ +\diffp{F}{x} +\] + in display style. Braces can be omitted for single token differentiands +and variables: \texttt{\textbackslash{}diffp Fx} does the job.\textbf{ +}As for \texttt{\textbackslash{}diff}, there is a slash form for more casual +use: \texttt{\textbackslash{}diffp F/x} displaying as $\diffp F/x$. Given +that \texttt{\textbackslash{}partial} takes 8 keystrokes to type, the slash +form \emph{does }economise on keystrokes for a partial derivative. +\par\end{flushleft} + +\begin{flushleft} +Again an optional argument allows the specification of the order of differentiation +and it may be numeric or algebraic or a mix of the two. For a second or, +indeed, an $n+4$th-order partial derivative, +\par\end{flushleft} +\begin{example} +\textbackslash{}diffp{[}n+4{]}\{F\}\{x\} $\Longrightarrow\quad{\displaystyle {\displaystyle \diffp[n+4]{F}{x},}}$\medskip{} + +\textbackslash{}diffp{[}n+4{]}\{F\}/\{x\} $\Longrightarrow\quad{\displaystyle {\displaystyle \diffp[n+4]{F}/{x},}}$ +\end{example} +In a subject like thermodynamics, there is a need to indicate which variable +or variables are held constant when the differentiation occurs. To show +this, append a final optional argument. Thus to differentiate the entropy +$S$ in the temperature $T$ while holding the volume $V$ constant, write +\begin{example} +\textbackslash{}diffp\{S\}\{T\}\{V\} $\Longrightarrow\quad{\displaystyle \diffp{S}{T}{V}}$ +\end{example} +As with \texttt{\textbackslash{}diff}\textbf{ }note how the final optional +argument is given in braces rather than square brackets, and that there +must be \emph{no space} before the argument: if used, it follows \emph{immediately} +on the second mandatory argument. This means that the differential coefficient +can be followed immediately by a mathematical expression wrapped in \textbackslash{}\{ +\textbackslash{}\}, or {[} {]}, without the expression being confused with +the (final) optional argument. + +We could save a few keystrokes by writing this last example as \texttt{\textbackslash{}diffp +ST\{V\}}. The braces around the optional argument can \emph{not} be dispensed +with (otherwise there is no way of knowing that it \emph{is} the final +optional argument and not part of a following expression). + +Note that for the slash form of the derivative it is anticipated that there +will be no trailing optional argument. If you \emph{do} use one, you will +need to change the nudge value either with the \texttt{\textbackslash{}diffset} +command or, better, by including a spacing command in the third argument: +\begin{example} +\textbackslash{}diffp\{S\}/\{T\}\{\textbackslash{};V\} $\Longrightarrow\quad{\displaystyle \diffp{S}/{T}{\;V}}$ +\end{example} +Without the spacing command, the subscript encroaches on the right parenthesis. + +\subsubsection{Appending the differentiand} + +If you want to remove the differentiand from the numerator to instead follow +the derivative, one way, as for ordinary derivatives, is to leave the first +mandatory argument empty and manually append the differentiand: +\begin{example} +\textbackslash{}diffp{[}n{]}\{\}xf(x) $\Longrightarrow\quad{\displaystyle \diffp[n]{}xf(x).}$ +\end{example} +However, you may wonder how that would look with the differentiand in the +numerator, which is a good reason for preferring the starred form of the +\texttt{\textbackslash{}diffp} command to achieve an appended derivative: +\begin{example} +\textbackslash{}diffp{*}{[}n{]}\{f(x)\}x $\Longrightarrow\quad{\displaystyle \diffp*[n]{f(x)}x.}$ +\end{example} +Now it is easy to switch between an appended differentiand and one in the +numerator simply by inserting or deleting the asterisk. In the slash form, +parentheses are automatically inserted around the differential operator: +\begin{example} +\textbackslash{}diffp{*}{[}n{]}\{f(x)\}/x $\Longrightarrow\quad{\displaystyle \diffp*[n]{f(x)}/x.}$ +\end{example} +It also happens, for example in thermodynamics, that you may wish to both +append the differentiand \emph{and} indicate variables held constant. In +that case, the starred \texttt{\textbackslash{}diffp} command is much easier +to use. Thus, to express a relation in thermodynamics, +\begin{example} +\textbackslash{}diffp{*}\{\textbackslash{}frac \{P\}\{T\}\}\{U\}\{V\} = +\textbackslash{}diffp{*}\{\textbackslash{}frac\{1\}\{T\}\}\{V\}\{U\} $\Longrightarrow\quad{\displaystyle \diffp*{\frac{P}{T}}{U}{V}=\diffp*{\frac{1}{T}}{V}{U}}$ +\end{example} +\noindent where the starred form automatically takes care of the parentheses +and subscripts. Again, not all the braces are necessary, with some help +to readability: +\begin{example} +\textbackslash{}diffp{*}\{\textbackslash{}frac PT\}U\{V\} = \textbackslash{}diffp{*}\{\textbackslash{}frac +1T\}V\{U\} $\Longrightarrow\quad{\displaystyle \diffp*{\frac{P}{T}}U{V}=\diffp*{\frac{1}{T}}V{U}}$ +\end{example} + +\subsection{Mixed partial derivatives} + +The new thing with partial derivatives, not present with ordinary derivatives, +is \emph{mixed} partial derivatives, where there is more than one variable +of differentiation. If each variable is differentiated only to the first +order, then it is easy to specify the derivative. Say $f(x,y,z)$ is a +function of three variables, as indicated. Then +\begin{example} +\textbackslash{}diffp\{f\}\{x\c{,}y,z\} $\Longrightarrow\quad{\displaystyle \diffp{f}{x,y,z}}.$ +\end{example} +The variables of differentiation are listed in order in a comma list forming +the second mandatory argument. The total order of differentiation (3 in +this example) is inserted automatically \textendash{} \texttt{diffcoeff} +does the calculation itself. There is also a slash form: +\begin{example} +\textbackslash{}diffp\{f\}/\{x\c{,}y,z\} $\Longrightarrow\quad{\displaystyle \diffp{f}/{x,y,z}}.$ +\end{example} +If we want to differentiate variables to higher order, then their orders +need to be specified explicitly. To do so use a comma list also in the +\emph{optional} argument: +\begin{example} +\textbackslash{}diffp{[}2,3{]}\{f\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[2,3]{f}{x,y,z}.}$ +\end{example} +\noindent Notice that the overall order of the derivative \textendash{} +6 \textendash{} is again automatically calculated and inserted as a superscript +on the $\partial$ symbol in the numerator. In this example, the comma +list of orders has only two members, even though there are three variables. +It is assumed that the orders given in the comma list of orders apply in +sequence to the variables, the first order to the first variable, the second +to the second variable, and so on, and that any subsequent orders not listed +in the optional argument are, by default, 1. Thus we need to specify only +2 and 3 in the example; the order of $z$ is 1 by default. + +But you \emph{cannot} use an order specification like \texttt{{[},,2{]}}. +This will be treated as if it were \texttt{{[}2{]}}. (This is a feature +of comma lists in the expl3 language used by \texttt{diffcoeff.sty}.) Instead +write \texttt{{[}1,1,2{]}}.\textbf{ }It is only the \emph{tail} of an order +specification which can be omitted. + +The automatic calculation of the overall order of differentiation remains +true even when some or all of the orders for the individual variables are +algebraic. For example, differentiating in three variables with orders +\texttt{2k}, \texttt{m-k-2}, \texttt{m+k+3}, we have +\begin{example} +\textbackslash{}diffp{[}2k-1,m-k-2,m+k+3{]}\{F(x,y,z)\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[2k-1,m-k-2,m+k+3]{F(x,y,z)}{x,y,z}}$, +\end{example} + +\subsection{The order-override option} + +In this example the overall order is presented as \texttt{2k+2m}. You might +prefer this to be presented as \texttt{2(k+m)} instead. Although \texttt{diffcoeff} +takes some steps to present the overall order appropriately, it does not +factorise expressions. If you want to present the order in a manner distinct +from that of \texttt{diffcoeff}, use the\emph{ order-override option}, +which is a second optional argument immediately following the first: +\begin{example} +\textbackslash{}diffp{[}2k-1,m-k-2,m+k+3{]}{[}2(k+m){]}\{F(x,y,z)\}\{x,y,z\} +$\Longrightarrow\quad{\displaystyle \diffp[2k-1,m-k-2,m+k+3][2(m+k)]{F(x,y,z)}{x,y,z}}$. +\end{example} +The order-override option does exactly that: overrides the presentation +of the calculated order with the manually given one. (In fact the algorithm +does not get called at all.) + +\subsubsection{Order specifications beyond the scope of \texttt{diffcoeff.sty}} + +The order specification can include signed integers, variables like $k$ +and $\alpha$ with signed integer coeffients, and products of any number +of variables like $mn$ or $kmn$ with signed integer coefficients. The +algorithm that calculates the overall order in \texttt{diffcoeff.sty} \emph{cannot} +handle\texttt{ }exponents, subscripts or parentheses. For such constructs, +or more exotic ones, the order-override option is always available. If +it is present (even if empty), the algorithm is bypassed completely and +one can include `anything' there without causing error. + +I doubt that these limitations matter in any practical sense. We are in +`overkill' territory here. Mixed partial derivatives are used far more +rarely than the `pure' ones, and mixed partial derivatives to `exotic' +orders of differentiation are used \emph{vanishingly} rarely, and in any +case the order-override option is always available. But should you, in +some freak circumstance, find yourself needing to write such things, then +I suggest you use \texttt{diffcoeffx.sty}, which is \texttt{diffcoeff.sty} +`on steroids'. It can handle the situations described above that are +beyond the scope of \texttt{diffcoeff.sty}, and it uses exactly the same +commands so there is nothing new to remember. It also provides additonal +functionality for the trailing optional argument. + +\subsubsection{Presentation of the overall order} + +To take a grotesque example, that will never arise in practice, consider +the following: +\begin{example} +\textbackslash{}diffp{[}kmn-mn+n-1,2kmn-mn+2n-1,n+1{]}\{f\}\{x,y,z,w\} +$\Longrightarrow{\displaystyle \diffp[kmn-mn+n-1,2kmn-mn+2n-1,n+1]{f}{x,y,z,w}}.$ +\end{example} +As noted earlier, since the final variable $w$ is differentiated only +to order 1, there is no need to specify it in the comma list of orders. +The implicit 1 contributes to the vanishing of the numerical part in the +overall order of differentiation. In this example, the overall order contains +multivariable terms, $kmn$ and $mn$. \texttt{diffcoeff} initially organises +these in the sequence: \ldots{} 3-variable terms before 2-variable terms +before single-variable terms, generally before the numerical term. However +if a minus sign precedes the first many-variable term, and the numerical +term is positive, it will be presented first: +\begin{example} +\textbackslash{}diffp{[}12-2km,k-1,km+1{]}\{f\}\{x,y,z,w\} $\Longrightarrow{\displaystyle \diffp[12-2km,km-1,k+1]{f}{x,y,z,w}}.$ +\end{example} +Should the numerical term either vanish or be negative and the leading +algebraic term is preceded by a minus sign, \texttt{diffcoeff} will look +for an algebraic term with a preceding $+$ sign and put that first: +\begin{example} +\textbackslash{}diffp{[}2km-3k-1,2k-1,-3km+4k+1{]}\{f\}\{x,y,z,w\} $\Longrightarrow{\displaystyle \diffp[2km-3k-1,2k-1,-3km+4k+1]{f}{x,y,z,w}}.$ +\end{example} + +\subsection{\textbackslash{}diffset: formatting tweaks} + +As with ordinary derivatives, there are a number of tweaks one can make +to the display of a partial derivative. + +You may want more space between the $\partial$ symbol in the numerator +of a partial derivative and the superscripted order of the derivative. +The separation can be altered by using the +\begin{example} +\textbackslash{}diffset{[}p-sep = $n${]} +\end{example} +\noindent command which adds an extra $n$~mu to \TeX{}'s spacing. The +default value is 1 (i.e. 1~mu). The new separation will affect all derivatives +following the new setting. Put in the preamble, the new separation will +be document-wide. + +You may also want to adjust the spacing between the terms in the denominator. +This can be done with the command +\begin{example} +\textbackslash{}diffset{[}sep=$n${]} +\end{example} +which adds an extra $n$~mu to \TeX{}'s spacing. The default value is +2~mu. + +If you wish to indicate the point at which a partial derivative is evaluated, +you may not want to use parentheses, since these when subscripted are widely +held to indicate variables held constant. To change the delimiter on the +right to a vertical line, use +\begin{example} +\textbackslash{}diffset{[}p-delims = . | {]}\textmd{,} +\end{example} +the dot suppressing the delimiter on the left. (Note that to use \texttt{\textbackslash{}\{} +and \texttt{\textbackslash{}\}} as delimiters, \texttt{\textbackslash{}diffset +}must be placed between maths delimiters.) + +Changing the delimiters will usually require a repositioning of the subscript. +The command is +\begin{example} +\textbackslash{}diffset{[}p-nudge = $n${]}\textmd{.} +\end{example} +For parentheses the default value of $n$ is $-6$, but for the vertical +rule a zero value is appropriate. Thus the overall command for . | would +be +\begin{example} +\textbackslash{}diffset{[}p-delims = . |, p-nudge = 0{]} \textmd{.} +\end{example} +Writing +\begin{example} +\textbackslash{}diffset +\end{example} +will return all settings to their default values. + +\subsection{Variations} + +\subsubsection{Multi-character variables of differentiation} + +In thermodynamics one may want to differentiate in the reciprocal of the +temperature, $1/T$. In tensor calculus the differentiations are almost +always in terms of super- or subscripted coordinates, and in many other +contexts this is the case too. This is why a comma list is used in \texttt{diffcoeff} +for specifying the variables of differentiation for partial derivatives. +Although it would be nice to write the minimal \texttt{\{xy\}} for this +rather than \texttt{\{x,y}\}, the extra writing is trivial and the comma +list allows the simplest handling of multi-character variables: +\begin{example} +\textbackslash{}diffp\{A\_i\}\{ x\textasciicircum{}j,x\textasciicircum{}k +\} $\Longrightarrow{\displaystyle \diffp{A_{i}}{x^{j},x^{k}},}$ +\end{example} +taken from tensor calculus, or this strange object taken from statistical +mechanics: +\begin{example} +\textbackslash{}diffp{[}2{]}q\{\textbackslash{}frac 1\textbackslash{}Theta\} +$\Longrightarrow{\displaystyle \diffp[2]q{\frac{1}{\Theta}}}$. +\end{example} +The parentheses have been inserted automatically by \texttt{diffcoeff} +to clarify exactly what the variable of differentiation is. + +\subsubsection{Use of phantoms when appending differentiands} + +As for ordinary derivatives, when appending a differentiand you may want +to include a phantom (\texttt{\textbackslash{}hphantom} etc.) in the numerator +of the differential coefficient to alter the placement of the $\partial$ +symbol. This may be particularly relevant if the order of differentiation +is a multi-character symbol or if there are a number of variables of differentiation. + +Either means of achieving the appended differentiand achieve the same result: +\begin{example} +\textbackslash{}diffp{[}m,2{]}\{\textbackslash{}hphantom\{\textbackslash{}partial +y \textbackslash{}partial \}\}\{x,y,z\} (\textbackslash{}ln \textbackslash{}cos +x + \textbackslash{}ln \textbackslash{}sin y)z $\Longrightarrow\quad{\displaystyle \diffp[m,2]{\hphantom{\partial y\partial}}{x,y,z}}(\ln\cos x+\ln\sin y)z,$\medskip{} + +\textbackslash{}diffp{*}{[}m,2{]}\{\textbackslash{}hphantom\{\textbackslash{}partial +y \textbackslash{}partial \}(\textbackslash{}ln \textbackslash{}cos x + +\textbackslash{}ln \textbackslash{}sin y)z\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp*[m,2]{\hphantom{\partial y\partial}(\ln\cos x+\ln\sin y)z}{x,y,z}},$ +\end{example} +which is to be compared with the derivative without the phantom, +\begin{example} +\textbackslash{}diffp{*}{[}m,2{]}\{(\textbackslash{}ln \textbackslash{}cos +x + \textbackslash{}ln \textbackslash{}sin y)z\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp*[m,2]{(\ln\cos x+\ln\sin y)z}{x,y,z}}.$ +\end{example} +\noindent In the starred form, \texttt{diffcoeff} understands that the +phantom is not appended with the differentiand but stays in the numerator. +(But a \emph{second} phantom would be appended.) + +\subsubsection{Iterated derivatives} + +Partial derivatives can be iterated. For example, +\begin{example} +\textbackslash{}diffp f\{x,y\} = \textbackslash{}diffp{*}\{\textbackslash{}diffp +fy\}x $\Longrightarrow{\displaystyle \diffp f{x,y}=\diffp*{\diffp fy}x,}$\medskip{} + +\textbackslash{}diffp f\{x,y\} = \textbackslash{}diffp\{\textbackslash{}diffp +fy\}x $\Longrightarrow{\displaystyle \diffp f{x,y}=\diffp{\diffp fy}x.}$ +\end{example} +It is easy to switch between these forms by inserting or deleting the asterisk. + +\subsection{Jacobians} + +\texttt{diffcoeff} provides a command \texttt{\textbackslash{}jacob} for +constructing Jacobians. For example +\begin{example} +\textbackslash{}jacob\{u,v,w\}\{x,y,z\} $\Longrightarrow{\displaystyle \jacob{u,v,w}{x,y,z}.}$ +\end{example} +The comma lists can contain any number of variables. \texttt{\textbackslash{}jacob} +does \emph{not} check that the two arguments contain the same number of +variables, so it is perfectly possible to form an object like +\begin{example} +\textbackslash{}jacob\{u,v,w\}\{x,y\} , +\end{example} +which as far as I know has no meaning. + +\section{Discussion of the code} + +I set about creating this package when faced with trying to parse \LaTeX{} +expressions involving derivatives for another program I was working on. +Trying to parse \texttt{\textbackslash{}frac\{d<something>\}\{d<something +else>\}}, perhaps with \texttt{\textbackslash{}mathrm\{d\}}'s, and a superscript +on the first \texttt{d}, perhaps with a \texttt{\textbackslash{}tfrac} +or \texttt{\textbackslash{}dfrac} for the \texttt{\textbackslash{}frac},\textbf{ +}wasn't quite hopeless, but it was certainly \emph{messy}. (I used regular +expressions to transform the fraction into something more systematic.) + +\subsection{Other packages} + +Looking through the MiK\TeX{} distribution and, less assiduously, through +CTAN, produced the following packages which provide macros for derivatives. +(Strangely, AMS packages do not touch this subject, as far as I can see.) +\begin{itemize} +\item \texttt{bropd} +\begin{itemize} +\item \texttt{\textbackslash{}od{[}n{]}\{y\}\{x\}} and \texttt{\textbackslash{}pd{[}n{]}\{y\}\{x\}} +for ordinary and partial derivatives of order \texttt{n} in one variable +\item \texttt{\textbackslash{}pd\{u\}\{x,x,t\}} for a mixed partial derivative, +order 2 in \texttt{x}, 1 in \texttt{t} +\item \texttt{\textbackslash{}pd\{\}\{z\}\{x+y\}} for appending \texttt{(x+y)} +\item \texttt{\textbackslash{}pd\{!\}\{z\}\{x+y\}} for appending \texttt{x+y} +\end{itemize} +\item \texttt{commath} +\begin{itemize} +\item \texttt{\textbackslash{}od{[}n{]}\{y\}\{x\}} and \texttt{\textbackslash{}pd{[}n{]}\{y\}\{x\}} +for ordinary and partial derivatives of order \texttt{n} in one variable +\item \texttt{\textbackslash{}md\{f\}\{5\}\{x\}\{2\}\{y\}\{3\}} for a 5th order +mixed partial derivative +\item \texttt{\textbackslash{}tmd}, \texttt{\textbackslash{}dmd} and similar +commands for forcing text and display styles +\end{itemize} +\item \texttt{esdiff} +\begin{itemize} +\item \texttt{\textbackslash{}diff{[}n{]}\{y\}\{x\}}\textbf{ }and \texttt{\textbackslash{}diffp{[}n{]}\{y\}\{x\}}\textbf{ +}for ordinary and partial derivatives of order \texttt{n} in one variable +\item \texttt{\textbackslash{}diffp\{f\}\{\{x\textasciicircum{}2\}\{y\}\{z\textasciicircum{}3\}\}} +for a mixed partial derivative of order 6 in three variables +\item \texttt{\textbackslash{}diff{*}{[}n{]}\{y\}\{x\}\{0\}} for indicating the +point of evaluation of the derivative (using a subscript on parentheses) +\item \texttt{\textbackslash{}diffp{*}\{P\}\{T\}\{V\}} to indicate a variable +held constant +\end{itemize} +\item \texttt{physymb} +\begin{itemize} +\item \texttt{\textbackslash{}ud\{y\}\{x\}} and \texttt{\textbackslash{}pd\{y\}\{x\}} +for ordinary and partial derivatives of first order +\item \texttt{\textbackslash{}udd\{y\}\{x\}}, \texttt{\textbackslash{}uddd\{y\}\{x\}} +and \texttt{\textbackslash{}pdd\{y\}\{x\}}, \texttt{\textbackslash{}pddd\{y\}\{x\}} +for second and third order ordinary and partial derivatives +\item higher order derivatives not catered for +\end{itemize} +\end{itemize} +None of the packages quite gave what I wanted (but for all that, I suspect +cope with well over 90\% of use cases). \texttt{esdiff} comes closest but +failed when it came to combining algebraic and numeric orders of differentation +in a mixed partial derivative. Also the need to em-brace variables in a +mixed partial derivative in \texttt{esdiff}\textbf{ }was another (small) +count against it. + +\subsection{diffcoeff.sty} +\begin{itemize} +\item The distinctive feature of \texttt{diffcoeff.sty} is that it will automatically +form the overall order of a mixed partial derivative, including those containing +both algebraic and numeric contributions to the order: +\end{itemize} +\begin{example} +\textbackslash{}diffp{[}m-k-1,m+k{]}\{F(x,y,z)\}\{x,y,z\} $\Longrightarrow{\displaystyle \diffp[m-k-1,m+k]{F(x,y,z)}{x,y,z}}.$ +\end{example} +\begin{itemize} +\item Ease of use was another major consideration, trying to avoid the unnecessary +writing of superscripts and subscripts and brace pairs. In this example, +no superscripts are written and only the two inescapable brace pairs are +required. +\begin{itemize} +\item The use of a comma list for the second mandatory argument in a partial +derivative is another example. That makes differentiations in super- or +subscripted symbols easier to both write and read by avoiding `entanglements' +of braces. +\end{itemize} +\item I've also tried to make the options `natural' and consistent across both +ordinary and partial derivatives. Looking at the other packages listed +above, writing something like \texttt{\textbackslash{}diff{[}n{]}\{f\}\{x\}} +(which can be trimmed to \texttt{\textbackslash{}diff{[}n{]}fx} in this +instance) seems `natural' \textendash{} only \texttt{physymb} deviates +from the pattern. It seems consistent with this pattern to use a comma +list as an optional argument for mixed partial derivatives. +\item I debated whether to include provision for points of evaluation and variables +held constant into the \texttt{\textbackslash{}diff} and \texttt{\textbackslash{}diffp} +commands. \texttt{esdiff} certainly allows this. I think a case can be +made, in subjects like thermodynamics, to consider the parentheses and +subscript as part of the overall symbol. The partial derivative itself +doesn't give the full story; it is ambiguous. Hence provision for these +extra elements was included in \texttt{\textbackslash{}diff} and \texttt{\textbackslash{}diffp}. +It's positioning as a final optional argument also felt natural given the +position of the resulting symbol in the displayed derivative: +\end{itemize} +\begin{example} +\textbackslash{}diffp ST\{V\} $\Longrightarrow\quad{\displaystyle \diffp ST{V}}$ +\end{example} +\begin{itemize} +\item Although initially I used standard square brackets for this trailing optional +argument, the possibility of an immediately following mathematical expression +being enclosed in square brackets convinced me to use braces for the argument. +An immediately following expression can now be enclosed in \texttt{{[} +{]}}, or \texttt{\textbackslash{}\{ \textbackslash{}\}}, without ambiguity. +\item The star option also prompted the reflection: is it needed? One can always +leave the first mandatory argument empty and append the differentiand `by +hand'. But once the provision for points of evaluation or variables held +constant was incorporated into the \texttt{\textbackslash{}diff} and \texttt{\textbackslash{}diffp} +commands, the star option became the simplest way of handling appended +differentiands using the extra provision. (Note that it conflicts with +the star option in \texttt{esdiff}, but I can't see the packages ever being +used together.) And once the option is available, it provides a simple +way to switch between differentiand in the numerator/differentiand appended. +\item The final option added to the package was the slash option. This was prompted +after seeing the expression $\diff*{[\log f(z)]}/z$ in a text on statistical +mechanics. Alerted to the form, I then skimmed through various texts and +found this form of the derivative was used sufficiently often to justify +inclusion. The placement of the slash, between the two mandatory arguments, +seemed more-or-less self-evident. +\end{itemize} + +\subsection{The mixed partial derivatives algorithm} + +It occurred to me, after I had created an algorithm for splitting a linear +expression composed of signs, integers and variables into its numerical +and algebraic parts, that the same algorithm could be used in a recursive +way to simplify the algebraic part of the expression. + +Given an order specification like, say, \textbf{\strong{\textbf{{[}2m+k\textendash 1,2m\textendash k+1,2k,1{]}}}}, +the idea is to concatenate the terms with intervening \textbf{+} signs, +thus \textbf{\strong{\textbf{2m+k\textendash 1+2m\textendash k+1+ 2k+1}}}, +then split this expression into numeric and algebraic parts, giving \textbf{\strong{\textbf{\textendash 1+ 1+1}}} +for the numeric part and \textbf{\strong{\textbf{2m+k+2m\textendash k+2k}}} +for the algebraic part. The numeric part, assumed to be a combination of +integers, is evaluated and the result stored. For the algebraic part, remove +throughout all instances of one of the variables, say \textbf{\strong{\textbf{m}}}. +The result is \textbf{\strong{\textbf{2+k+2\textendash k+2k}}}. Split +this into numeric and algebraic parts: \textbf{\strong{\textbf{2+2}}} +for the numeric part and \textbf{\strong{\textbf{k\textendash k+2k}}} +for the algebraic part. Evaluate the numeric part, \textbf{\strong{\textbf{+4}}}, +and you have the overall coefficient of the variable \textbf{\strong{\textbf{m}}}. +Repeat the process for the next variable, and so on until all variables +have been accounted for. + +In fact repeating the process for the next variable, \strong{k} in this +example, immediately reveals a problem. Removing \strong{k} from \textbf{\strong{\textbf{k\textendash k+2k}}} +leaves \strong{\textendash +2} which evaluates to \strong{\textendash 2} +whereas the correct coefficient for \strong{k} should be \strong{+2}. +The solution is to insert \strong{1} before any `bare' variable \textendash{} +a variable preceded only by a sign rather than a number. In that case the +expression we remove \strong{k} from is \strong{1k\textendash 1k+2k} +giving the correct overall coefficient \strong{+2}. + +A second problem may arise if there are terms involving products of variables +as in the order specification \strong{{[}mk\textendash 2,2m+1,2k+1{]}}. +This splits into a numeric part \strong{\textendash 2+1+1} evaluating +to \strong{0}, and an algebraic part \strong{mk+2m+2k}. If we choose +\strong{m} as the first variable to remove from this expression, we get +\strong{+2} for the numeric part (and hence the overall coefficient of +\strong{m}) and \strong{k+2k} for the algebraic part, which is wrong, +since that will lead to the wrong overall coefficient \strong{+3} for +\strong{k}, and the 2-variable term \strong{mk} will not get treated +at all. The cure is to treat \strong{mk} as a variable itself, count the +number of tokens in each such product and start the removal process with +the largest. + +\subsubsection{The splitting algorithm} + +Write $s$ for a sign, one of \strong{+}, \strong{\textendash{}}, and +\strong{s} for the state of assembling a signed term; a signed term is +a string of one or more signs. Write $d$ for a digit, one of 0123456789, +and \strong{n} for the state of assembling a numeric term; a numeric term +is a signed term followed by a string of one or more digits. Write $v$ +for a variable, usually a letter from the roman alphabet but in principle +any single token that is not a sign or a digit, and \strong{a} for the +state of assembling an algebraic term; an algebraic term is a numeric term +followed by a string of one or more variables. Rather than referring to +a signed-term-assembling state, we shall (obviously) simply refer to a +\emph{signed state}, and similarly to a \emph{numeric state} and an \emph{algebraic +state}. + +\begin{table} +\noindent \centering{}\caption{\label{tab:Input-output-states}State transitions} +\medskip{} +\begin{tabular}{ccccc} +\cmidrule{2-5} + & Curr. state & Curr. token & Action & Next state\tabularnewline +\cmidrule{2-5} +1 & \strong{s} & $s$ & $Ts\to s'$; $T=s'$ & \strong{s}\tabularnewline +\cmidrule{2-5} +2 & \strong{s} & $d$ & $Td$ & \strong{n}\tabularnewline +\cmidrule{2-5} +3 & \strong{s} & $v$ & $Vv$; $T1v$ & \strong{a}\tabularnewline +\cmidrule{2-5} +4 & \strong{n} & $s$ & $\mathbf{N}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-5} +5 & \strong{n} & $d$ & $Td$ & \strong{n}\tabularnewline +\cmidrule{2-5} +6 & \strong{n} & $v$ & $Vv$; $Tv$ & \strong{a}\tabularnewline +\cmidrule{2-5} +7 & \strong{a} & $s$ & $\mathbf{V}V,$; $V=\textrm{Ø}$; $\mathbf{A}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-5} +8 & \strong{a} & $d$ & error & \strong{!!}\tabularnewline +\cmidrule{2-5} +9 & \strong{a} & $v$ & $Vv$; $Tv$ & \strong{a}\tabularnewline +\cmidrule{2-5} +\end{tabular} +\end{table} +We also want to record the variables in the extended sense of products +of same. Call a one-token variable a prime variable. Then in this desired +sense, a variable is a string of one or more prime variables. + +Let $\mathbf{E}$ be the initial expression. Let $\mathbf{A}$ be a container +for the algebraic part of $\mathbf{E}$; let $\mathbf{N}$ be a container +for the numeric part of $\mathbf{E}$; and let $\mathbf{V}$ be a container +for the extended variables in $\mathbf{E}$. Let $T$ be a container in +which to accumulate the current term, and $V$ a container in which to +accumulate the current extended variable (if any). Initially all these +containers are empty ($\textrm{Ø}$). + +We work through $\mathbf{E}$ token by token from the left. The table shows +the alternatives. +\begin{itemize} +\item Row 1. The current token is a sign $s$ and the system is in a signed state +\strong{s}. We append $s$ to the current term, $Ts$, then resolve the +juxtaposition of signs according to the familiar rules: $++\to+$, $--\to+$, +$+-\to-$, $-+\to-$, so that $T$ contains only the resolved sign $s'$. +The system remains in a signed state. +\item Row 2. The current token is a digit $d$ and the system is in a signed- +state \strong{s}. We append $d$ to the current term, $Td$ (which will +now consist of a sign and a digit), and the system shifts to a numeric +state \strong{n}. +\item Row 3. The current token is a prime variable $v$ and the system is in +a signed state \strong{s}. We start assembling an extended variable, +$Vv$, and append $1v$ to the current term, $T1v$, where the $1$ is +necessary as discussed earlier (and in any case `sign variable' is not +a recognised \emph{term} \textendash{} neither signed, numeric or algebraic). +The system shifts to an algebraic state \strong{a}. +\item Row 4. The current token is a sign $s$ and the system is in a numeric +state \strong{n}. The current term is a numeric term, a sign followed +by at least one digit, and is complete. We append it to the numeric part +$\mathbf{N}$ of $\mathbf{E}$, $\mathbf{N}T$, then initialise $T$ to +$s$. The system shifts to a signed state. +\item Row 5. The current token is a digit $d$ and the system is in a numeric +state \strong{n}. We append $d$ to the current term, $Td$, and remain +in a numeric state. +\item Row 6. The current token is a prime variable $v$ and the system is in +a numeric state \strong{n}. We start assembling a variable, $Vv$, and +also append $v$ to the current term, $Tv$. The system shifts to an algebraic +state \strong{a}. +\item Row 7. The current token is a sign $s$ and the system is in an algebraic +state \strong{a}. The current term is an algebraic term, a sign followed +by at least one digit followed by at least one prime variable, and is complete. +We append it to the algebraic part $\mathbf{A}$ of $\mathbf{E}$, $\mathbf{A}T$, +then initialise $T$ to $s$. We also append $V$, in which we have been +accumulating the (extended) variable, to $\mathbf{V}$, $\mathbf{V}V,$, +then empty $V$ in preparation for the next (extended) variable. Attention +is drawn to the comma following $V$ also appended to $\mathbf{V}$, so +that we can distinguish where one variable ends and the next begins. The +system shifts to a signed state. +\item Row 8. The current token is a digit $d$ and the system is in an algebraic +state \strong{a}. This situation should not arise.\emph{ }We don't write +$k2$; we write $2k$ \textendash{} number precedes variable. An error +is generated. +\item Row 9. The current token is a variable $v$ and the system is in an algebraic +state \strong{a}. We append $v$ to the current extended variable, $Vv$, +and also append $v$ to the current term, $Tv$. The system remains in +an algebraic state \strong{a}. +\end{itemize} +To get things under way, an initial plus sign is put in $T$, $T=+$, and +the system is set to the signed state \strong{s}. In order that \emph{all} +terms of $\mathbf{E}$ are recorded in either $\mathbf{N}$ or $\mathbf{A}$, +and all extended variables in $\mathbf{V}$, we append a plus sign to $\mathbf{E}$: +$\mathbf{E}+$. Since an expression doesn't end with a trailing sign (we +don't write, e.g., \textbf{\strong{\textbf{2m+k\textendash{}}}}), the +process necessarily terminates either in row 4 or row 7 with the final +term appended either to $\mathbf{N}$ or $\mathbf{A}$ and with $T=+$; +if it terminates in row 7, the final extended variable is appended to $\mathbf{V}$, +$\mathbf{V}V$ (and $V$ is emptied, although that hardly matters at this +point). + +\subsubsection{An enlarged scheme?} + +Row 8 of our table generates an error: a digit following a variable. But +having allowed products of variables like \texttt{mn} ($mn$), it is very +tempting to allow \texttt{mm}, i.e. \texttt{m\textasciicircum{}2} ($m^{2}$) +and, indeed, \texttt{m\textasciicircum{}n} ($m^{n}$). And if we allow +\texttt{m\textasciicircum{}2} and \texttt{m\textasciicircum{}n}, how can +we say no to subscripted forms like \texttt{k\_2} ($k_{2}$) and \texttt{k\_n} +($k_{n}$)? Or, for that matter, \texttt{k\_+} ($k_{+}$) and \texttt{k\_-} +($k_{-}$), and therefore \texttt{m\textasciicircum{}+} ($m^{+}$) and +\texttt{m\textasciicircum{}-} ($m^{-}$)? And having extended the scheme +in this way to exponents of \emph{variables}, surely it should also encompass +exponents of \emph{numbers}, not only an obvious case like \texttt{2\textasciicircum{}2} +($2^{2}$) but less obviously, yet still compellingly, \texttt{2\textasciicircum{}n} +($2^{n}$)? + +Each of these extensions produces its own problems, but they can all be +accommodated within an enlarged scheme, as can the use of parentheses (with +numerical coefficients). Table~\ref{tab:Input-output-states} translates +neatly into code. Rather than add these complications to \texttt{diffcoeff.sty}, +I have transferred the enlarged scheme to \texttt{diffcoeff.sty}'s `big +brother', \texttt{diffcoeffx.sty}. The comparable table and routine resulting +from it in \texttt{diffcoeffx.sty} is much bigger and less obvious than +in \texttt{diffcoeff.sty}. + +\subsubsection{Some code details} + +In the code, the states are distinguished by integers as indicated in Table~\ref{tab:State-integers}. +Tokens are assigned similar integer indexes, as indicated in the table. +The relevant routine is \texttt{\textbackslash{}\_\_diffco\_get\_curr\_index:NN}. +The actions embodied in Table~\ref{tab:Input-output-states} are encoded +in \texttt{\textbackslash{}\_\_diffco\_compare\_states:NNNNN} which is +a direct translation of the table into expl3 code. + +\begin{table}[h] +\caption{Some code details} + +\noindent \centering{}\subfloat[\label{tab:State-integers}State integers]{\centering{}% +\begin{tabular}{|c|c|c|} +\hline +State & Index & Tokens\tabularnewline +\hline +\hline +signed & 0 & $+$ $-$\tabularnewline +\hline +numeric & 1 & 0123456789\tabularnewline +\hline +algebraic & 2 & variables\tabularnewline +\hline +\end{tabular}}~~~\subfloat[Translations]{ +\centering{}% +\begin{tabular}{|c|c|} +\hline +Symbol & Code variable\tabularnewline +\hline +\hline +$s$,$d$,$v$ & \texttt{\textbackslash{}l\_\_diffco\_curr\_tok\_tl}\tabularnewline +\hline +$T$ & \texttt{\textbackslash{}l\_\_diffco\_curr\_term\_tl}\tabularnewline +\hline +$V$ & \texttt{\textbackslash{}l\_\_diffco\_curr\_var\_tl}\tabularnewline +\hline +$\mathbf{N}$ & \texttt{\textbackslash{}l\_\_diffco\_nos\_tl}\tabularnewline +\hline +$\mathbf{A}$ & \texttt{\textbackslash{}l\_\_diffco\_alg\_tl}\tabularnewline +\hline +$\mathbf{V}$ & \texttt{\textbackslash{}l\_\_diffco\_vars\_prop}\tabularnewline +\hline +\end{tabular}} +\end{table} +A property list is used to store the variables, organised by size \textendash{} +the number of tokens composing an extended variable. This enables the sorting +by size needed for the determination of the overall coefficients of variables +by removing them in turn from the algebraic part of the expression. That +process is conducted in the routine \texttt{\textbackslash{}\_\_diffco\_eval\_vars:NN}. +The variables are recorded only on the first scan through the order specification +expression. This is the function of the boolean \texttt{\textbackslash{}l\_\_diffco\_vars\_noted\_bool} +which is set in \texttt{\textbackslash{}\_\_diffco\_eval\_vars:NN}. Evaluation +of the numeric parts of expressions is provided by \texttt{\textbackslash{}\_\_diffco\_eval\_nos:N}. + +\section{Summary of main commands} + +\subsubsection{Ordinary derivatives} + +The syntax is +\begin{example} +{\small{}\textbackslash{}diff{[}order{]}\{differentiand\}\{variable\}\{point +of evaluation\}}{\small \par} +\end{example} +for the differentiand in the numerator, and where the final argument, although +using braces, is an \emph{optional} argument. A starred form appends the +differentiand: +\begin{example} +{\small{}\textbackslash{}diff{*}{[}order{]}\{differentiand\}\{variable\}\{point +of evaluation\}}{\small \par} +\end{example} +No space must occur between the final optional argument, if it is used, +and the second mandatory argument. + +There are also slash forms of both these commands: +\begin{example} +{\small{}\textbackslash{}diff{[}order{]}\{differentiand\}/\{variable\}\{point +of evaluation\}}{\small \par} + +{\small{}\textbackslash{}diff{*}{[}order{]}\{differentiand\}/\{variable\}\{point +of evaluation\}}{\small \par} +\end{example} +For the starred form, the differential coefficient is enclosed in parentheses. + +Precisely similar definitions, but without the slash forms, apply to \texttt{\textbackslash{}Diff}, +forming a differential coefficient with $D$, \texttt{\textbackslash{}diffd}, +forming a differential coefficient with $\delta$, and \texttt{\textbackslash{}Diffd}, +forming a differential coefficient with $\Delta$. + +\subsubsection{Partial derivatives} + +The syntax is +\begin{example} +{\small{}\textbackslash{}diffp{[}order spec.{]}{[}order override{]}\{differentiand\}\{variables\}\{constant +variables\}}{\small \par} +\end{example} +for the differentiand in the numerator and where the final argument, although +in braces, is an \emph{optional} argument. No space must occur between +the final optional argument, if it is used, and the second mandatory argument. +The \textbf{\strong{\textbf{order spec.}}} is a comma-separated list; +the \strong{variables} is also a comma-separated list. A starred form +appends the differentiand: +\begin{example} +{\small{}\textbackslash{}diffp{*}{[}order{]}{[}order~override{]}\{differentiand\}\{variables\}\{constant +variables\}}{\small \par} +\end{example} +Slash forms exist also for these commands: +\begin{example} +{\small{}\textbackslash{}diffp{[}order spec.{]}{[}order override{]}\{differentiand\}/\{variables\}\{constant +variables\}}{\small \par} + +{\small{}\textbackslash{}diffp{*}{[}order{]}{[}order~override{]}\{differentiand\}/\{variables\}\{constant +variables\}}{\small \par} +\end{example} +For the starred version of the slash form, the differential coefficient +is enclosed in parentheses. + +\subsubsection{Settings} +\begin{example} +\textbackslash{}diffset{[}option1=<value1>,option2=<value2>,...{]} +\end{example} +All numerical values should be integers (\texttt{diffcoeff} interprets +this in units of mu, 1/18 of an em). To return all options to default values, +write +\begin{example} +\textbackslash{}diffset +\end{example} +The options and defaults are +\begin{description} +\item [{\strong{roman = false}}] \textbf{\strong{\textbf{true}}} gives upright +(roman) \textbf{\strong{\textbf{d}}} and \textbf{\strong{\textbf{D}}} +\item [{\strong{d-delims = . |}}] delimiters which, when subscripted, indicate +the point of evaluation of an ordinary derivative +\item [{\strong{p-delims = ( )}}] delimiters which, when subscripted, indicate +variables held constant for partial derivatives +\item [{\strong{d-nudge = 0}}] adjustment for positioning the subscript to +the preceding delimiters +\item [{\strong{p-nudge = $-$6}}] adjustment for positioning the subscript +to the preceding delimiters +\item [{\strong{d-sep = 1}}] additional separation between the \textbf{$d$ +}and its superscript in the numerator of a second or higher order ordinary +derivative +\item [{\strong{p-sep = 1}}] additional separation between the \textbf{$\partial$} +and its superscript in the numerator of a second or higher order partial +derivative +\item [{\strong{sep = 2}}] additional separation between the terms in the denominator +of a mixed partial derivative +\end{description} + +\end{document} |