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diff --git a/Master/texmf-dist/doc/latex/diffcoeff/README.txt b/Master/texmf-dist/doc/latex/diffcoeff/README.txt new file mode 100644 index 00000000000..e69b3a12ff5 --- /dev/null +++ b/Master/texmf-dist/doc/latex/diffcoeff/README.txt @@ -0,0 +1,23 @@ +README file for LaTeX package: diffcoeff + +Author: Andrew Parsloe (aparsloe@clear.net.nz) 2016-06-27 + +diffcoeff.sty allows the easy writing of ordinary and partial differential +coefficients of arbitrary order. For mixed partial derivatives, the overall +order (algebraic or numeric) 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, \Delta and \delta and a +command for writing Jacobians. + +diffcoeffx.sty is diffcoeff.sty 'on steroids', extending that package in ways +that are probably unnecessary but have a certain logical forece. + +The packages require the LaTeX3 bundles, l3kernel and l3packages. + +There are six files plus this README: diffcoeff.tex, diffcoeff.pdf (compiled +from diffcoeff.tex), and diffcoeff.sty; and diffcoeffx.tex, diffcoeffx.pdf +and diffcoeffx.sty. The tex and pdf files should be placed in texmf/doc/ +latex/diff, and the sty files in texmf/tex/latex/diff.
\ No newline at end of file diff --git a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.pdf b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.pdf Binary files differnew file mode 100644 index 00000000000..e62bee30a39 --- /dev/null +++ b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.pdf 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} diff --git a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.pdf b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.pdf Binary files differnew file mode 100644 index 00000000000..83ee846ff98 --- /dev/null +++ b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.pdf diff --git a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.tex b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.tex new file mode 100644 index 00000000000..d7b48cac28d --- /dev/null +++ b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeffx.tex @@ -0,0 +1,656 @@ +%% 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[twoside,english]{article} +\usepackage{lmodern} +\renewcommand{\sfdefault}{lmss} +\renewcommand{\ttdefault}{lmtt} +\usepackage[T1]{fontenc} +\usepackage[latin9]{inputenc} +\usepackage{geometry} +\geometry{verbose,lmargin=4cm,rmargin=3.5cm} +\setcounter{secnumdepth}{2} +\setcounter{tocdepth}{1} +\usepackage{wrapfig} +\usepackage{booktabs} +\usepackage{amstext} +\usepackage{esint} + +\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{diffcoeffx} + +\@ifundefined{showcaptionsetup}{}{% + \PassOptionsToPackage{caption=false}{subfig}} +\usepackage{subfig} +\makeatother + +\usepackage{babel} +\begin{document} + +\title{\texttt{diffcoeffx}~\\ +extending the \texttt{diffcoeff} package} + +\author{Andrew Parsloe\\ +{\small{}(aparsloe@clear.net.nz)}} +\maketitle +\begin{abstract} +\noindent \texttt{diffcoeffx.sty} is \texttt{diffcoeff.sty} `on steroids'. +It provides additional functionality for the trailing optional argument +and extends the algorithm used to calculate the overall order of differentiation +of mixed partial derivatives. That now accepts order-of-differentiation +specifications that include powers of numbers and variables, subscripts +on variables, and (possibly nested) parentheses with numerical coefficients. +The enhancements come under the category of `gilding the lily'. +\end{abstract} + +\section{The \texttt{diffcoeffx} package} + +The \texttt{diffcoeffx} package is \texttt{diffcoeff} `on steroids', +providing exactly the same commands but with some extra functionality.\texttt{ }It +is called in the usual way in the LaTeX preamble: +\begin{lyxcode} +\textbackslash{}usepackage\{diffcoeffx\} +\end{lyxcode} +It is assumed that you are familiar with the \texttt{diffcoeff} package +and its manual.\texttt{ }There are two enhancements to that package: \texttt{diffcoeffx} +takes the calculation of the overall order of mixed partial derivatives +deep into `overkill' territory, accepting single-token powers of numbers +and variables, single-token subscripts on variables, and possibly nested +parentheses with numerical coefficients. The \texttt{\textbackslash{}times} +token ($\times$) can also be used in an order specification. The other +enhancement is an extension to the capabilities of the trailing optional +argument. + +\subsection[Exploiting the final argument]{Exploiting the trailing optional argument} + +For \texttt{diffcoeff }there was an attempt to give a `natural feel' +to the design choices made and their use. By comparison the additional +functionality that the trailing optional argument acquires in \texttt{diffcoeffx.sty} +is in the nature of a \emph{hack}. It works, but I'm not sure that it should +be encouraged. + +In \texttt{diffcoeff} if you write \texttt{\textbackslash{}diff yx\{\}} +the trailing but \emph{empty} optional argument is ignored. Not so in \texttt{diffcoeffx}: +\begin{example} +\textbackslash{}diffp yx\{\}${\displaystyle \Longrightarrow\quad\diffp yx{}}$ +\end{example} +The parentheses are inserted without a subscript. Thus we can write (for +instance) Lagrange's equations of motion in analytical mechanics in the +manner: +\begin{example} +\textbackslash{}diffp L\{q\_k\}-\textbackslash{}diff{*}\{\textbackslash{}diffp +L\{\textbackslash{}dot\{q\}\_k\}\{\}\}t = 0 $\Longrightarrow\quad{\displaystyle \diffp L{q_{k}}-\diff*{\diffp L{\dot{q}_{k}}{}}t}=0$, +\end{example} +without having to bother with inserting \texttt{\textbackslash{}left(} +and \texttt{\textbackslash{}right}).\texttt{ }The empty trailing optional +argument and the default delimiters for partial derivatives do the job +for us. + +There are many other places in analytical mechanics where using an empty +trailing optional argument is a similarly convenient way of writing large +parentheses, for instance, +\begin{example} +\textbackslash{}dot\{q\_k\}=\textbackslash{}diffp H\{\textbackslash{}diffp +S\{q\_k\}\{\}\} ${\displaystyle \Longrightarrow\quad\dot{q_{k}}=\diffp H{\diffp S{q_{k}}{}}}\!.$ +\end{example} +An application of Lagrange's equations (to a one-dimensional elastic solid) +gives rise to a Langrangian density function, +\begin{example} +\textbackslash{}frac 12\textbackslash{}left\textbackslash{}\{ \textbackslash{}rho\textbackslash{}dot\{\textbackslash{}eta\}\textasciicircum{}2-E\textbackslash{}diff\textbackslash{}eta +x\{;2;()\}\textbackslash{}right \textbackslash{}\} $\Longrightarrow\quad{\displaystyle \frac{1}{2}\left\{ \rho\dot{\eta}^{2}-E\diff\eta x{;2;()}\right\} }.$ +\end{example} +Another application of those equations (the acoustic approximation to the +irrotational motion of a compressible non-viscous fluid) produces a Lagrangian +density +\begin{example} +\textbackslash{}frac 12\textbackslash{}rho\textbackslash{}left\textbackslash{}\{(\textbackslash{}nabla\textbackslash{}psi)\textasciicircum{}2-\textbackslash{}frac +1\{c\textasciicircum{}2\}\textbackslash{}diff\textbackslash{}psi t\{;2;()\}\textbackslash{}right\textbackslash{}\} +$\Longrightarrow\quad{\displaystyle \frac{1}{2}\rho\left\{ (\nabla\psi)^{2}-\frac{1}{c^{2}}\diff\psi t{;2;()}\right\} }.$ +\end{example} +In both examples, the trailing optional argument of the \emph{ordinary} +derivative has been filled by a semicolon-delimited list: \texttt{\{;2;()\}}. +The initial slot where a subscript is specified is empty but the semicolon +is necessarily included. The second spot specifies a \emph{superscript} +and the third slot the delimiters to use. Since parentheses are not the +default delimiters for an ordinary derivative we needed to specify them +explicitly here. However, this does not change the default delimiters which +remain \texttt{.|} for an ordinary derivative and can only be changed by +means of the \texttt{\textbackslash{}diffset} command. + +Both subscript and superscript can be used at the same time. In a text +on ordinary differential equations, an example employing Green's functions +gives rise to +\begin{example} +\textbackslash{}diff{[}n-1{]}Gx\{\textbackslash{}xi-\textbackslash{}epsilon;\textbackslash{}xi+\textbackslash{}epsilon;{[}{]}\} +$\Longrightarrow\quad{\displaystyle \diff[n-1]Gx{\xi-\epsilon;\xi+\epsilon;[]}}$ +\end{example} +the derivative being evaluated at both superscript and subscript values +and the difference taken. Here the trailing optional argument has its first +three slots filled, with square brackets explicitly specified. The same +book includes the example +\begin{example} +\textbackslash{}diff{*}{[}p-1{]}\{x\textasciicircum{}\textbackslash{}alpha\}\textbackslash{}alpha\{\textbackslash{}alpha=a;;\textbackslash{}\{\textbackslash{}\}\} +$\Longrightarrow\quad{\displaystyle \diff*[p-1]{x^{\alpha}}\alpha{\alpha=a;;\{\}}}$ +\end{example} +where, this time braces are specified in the trailing optional argument.\footnote{For LyX users, the braces \textbackslash{}\{ and \textbackslash{}\} are +inserted into a formula in the maths editor simply by typing the braces +without the backslashes. LyX takes care of the latter.} + +This argument can be used to form the absolute value of a derivative, +\begin{example} +\textbackslash{}diff yx\{;;||\} ${\displaystyle \Longrightarrow\quad\diff yx{;;||}}$ +\end{example} +where both initial slots, subscript and superscript, are empty and two +semicolons necessarily included in the trailing optional argument: \texttt{\{;;||\}}. +It also provides an alternative way, indeed \emph{two} alternative ways, +of forming a quotient of derivatives: +\begin{example} +\textbackslash{}diff yx\{;;./\}\textbackslash{}diff xy=\textbackslash{}diff +yx\textbackslash{}diff xy\{;;/.\} ${\displaystyle {\displaystyle \Longrightarrow\quad\diff yx{;;./}}\diff xy={\displaystyle \diff yx}\diff xy{;;/.}}$ +\end{example} +where the delimiter specification \texttt{./} on the left has been changed +to \texttt{/.} on the right. The spacing in the two quotients is not quite +identical, which might be relevant in some contexts. As a more realistic +example of use of the same construct, if $F(x,t)$ is a function of $x$ +and $t$ and $x=x(t)$, then if $\diff Ft=0$, +\begin{example} +\textbackslash{}diff xt=-\textbackslash{}diffp Ft\{;;./\}\textbackslash{}diffp +Fx $\Longrightarrow\quad{\displaystyle \diff xt=-\diffp Ft{;;./}\diffp Fx}$ +\end{example} +For an inline use, you may prefer to use the slash form of the derivative +$\diff y/z{0;;();-1}$. In this case a \emph{fourth} slot in the trailing +optional argument has been filled, the \texttt{nudge override} slot, since +the default nudge is designed to position the subscript relative to the +\emph{displaystyle} delimiters. + +The complete specification of what is available in the trailing optional +argument is: +\begin{example} +\{ subscript; superscript; delimiters; nudge override \} +\end{example} +\begin{itemize} +\item In `normal' use, the \texttt{subscript} is the point of evaluation (ordinary +derivatives), or list of variables held constant (partial derivatives). +Since the list of variables held constant is likely to be comma-separated, +so we have the need for semicolons to separate items in the larger list. +\item The \texttt{superscript} is generally a power to which the derivative is +raised but, as instanced by the Green's function example, it can also be +another point of evaluation of the derivative. +\item The \texttt{delimiters} are, by default, \texttt{.|} for ordinary derivatives +and \texttt{()} for partial derivatives. These are not always the right +ones for a particular task. Rather than changing them \emph{globally} as +the use of \texttt{\textbackslash{}diffset} entails, they can be changed +\emph{locally} for the particular instance by specifying them in this slot. +The global choices are unaffected. +\item If the built-in placement of sub- or superscript relative to the right +delimiter is unsatisfactory, a value specified in the \texttt{nudge override} +slot\texttt{ }overrides the default value locally. The value is a pure +number which \texttt{diffcoeffx} treats as that number of mu (1/18 of an +em). (For comparison, a thin space \textbackslash{}, and a negative thin +space \textbackslash{}! are 3/18 of an em.) The default nudges are shown +in Table~\ref{tab:Default-nudges}. They are intended for displaystyle +presentation, and are not affected by any value included in this slot. +\end{itemize} +\noindent\begin{minipage}[t]{1\columnwidth}% +\begin{wraptable}[10]{o}{0.35\columnwidth}% +\centering{}\caption{\label{tab:Default-nudges}Default nudges} +\begin{tabular}{|c|c|} +\hline +right delimiter & nudge\tabularnewline +\hline +\hline +), > & -6\tabularnewline +\hline +\textbackslash{}\} & -4\tabularnewline +\hline +|, {]} & 0\tabularnewline +\hline +other & 0\tabularnewline +\hline +\end{tabular}\end{wraptable}% +Note that if one wants to use the nudge override with the default delimiters, +it is necessary to indicate all preceding slots, even if they are empty, +e.g., \texttt{\{;;;-3\}}. Similarly, to change the delimiters, to parentheses +say, without sub- or superscript, it is necessary to indicate all preceding +empty slots, but the following one does not need to be indicated: \texttt{\{;;()\}}. +If one wants to specify a superscript, 2 say, but leave all else unchanged, +it is only necessary to specify the one preceding empty slot: \texttt{\{;2\}}. +Trailing empty slots can be omitted, which is why, if one wants to use +the trailing empty argument `as nature intended', i.e., to specify a +point of evaluation or variables held constant, one can close one's mind +to the other potential slots and simply write (for instance) \texttt{\{0\}} +or \texttt{\{x=1\}}.% +\end{minipage} + +\subsection{The enhanced mixed partial derivative algorithm\label{subsec:The-enhanced-mixed}} + +In the documentation for \texttt{diffcoeff.sty} I discussed the transition +table, Table~\ref{tab:Input-output-states}, in which signed \strong{s}, +numeric \strong{n}, or algebraic \strong{a} states changed to one of +the others, or not, depending on the nature of the current token: sign, +digit or variable. Signs and digits were explicitly defined; anything and +everything else was called a (prime) variable. (Not quite true: in fact +\texttt{diffcoeff.sty} checked for \texttt{(}, \texttt{\textasciicircum{}} +and \texttt{\_} and raised an error if they were encountered.)\texttt{ } + +\begin{table}[h] +\noindent \begin{centering} +\caption{A first enhancement} +\subfloat[\label{tab:Input-output-states}State transitions]{\noindent \centering{}\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}} +\par\end{centering} +\noindent \centering{}\subfloat[\label{tab:Allowing-powers-variables}Allowing powers of variables]{\centering{}% +\begin{tabular}{ccccc} +\cmidrule{2-5} + & Curr. state & Curr. token & Action & Next state\tabularnewline +\cmidrule{2-5} +8 & \strong{a} & $d$ & $Vd$; $Td$ & \strong{a}\tabularnewline +\cmidrule{2-5} +\end{tabular}} +\end{table} +There is a certain inner logic at play here. Multi-token variables like +$kmn$ are included in the above scheme. But having accommodated $mn$, +surely one should be able to handle $mm$, i.e. $m^{2}$? And if $m^{2}$, +then why not $m^{n}$? In fact it is easy to do so. Since the superscript +token \textasciicircum{} is neither sign nor digit, no longer raise an +error if it is encountered but treat it, among the `everything else' +tokens, as a variable. If we change row 8 of the table as in Table~\ref{tab:Allowing-powers-variables} +we have enlarged our scheme to include powers of variables \textendash{} +not only numerical powers (row 8) but also algebraic powers (row 9). As +a side-effect, if we also suppress the raising of an error when the subscript +token \texttt{\_} is encountered, it too will be classified as a variable +and allow numeric and algebraic subscripts on variables: things like $k_{2}$ +or $k_{n}$. + +Implicit in this discussion is the understanding that exponents and subscripts +are restricted to \emph{single tokens}. Coping with multi-token quantities +in those positions would entail changes to other parts of the code, which +I have chosen not to do. + +This is a simple way of enlarging the range of tokens acceptable to the +overall-order algorithm, but it does assume that the user does \emph{not} +include a sign as a superscript or subscript. If they do, then when the +algorithm meets the sign it arrives at row 7 of the table and stores what +is clearly an unintended variable, something like \texttt{k\textasciicircum{}} +or \texttt{k\_}. So, we need to check when a sign is met whether the previous +token was one of \texttt{\textasciicircum{}} or \texttt{\_} and raise an +error if it was. But then the thought arises: if we are going to the trouble +of checking for sub- or superscript tokens, why just raise an error? Why +not incorporate signs in sub- or superscript positions into the scheme? + +To this end, we might introduce a fourth state, the \emph{script} state, +denoted by \strong{p}. A script token, denoted $p$, is one of \texttt{\textasciicircum{}} +or \texttt{\_}. There is only one way to enter a script state, and that +is by appending a script token to a \emph{variable}. Appending a script +token to a sign or number (or, indeed, another script token) raises an +error. Table~\ref{tab:Enlarged-scheme-transitios} is the result. In this +scheme, signs can be used as sub- or superscripts to variables, but not +to numbers. We might console ourselves with the thought that this is, in +any case, a limitation of the calculational engine used to evaluate our +integer expressions. The \texttt{l3int} module of the LaTeX3 bundle \texttt{l3kernel} +cannot handle powers of integers. + +\begin{table}[h] +\caption{\label{tab:Enlarged-scheme-transitios}Transition states for an enlarged +scheme} + +\noindent \centering{}% +\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{s} & $p$ & error & \strong{!!}\tabularnewline +\cmidrule{2-5} +5 & \strong{n} & $s$ & $\mathbf{N}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-5} +6 & \strong{n} & $d$ & $Td$ & \strong{n}\tabularnewline +\cmidrule{2-5} +7 & \strong{n} & $v$ & $Vv$; $Tv$ & \strong{a}\tabularnewline +\cmidrule{2-5} +8 & \strong{n} & $p$ & error & \strong{!!}\tabularnewline +\cmidrule{2-5} +9 & \strong{a} & $s$ & $\mathbf{V}V,$; $V=\textrm{Ø}$; $\mathbf{A}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-5} +10 & \strong{a} & $x\in\{dv\}$ & $Vx$; $Tx$ & \strong{a}\tabularnewline +\cmidrule{2-5} +11 & \strong{a} & $p$ & $Vp$; $Tp$ & \strong{p}\tabularnewline +\cmidrule{2-5} +12 & \strong{p} & $x\in\{sdv\}$ & $Vx$; $Tx$ & \strong{a}\tabularnewline +\cmidrule{2-5} +13 & \strong{p} & $p$ & error & \strong{!!}\tabularnewline +\cmidrule{2-5} +\end{tabular} +\end{table} +Yet this still leaves an unfinished feeling. While attaching a script token +to a sign or other script token is a nonmathematical usage, attaching a +superscript token to a number is a basic mathematical use, and so two of +the errors raised can really be ignored. For the other, the question nags: +why should we have to remember that although variables can be raised to +powers, numbers cannot be? The urge to enlarge the scheme again is irresistible. +Exponents on numbers should be accepted; but subscripts should not. The +latter is a nonmathematical usage or is used only in special contexts remote +from the present one. But that means we need to distinguish sub- and superscript +tokens. We can't lump them together as `script tokens'. + +\subsubsection{Raising numbers to powers: new states} + +So a first step is to enlarge the number of states. We need an \emph{exponent} +state \strong{e} when we encounter the token \textasciicircum{} and a +\emph{subscript }state \strong{b} when we encounter the token \_. That +allows us to distinguish acceptable forms like \texttt{2\textasciicircum{}3} +($2^{3}$) from unacceptable ones like \texttt{2\_3} ($2_{3}$). But how +do we know which state to transition to when we meet the \texttt{3} in +\texttt{2\textasciicircum{}3}? The current state is the exponent one \strong{e} +and the \texttt{3} could be decorating either a variable or a number. We +need to know the \emph{previous} state as well as the current one. If the +previous state is numeric we transition to a numeric state; if it is algebraic, +we transition to an algebraic state. + +But that also introduces a problem. It is perfectly acceptable to add a +digit to a term in a numeric state. Normally, this is how a multi-digit +number is accumulated: \texttt{23}4, two hundred and thirty four. That +is a very different meaning from \texttt{2\textasciicircum{}34} which means +$2^{3}4$ to us (rather than $2^{34}$ since we accept only single-token +superscripts). At this point, the syntax required by the underlying engine +used for evaluating numerical expressions comes into play. For all numerical +evaluations except those involving exponents, \texttt{l3int} of the LaTeX3 +kernel is used; for expressions involving exponents, \texttt{l3fp} is used. +To \texttt{l3fp}, \texttt{2\textasciicircum{}34} is read as $2^{34}$. +We need to insert a multiplication token between the \texttt{3} and \texttt{4}, +which for \texttt{l3fp} is the asterisk, \texttt{{*}}. Considering the +different tokens that might follow \emph{that}, we are forced to introduce +a third new state, the \emph{multiplicative} state, \strong{m}. So, to +introduce powers of numbers means considering three new states and reference +to the previous state. + +That, of course, is \emph{numeric} powers of numbers. To also allow algebraic +powers, forms like $2^{n}$, introduces further complication. These can't +be evaluated numerically, so presumably they are to be classified as variables. +We need to consider terms like $+2^{n}$, $3*2^{n}$, $3^{m}2^{n}$, and +$2^{n}m$. The problem here is that we have something that looks as if +it is going to be a number (the digit 2) but then transforms into a variable, +$2^{n}$. Do we need a \emph{fourth} new state, the entangled state \strong{q} +(the `q' as in `quantum entanglement')? + +\begin{table} +\noindent \centering{}\caption{\label{tab:Revised-input-output}State transitions of the full scheme} +\medskip{} +\begin{tabular}{cccccc} +\cmidrule{2-6} + & $S_{-}$ & $S$ & $t\in\left\{ sdv\text{\textasciicircum\_}*\right\} $ & Action & $S_{+}$\tabularnewline +\cmidrule{2-6} +1 & & \strong{s} & $s$ & $Ts\to s'$; $T=s'$ & \strong{s}\tabularnewline +\cmidrule{2-6} +2 & & \strong{s} & $d$ & $Qd$; $Td$ & \strong{n}\tabularnewline +\cmidrule{2-6} +3 & & \strong{s} & $v$ & $Vv$; $T1v$ & \strong{a}\tabularnewline +\cmidrule{2-6} +4 & & \strong{n} & $s$ & $Q=\textrm{Ø}$; $\mathbf{N}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-6} +5 & & \strong{n} & $d$ & $Qd$; $Td$ & \strong{n}\tabularnewline +\cmidrule{2-6} +6 & & \strong{n} & $v$ & $Q=\textrm{Ø}$; $Vv$; $Tv$ & \strong{a}\tabularnewline +\cmidrule{2-6} +7 & & \strong{n} & $\text{\textasciicircum}$ & $Q\text{\textasciicircum}$; $T\text{\textasciicircum}$ & \strong{e}\tabularnewline +\cmidrule{2-6} +8 & & \strong{n} & $*$ & $Q=\textrm{Ø}$; $T*$ & \strong{m}\tabularnewline +\cmidrule{2-6} +9 & & \strong{a} & $s$ & $\mathbf{V}V,$; $V=\textrm{Ø}$; $\mathbf{A}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-6} +10 & \strong{e} & \strong{a} & $d$ & $Vd$; $Td$ & \strong{a}\tabularnewline +\cmidrule{2-6} +11 & & \strong{a} & $v$ & $Vv$; $Tv$ & \strong{a}\tabularnewline +\cmidrule{2-6} +12 & & \strong{a} & $\text{\textasciicircum}$ & $V\text{\textasciicircum}$; $T\text{\textasciicircum}$ & \strong{e}\tabularnewline +\cmidrule{2-6} +13 & & \strong{a} & $\text{\_}$ & $V\text{\_}$; $T\text{\_}$ & \strong{b}\tabularnewline +\cmidrule{2-6} +14 & \strong{a} & \strong{e} & $t\in\left\{ sdv\right\} $ & $Vt$; $Tt$ & \strong{a}\tabularnewline +\cmidrule{2-6} +15 & \strong{n} & \strong{e} & $d$ & $Q=\textrm{Ø}$; $Td*$ & \strong{m}\tabularnewline +\cmidrule{2-6} +16 & \strong{n} & \strong{e} & $v$ & $Qv$; $Tv$; $V=Q$; $Q=\textrm{Ø}$ & \strong{a}\tabularnewline +\cmidrule{2-6} +17 & \strong{a} & \strong{b} & $t\in\left\{ sdv\right\} $ & $Vt$; $Tt$ & \strong{a}\tabularnewline +\cmidrule{2-6} +18 & \strong{e} & \strong{m} & $s$ & $T1$; $\mathbf{N}T$; $T=s$ & \strong{s}\tabularnewline +\cmidrule{2-6} +19 & & \strong{m} & $d$ & $Qd$; $Td$ & \strong{n}\tabularnewline +\cmidrule{2-6} +20 & & \strong{m} & $v$ & $Vv$; $T1v$ & \strong{a}\tabularnewline +\cmidrule{2-6} +21 & & \strong{m} & $*$ & & \strong{m}\tabularnewline +\cmidrule{2-6} +\end{tabular} +\end{table} +In fact I find that these problems can all be dealt with not by creating +another state but by including another accumulator for \emph{potential} +variables. I'll call it $Q$ (from the quantum suggestion, or perhaps quasi-variable, +or even acqumulator). It stores numbers whose status has not been determined +yet: they might yet be followed by a superscript token which might in turn +be followed by a variable. Once resolved, $Q$ either transfers its contents +to $V$, the variable accumulator, and is emptied (row 16), or is emptied +forthwith (rows 4, 6, 8, 15). + +Table~\ref{tab:Revised-input-output} lists the transitions. I've denoted +the previous state by $S_{-}$, the present state by $S$, and the next +state by $S_{+}$. The final row of the table is intended: do nothing if +we meet a multiplicative token when in a multiplicative state. The first +scan through an order specification (to split it into numeric and algebraic +parts) may introduce a {*} token (rows 8 and 15). We don't want to introduce +a second such token in the recursive determination of the coefficients +of variables. Hence row 21: do nothing. Also, if in the order specification +we have something like $2^{3}*3^{2}$ (since \texttt{2\textasciicircum{}33\textasciicircum{}2} +looks weird), we don't want the manually inserted {*} to cause an error +because of the automatically inserted one (row 15). + +Possibilities not explicitly present in the table generally raise an error, +e.g. current state \strong{s} and current token $\textnormal{\textasciicircum}$, +or previous state \strong{n}, current state \strong{e} and current token +$s$ ($+$ or $-$), and so on. I have omitted them from the table in the +interests of space. The table is big enough already. + +With this table of transitions it is now possible to handle order specifications +that include components like $n^{2}$ or $n^{m}$ or $k^{+}$ or $k_{2}$ +or $k_{n}$ or $2^{2}$ or $2^{3}3^{2}$ or $2\times3^{n}$ or $2^{2}3^{n}$ +or \ldots{} + +Note that the \texttt{\textbackslash{}times} token is converted internally +by \texttt{diffcoeffx.sty} to the asterisk. They can be used interchangeably +but it certainly looks more elegant. + +So, what could be better on a cool winter's evening, snug before the warmth +of the fire, a glass of sustaining liquid to hand, than to do a few mixed +partial derivatives? Like this, +\begin{example} +\textbackslash{}diffp{[}3\textasciicircum{}22\textasciicircum{}22\textasciicircum{}n+m,12\textasciicircum{}n-3m+2\textasciicircum{}3k,5m+2\textbackslash{}times2\textasciicircum{}32\textasciicircum{}n{]}\{F(x,y,z)\}\{x,y,z\}$\Longrightarrow\quad{\displaystyle \diffp[3^{2}2^{2}2^{n}+m,12^{n}-3m+2^{3}k,5m+2\times2^{3}2^{n}]{F(x,y,z)}{x,y,z}}$ +\end{example} +or like this, +\begin{example} +\textbackslash{}diffp{[}k\textasciicircum{}+k\_-+1,2\textbackslash{}times +k\_-,3\textasciicircum{}2k\_-,3k\textasciicircum{}+{]}\{F(x,y,z,w)\}\{x,y,z,w\} +${\displaystyle \Longrightarrow\quad\diffp[k^{+}k_{-}+1,2\times k_{-},3^{2}k_{-},3k^{+}]{F(x,y,z,w)}{x,y,z,w}}$ +\end{example} +In the first example the \texttt{\textbackslash{}times} symbol is inserted +by \texttt{diffcoeffx} in the overall order of differentiation in the numerator +so as to prevent the formation $522^{n}$ which would be read as 522 raised +to the power $n$ \textendash{} and for a similar reason it was used in +specifying the order of differentiation of the variable $z$ in the denominator +in the first example, but could and should have been deleted from the order +of differentiation of the variable $y$ in the second example. + +\subsubsection{Parentheses} + +The other major shortcoming of the basic scheme outlined in \texttt{diffcoeff.sty} +was the inability to handle even the simplest instance of parentheses in +an order specification \textendash{} something like\texttt{ {[}m-(n-1),m+(n-1){]}} +which might well arise in a Taylor expansion. Indeed, there is more reason +for including these in our scheme than exponents of numbers or $+$ or +$-$ as sub- or superscripts. + +How might we fit parentheses to the scheme? We are not seeking a general +treatment. Rather we wish to be able to handle order specifications a little +more complicated (but only a little) than the one just given, say something +like \texttt{{[}m+2(n-1),m-(n-1){]}}, perhaps with nesting. In that case +the following stipulations meet our needs: +\begin{itemize} +\item a left parenthesis, (, either starts an item in the comma list, or is preceded +by a sign or a number or $*$ or (, but \emph{not} by a variable or \textasciicircum{} +or \_ or ); +\item a right parenthesis, ), either concludes an item in the comma list, or +is followed by a sign or ), but \emph{not} by a number or a variable or +\textasciicircum{} or \_ or $*$ or (. +\end{itemize} +These limitations allow nesting of parentheses but not products of parentheses. +The main limitation they impose is that a variable lie \emph{within} parentheses +but not adjoining-outside. They enable us to get away with the following +`cheap and cheerful' scheme. It means we do not need to add parenthesis +states to our scheme. The particular point to note are the $+0$ insertions. +When we start parsing an expression from the left we do not know what it +contains. In particular when we meet a left parenthesis, we have no foreknowledge +of whether the parenthesised expression will be numeric, algebraic or a +mix of both. We need to prepare for both by inserting a left parenthesis +to both numeric and algebraic parts. But that brings us up against a quirk +of \texttt{l3int}, the `engine' behind the numerical evaluations performed +in \texttt{diffcoeff} and \texttt{diffcoeffx}. \texttt{l3int} objects to +an empty pair of parentheses, \texttt{()}, which we would have should either +numeric or algebraic parts be missing from the parenthesised expression. +To avoid this we insert $+0$ and \texttt{l3int} is happy.\texttt{ } + +\begin{table} +\centering{}\caption{\label{tab:Parentheses}Parentheses} +\begin{tabular}{ccccc} +\cmidrule{2-5} + & Curr. state & Curr. token & Action & Next state\tabularnewline +\cmidrule{2-5} +1 & \strong{s} & ( & $T\text{1*(}$; $\mathbf{N}T$; $\mathbf{A}T$; $T=+$ & \strong{s}\tabularnewline +\cmidrule{2-5} +2 & \strong{s} & ) & $\mathbf{N})$; $\mathbf{A})$ & \strong{s}\tabularnewline +\cmidrule{2-5} +3 & \strong{n} & ( & $T*($; $\mathbf{N}T$; $\mathbf{A}T$; $T=+$ & \strong{s}\tabularnewline +\cmidrule{2-5} +4 & \strong{n} & ) & $T)$; $\mathbf{N}T$; $\mathbf{A}\text{+0)}$; $T=+$ & \strong{s}\tabularnewline +\cmidrule{2-5} +5 & \strong{a} & ) & $\mathbf{V}V,$; $V=\textrm{Ø}$; $\mathbf{N}\text{+0)}$; $T)$; $\mathbf{A}T$ +; $T=+$ & \strong{s}\tabularnewline +\cmidrule{2-5} +6 & \strong{m} & ( & $T($; $\mathbf{N}T$; $\mathbf{A}T$; $T=+$ & \strong{s}\tabularnewline +\cmidrule{2-5} +\end{tabular} +\end{table} + +\begin{itemize} +\item Row 1. Quirks of the \texttt{l3int} module of the \LaTeX{}3 kernel mean +we need to insert \strong{1{*}} before the left parenthesis.\footnote{Specifically, \texttt{\textbackslash{}int\_eval\{}\textbf{ }\texttt{\textendash (} +or \texttt{\textbackslash{}int\_eval\{}\textbf{ }\texttt{+(} throw errors.} Note that we add $T$ to \emph{both} the numeric and algebraic parts of +the expression. We are working through our expression $\mathbf{E}$ from +the left, token by token, and have no foreknowledge of what the parenthesised +expression contains, whether algebraic terms only or numeric terms only +or some combination of both. Hence the need to prepare for both. The system +shifts to a signed state \strong{s} with $T=+$, exactly the same as when +beginning to scan $\mathbf{E}$. After all, the parenthesised expression +is an expression in itself. +\item Row 2. This is to allow nested parentheses like )). It shouldn't arise +otherwise. Because of rows 4 and 5, the first right parenthesis puts the +system into a signed state. The current term will be $T=+$, but we ignore +it and store only a right parenthesis in both numeric and algebraic parts. +\item Row 3. We already have a number present in $T$; only the asterisk needs +inserting before the parenthesis. Again we add $T$ to \emph{both} the +numeric and algebraic parts of the expression, initialise $T$ to $+$ +and change the state to a signed one. +\item Row 4. We are in a numeric state. We append ) to the current term and the +current term to the numeric part of the expression. We append $+0)$ to +the algebraic part, and shift to a signed state \strong{s} with $T=+$, +as at the outset. The $+0)$ in the algebraic part is necessary to prevent +an empty parenthesis pair in $\mathbf{A}$ should the parenthesised expression +have contained \emph{no} algebraic term. +\item Row 5. We are in an algebraic state. We append ) to the current term and +the current term to the algebraic part of the expression. We append $+0)$ +to the numeric part and shift to the initial signed state again. The $+0)$ +in the numeric part is necessary to prevent an empty parenthesis pair in +$\mathbf{N}$ should the parenthesised expression have contained \emph{no} +numeric term. +\item Row 6. We are in the new state, the multiplicative state, and the current +token is a left parenthesis. We have already met and inserted an asterisk +(row 5); we don't need to insert another. We append ( to $T$, $T$ to +both numeric and algebraic parts and shift to the initial signed state +again. The use of the multiplicative state prevents a string of asterisks +arising (but we have been able to avoid introducing new states for left +and right parentheses). +\end{itemize} +To work through an example, suppose we have an order specificiation \strong{{[}m+(n\textendash (k\textendash 1)),m\textendash{} 2(n+(k\textendash 1)),k{]}}. +(Digit \strong{1} rather than a lower-case letter \strong{l} within the +parentheses!) Concatenating, with linking + signs gives \strong{m+(n\textendash (k-1))+m\textendash 2(n+(k\textendash 1))+k}. +Splitting into numeric and algebraic parts now results in \strong{+1{*}(\textendash 1{*}(\textendash 1))\textendash 2{*}(+1{*}(\textendash 1))} +for the numeric part, evaluating to \strong{3}, and \strong{+1m+1{*}(+1n\textendash 1{*}(+1k+0))+1m\textendash 2{*}(+1n+1{*}(+1k+0))+1k} +for the algebraic part. + +Removing \strong{m} from the latter and splitting into numeric and algebraic +parts gives \strong{+1+ 1{*}(\textendash 1{*}(+0))+1\textendash 2{*}(+1{*}(+0))} +for the numeric part, evaluating to \strong{2} which is the overall coefficient +of \strong{m}, and \strong{+1{*}(+1n\textendash 1{*}(+1k+0))\textendash 2{*}(+1n+1{*}(+1k+0))+1k} +for the algebraic part. + +Now remove \strong{n} from this resulting algebraic part and again split +into parts. The result is \strong{+1{*}(+1-1{*}(+0))\textendash 2{*}(+1+1{*}(+0))} +for the numeric part, evaluating to \strong{\textendash 1} which is the +overall coefficient of \strong{n}, and \strong{+1{*}(\textendash 1{*}(+1k+0))\textendash 2{*}(+1{*}(+1k+0))+1k} +for the algebraic part. + +Removing \strong{k} from this and splitting gives \strong{+1{*}(\textendash 1{*}(+1+0))\textendash 2{*}(+1{*}(+1+0))+1} +for the numeric part, evaluating to \strong{\textendash 2} which is the +overall coefficient of \strong{k}, and \strong{+1{*}(\textendash 1{*}(+0)) \textendash 2{*}(+1{*}(+0))} +for the algebraic part. But we have run out of variables and so the process +stops at this point: +\begin{example} +\textbackslash{}diffp{[}m+(n-(k-1)),m-2(n+(k-1)),k{]}F\{x,y,z\}$\Longrightarrow{\displaystyle \hspace*{1em}\diffp[m+(n-(k-1)),m-2(n+(k-1)),k]F{x,y,z}}$ +\end{example} + +\end{document} diff --git a/Master/texmf-dist/tex/latex/diffcoeff/diffcoeff.sty b/Master/texmf-dist/tex/latex/diffcoeff/diffcoeff.sty new file mode 100644 index 00000000000..bb18b85d6fa --- /dev/null +++ b/Master/texmf-dist/tex/latex/diffcoeff/diffcoeff.sty @@ -0,0 +1,711 @@ +% This is file `diffcoeff.sty'. +% +% This program is free software; you can redistribute it and/or modify it +% under the terms of the GNU General Public License as published by the +% Free Software Foundation; either version 2 of the License, or (at your +% option) any later version. +% +% Andrew Parsloe aparsloe@clear.net.nz +% +\RequirePackage{expl3} +\RequirePackage{xparse} +\ProvidesExplPackage{diffcoeff} {2016/06/27} {1.0} + {Write differential coefficients easily.} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\cs_generate_variant:Nn \tl_if_eq:nnF { no } +\cs_generate_variant:Nn \tl_if_eq:nnT { no } +\cs_generate_variant:Nn \tl_if_in:NnTF { NV } +\cs_generate_variant:Nn \prop_pop:NnNT { NV } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% variables +\tl_new:N \g_diffco_d_ldelim_tl +\tl_new:N \g_diffco_d_rdelim_tl +\tl_new:N \g_diffco_p_ldelim_tl +\tl_new:N \g_diffco_p_rdelim_tl + +\tl_new:N \l__diffco_entiand_tl +\tl_new:N \l__diffco_phantom_tl +\tl_new:N \g_diffco_dop_tl +\tl_new:N \l__diffco_denom_tl +\tl_new:N \l__diffco_coeff_tl + +\clist_new:N \l__diffco_order_clist +\tl_new:N \l__diffco_order_spec_tl + +\bool_new:N \g__diffco_erent_shape_bool +\bool_new:N \l__diffco_no_parens_bool +\bool_new:N \l__diffco_override_bool +\bool_new:N \l__diffco_vars_noted_bool + +\tl_new:N \l__diffco_orders_in_tl +\tl_new:N \l__diffco_order_tl + +\tl_new:N \l__diffco_curr_tok_tl +\tl_new:N \l__diffco_curr_term_tl + +\int_new:N \l__diffco_curr_tok_int +\int_new:N \l__diffco_curr_state_int + +\prop_new:N \l__diffco_vars_prop +\tl_new:N \l__diffco_vars_tl +\clist_new:N \l__diffco_vars_clist + +\tl_new:N \l__diffco_curr_var_tl +\tl_new:N \l__diffco_nos_tl +\tl_new:N \l__diffco_nos_aux_tl + +\tl_new:N \l__diffco_alg_tl +\tl_new:N \l__diffco_alg_aux_tl +\tl_new:N \l__diffco_alg_expr_tl + +\tl_const:Nn \c__diffco_digits_tl { 1234567890 } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\keys_define:nn { diffcoeff } + { + roman .bool_gset:N = \g_diffco_roman_bool, + roman .default:n = { false }, + d-delims .code:n = { + \tl_gset:Nx \g_diffco_d_ldelim_tl { \left \use_i:nn #1 } + \tl_gset:Nx \g_diffco_d_rdelim_tl + { \right \use_ii:nn #1 } + }, + d-delims .default:n = { .| }, + p-delims .code:n = { + \tl_gset:Nx \g_diffco_p_ldelim_tl { \left \use_i:nn #1 } + \tl_gset:Nx \g_diffco_p_rdelim_tl + { \right \use_ii:nn #1 } + }, + p-delims .default:n = { () }, + d-nudge .code:n = { \tl_gset:Nn \g_diffco_d_nudge_tl { \mskip #1 mu } }, + d-nudge .default:n = { 0 }, + p-nudge .code:n = { \tl_gset:Nn \g_diffco_p_nudge_tl { \mskip #1 mu } }, + p-nudge .default:n = { -6 }, + sep .code:n = { \tl_gset:Nn \g_diffco_sep_tl { \mskip #1 mu } }, + sep .default:n = { 2 }, + d-sep .code:n = { \tl_gset:Nn \g_diffco_d_sep_tl { \mskip #1 mu } }, + d-sep .default:n = { 1 }, + p-sep .code:n = { \tl_gset:Nn \g_diffco_p_sep_tl { \mskip #1 mu } }, + p-sep .default:n = { 1 }, + } + +\DeclareDocumentCommand \diffset { o } + { + \IfValueTF { #1 } + { + \clist_if_empty:nTF { #1 } + { \keys_set:nn { diffcoeff } { roman,d-delims,p-delims, + d-nudge,p-nudge,sep,d-sep,p-sep } } + { \keys_set:nn { diffcoeff } { #1 } } + } + { + \keys_set:nn { diffcoeff } { roman,d-delims,p-delims, + d-nudge,p-nudge,sep,d-sep,p-sep } + } + } + +% now set the defaults +\diffset + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Derivatives +% Ordinary differential coefficient. #1 is the order; #2 the +% differentiand; #3 the variable of differentiation; #4 the +% point of evaluation. +\NewDocumentCommand \diff{ s o m t/ m g } + { + \group_begin: + % roman or italic? + \bool_if:NF \g__diffco_erent_shape_bool + { \__diffco_shape:N d } + + \IfValueTF { #2 } + { + \tl_set:Nn \l__diffco_order_tl { ^{ \g_diffco_d_sep_tl #2 } } + \int_compare:nNnTF { \tl_count:n { #5 } } > { \c_one } + { \tl_set:Nn \l__diffco_denom_tl { (#5)^{ #2 } } } + { \tl_set:Nn \l__diffco_denom_tl { #5^{ #2 } } } + } + { + \tl_set:Nn \l__diffco_order_tl { } + \tl_if_in:nnTF { #5 } { ^ } + { \tl_set:Nn \l__diffco_denom_tl { (#5) } } + { \tl_set:Nn \l__diffco_denom_tl { #5 } } + } + +% position of differentiand: appended (star) or numerator (no star); +% slash or fraction? + \IfBooleanTF { #1 } + { % differentiand follows diff coeff + \__diffco_phantom:n { #3 } + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #4 } + { % slash (therefore no phantom) + (\g_diffco_dop_tl \l__diffco_order_tl + / + \g_diffco_dop_tl \l__diffco_denom_tl ) + \l__diffco_entiand_tl + } + { % frac + \frac { \g_diffco_dop_tl \l__diffco_order_tl \l__diffco_phantom_tl } + { \g_diffco_dop_tl \l__diffco_denom_tl } \l__diffco_entiand_tl + } + } + } + { % differentiand in numerator + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #4 } + { % slash + \g_diffco_dop_tl \l__diffco_order_tl #3 + / + \g_diffco_dop_tl \l__diffco_denom_tl + } + { % frac + \frac { \g_diffco_dop_tl \l__diffco_order_tl #3 } + { \g_diffco_dop_tl \l__diffco_denom_tl } + } + } + } + % pt of eval + \IfNoValueTF { #6 } + { + \l__diffco_coeff_tl + } + { % trailing arg: pt of eval. + \tl_if_empty:nTF { #6 } + { + \l__diffco_coeff_tl + } + { + \g_diffco_d_ldelim_tl \l__diffco_coeff_tl \g_diffco_d_rdelim_tl + \c_math_subscript_token { \g_diffco_d_nudge_tl #6 } + } + } + \group_end: + } + +\NewDocumentCommand \Diff { o m m g } + { + \group_begin: + \__diffco_shape:N D + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\NewDocumentCommand \diffd { o m m g } + { + \group_begin: + \__diffco_shape:N \delta + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\NewDocumentCommand \Diffd { o m m g } + { + \group_begin: + \__diffco_shape:N \Delta + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\cs_new_nopar:Npn \__diffco_shape:N #1 + { + \bool_if:NTF \g_diffco_roman_bool + { \tl_gset:Nn \g_diffco_dop_tl { \mathrm{ #1} } } + { \tl_gset:Nn \g_diffco_dop_tl { #1 } } + } +%%%%% +\cs_new:Npn \__diffco_append:nn #1#2 + { + \g_diffco_d_ldelim_tl \l__diffco_coeff_tl \g_diffco_d_rdelim_tl + \c_math_subscript_token { \mskip #1 mu #2 } + } + +\cs_new:Npn \__diffco_phantom:n #1 + { + \bool_if:nTF + { + \tl_if_head_eq_meaning_p:nN { #1 } \hfill || + \tl_if_head_eq_meaning_p:nN { #1 } \hfil + } + { + \l__diffco_phantom_aux:nn #1\q_stop + } + { + \bool_if:nTF + { + \tl_if_head_eq_meaning_p:nN { #1 } \hspace || + \tl_if_head_eq_meaning_p:nN { #1 } \hphantom + } + { + \l__diffco_phantom_aux:nnn #1\q_stop + } + { + \tl_set:Nn \l__diffco_entiand_tl { #1 } + } + } + } + +\cs_new:Npn \l__diffco_phantom_aux:nn #1#2\q_stop + { + \tl_set:Nn \l__diffco_phantom_tl { #1 } + \tl_set:Nn \l__diffco_entiand_tl { #2 } + } + +\cs_new:Npn \l__diffco_phantom_aux:nnn #1#2#3\q_stop + { + \tl_set:Nn \l__diffco_phantom_tl { #1 { #2 } } + \tl_set:Nn \l__diffco_entiand_tl { #3 } + } +%%%%%%%%%%%%%%%%%%%% +% Partial differential coefficient. +% #1 (star) = append differentiand boolean +% #2 = comma list of the orders of diff. in each var.; +% #3 = order override for the numerator; #4 = differentiand; +% #5 slash / boolean; #6 = comma list of vars of diff.; +% #7 = list of vars held const. +\NewDocumentCommand \diffp{ s o o m t/ m g } + { + \group_begin: + + % diff. order list + \tl_clear:N \l__diffco_order_clist + \IfValueT { #2 } + { + \clist_set:Nn \l__diffco_order_clist { #2 } + \tl_set:Nx \l__diffco_order_spec_tl + { [\clist_use:Nn \l__diffco_order_clist { , }]} + } + + % order list override + \IfValueT { #3 } + { + \bool_set_true:N \l__diffco_override_bool + \tl_set:Nn \l__diffco_order_tl { #3 } + } + + \__diffco_orders:Nn \l__diffco_order_clist { #6 } + + % position of differentiand + \IfBooleanTF { #1 } + { % appended + \__diffco_phantom:n { #4 } + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #5 } + { % slash (therefore no phantom) + ( \partial \__diffco_omit_order_if_i:N + \l__diffco_order_tl + / + \l__diffco_denom_tl ) \l__diffco_entiand_tl + } + { % frac + \frac { \partial \__diffco_omit_order_if_i:N + \l__diffco_order_tl \l__diffco_phantom_tl } + { \l__diffco_denom_tl } \l__diffco_entiand_tl + } + } + } + { % in numerator + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #5 } + { + \partial \__diffco_omit_order_if_i:N \l__diffco_order_tl #4 + / \l__diffco_denom_tl + } + { + \frac { \partial + \__diffco_omit_order_if_i:N \l__diffco_order_tl + #4 } { \l__diffco_denom_tl } + } + } + } + + \IfNoValueTF { #7 } + { + \l__diffco_coeff_tl + } + { + \tl_if_empty:nTF { #7 } + { + \l__diffco_coeff_tl + } + { + \g_diffco_p_ldelim_tl \l__diffco_coeff_tl \g_diffco_p_rdelim_tl + \c_math_subscript_token { \g_diffco_p_nudge_tl #7 } + } + } + \group_end: + } % end of \diffp + +\NewDocumentCommand \jacob { m m } + { % need extra braces in denom for clist manipulations + \diffp{ (#1) }{ { {{(#2)}} } } + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% #1 is the clist of diff. orders +% #2 is the tl of variables (if commas present treat as clist) +\cs_new:Npn \__diffco_orders:Nn #1#2 + { + \int_zero:N \l_tmpa_int + \tl_clear:N \l_tmpb_tl + \clist_clear:N \l_tmpa_clist + \clist_clear:N \l_tmpb_clist + \clist_clear:N \l__diffco_vars_clist + + \clist_set:Nn \l__diffco_vars_clist { #2 } + \clist_map_inline:Nn \l__diffco_vars_clist + { + \bool_set_false:N \l__diffco_no_parens_bool + + % get order of each variable + \clist_pop:NNF #1 \l_tmpa_tl + { \tl_set:Nn \l_tmpa_tl { 1 } } + + \tl_if_eq:noT { 1 } \l_tmpa_tl + { \bool_set_true:N \l__diffco_no_parens_bool } + + \clist_put_right:NV \l_tmpa_clist \l_tmpa_tl + + % create the denominator + \tl_clear:N \l_tmpb_tl + \bool_if:nTF + { + \int_compare_p:nNn { \tl_count:n { ##1 } } = { \c_one } + || + \l__diffco_no_parens_bool + } + { \tl_put_right:Nn \l_tmpb_tl { \partial ##1 } } + { \tl_put_right:Nn \l_tmpb_tl { \partial (##1) } } + + \tl_if_eq:noF { 1 } \l_tmpa_tl + { + \tl_put_right:Nn \l_tmpb_tl { ^ } + \tl_put_right:Nx \l_tmpb_tl { { \l_tmpa_tl } } + } + + \clist_put_right:No \l_tmpb_clist \l_tmpb_tl + } % end of inline map + + % numerator order + \bool_if:NF \l__diffco_override_bool + { + \tl_set:Nx \l__diffco_orders_in_tl { \clist_use:Nn \l_tmpa_clist {+} } + \diff_simplify_orders:NN \l__diffco_orders_in_tl \l__diffco_order_tl + } + + % denominator + \tl_set:Nx \l__diffco_denom_tl { \clist_use:Nn + \l_tmpb_clist { \g_diffco_sep_tl } } + } + +% don't display order 1 superscript +\cs_new:Npn \__diffco_omit_order_if_i:N #1 + { + \tl_if_eq:noF { 1 } #1 + { ^{ \g_diffco_p_sep_tl \l__diffco_order_tl } } + } +%%%%%%%%%%%%%%%%%%%%%% +% #1 is expr. in; #2 is expr. out +\cs_new:Npn \diff_simplify_orders:NN #1 #2 + { + \tl_clear:N \l__diffco_nos_tl + + \__diffco_digest_expr:NNNN \l__diffco_orders_in_tl \l__diffco_nos_tl + \l__diffco_alg_tl \l__diffco_vars_prop + + \__diffco_eval_nos:N \l__diffco_nos_tl + + \tl_if_empty:NF \l__diffco_alg_tl + { \__diffco_eval_vars:NN \l__diffco_alg_tl \l__diffco_vars_prop } + + \__diffco_manicure_result:NNN \l__diffco_nos_tl \l__diffco_alg_tl #2 + } + +% #1 is the expression; #2 (tl) receives the numerical part; +% #3 (tl) receives the algebraic part; #4 (prop) receives the vars +\cs_new:Npn \__diffco_digest_expr:NNNN #1#2#3#4 + { + \tl_put_right:Nn #1 { + } + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set:Nn \l__diffco_curr_state_int { \c_zero } + \int_set:Nn \l__diffco_curr_tok_int { \c_zero } + \tl_clear:N #3 + \prop_clear:N #4 + + \tl_map_inline:Nn #1 + { + \tl_set:Nn \l__diffco_curr_tok_tl { ##1 } + \__diffco_get_curr_index:NN ##1 \l__diffco_curr_tok_int + \__diffco_compare_states:NNNNN \l__diffco_curr_state_int + \l__diffco_curr_tok_int #2#3#4 + } + \tl_if_empty:NT #2 { \tl_set:Nn #2 { 0 } } + } + +% #1 contains current token (tl); #2 receives current state (int) +\cs_new:Npn \__diffco_get_curr_index:NN #1#2 + { + \tl_if_in:NnTF \c__diffco_digits_tl { #1 } + { % digit + \int_set:Nn #2 { \c_one } + } + { + \tl_case:NnF #1 + { + + { \int_set_eq:NN #2 \c_zero } + - { \int_set_eq:NN #2 \c_zero } + ( { + \msg_error:nnxx { diffcoeff } { unknown_token } + { \l__diffco_order_spec_tl } { parentheses } + } + ^ { + \msg_error:nnxx { diffcoeff } { unknown_token } + { \l__diffco_order_spec_tl } { superscript } + } + } + { + \token_if_math_subscript:NTF #1 + { + \msg_error:nnxx { diffcoeff } { unknown_token } + { \l__diffco_order_spec_tl } { subscript } + } + { % var + \int_set_eq:NN #2 \c_two + } + } + } + } + +% #1 (int) is curr. state; #2 (int) is curr token index; +% #3 (tl) receives the numerical part; +% #4 (tl) receives the algebraic part; +% #5 (prop) receives the vars +\cs_new:Npn \__diffco_compare_states:NNNNN #1#2#3#4#5 + { + \int_case:nn { #1 } + { + { \c_zero } % sgn + { + \int_case:nn { #2 } + { + { \c_zero } % tok = s + { + \tl_if_eq:NNTF \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + { \tl_set:Nn \l__diffco_curr_term_tl { + } } + { \tl_set:Nn \l__diffco_curr_term_tl { - } } + % \int_set:Nn #1 { \c_zero } + } + { \c_one } % tok = d + { \__diffco_term_append:NN #1#2 } + { \c_two } % tok = v + { + \tl_put_right:Nn \l__diffco_curr_term_tl { 1 } + \__diffco_term_var_append:NN #1#2 + } + } + } + { \c_one } % num + { + \int_case:nn { #2 } + { + { \c_zero } % tok = s + { + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_clear:N \l__diffco_curr_term_tl + \__diffco_term_append:NN #1#2 + } + { \c_one } % tok = d + { \__diffco_term_append:NN #1 \c_one } + { \c_two } % tok = v + { \__diffco_term_var_append:NN #1#2 } + } + } + { \c_two } % alg + { + \int_case:nn { #2 } + { + { \c_zero } % tok = s + { + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_clear:N \l__diffco_curr_term_tl + \bool_if:NF \l__diffco_vars_noted_bool + { \__diffco_store_var:NN #5 \l__diffco_curr_var_tl } + \tl_clear:N \l__diffco_curr_var_tl + \__diffco_term_append:NN #1#2 + } + { \c_one } % tok = d + { + \msg_error:nnx { diffcoeff } { variable_digit_sequence } + { \l__diffco_order_spec_tl } + } + { \c_two } % tok = v + { \__diffco_term_var_append:NN #1 \c_two } + } + } + } + } + +% #1 current state; #2 current token index +\cs_new:Npn \__diffco_term_var_append:NN #1#2 + { + \tl_put_right:NV \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + \tl_put_right:NV \l__diffco_curr_var_tl \l__diffco_curr_tok_tl + \int_set_eq:NN #1 #2 + } + +% #1 current state; #2 current token index +\cs_new:Npn \__diffco_term_append:NN #1#2 + { + \tl_put_right:NV \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + \int_set_eq:NN #1 #2 + } + +% #1 is prop list; #2 is tl containing var +\cs_new:Npn \__diffco_store_var:NN #1 #2 + { + \int_set:Nn \l_tmpa_int { \exp_args:NV \tl_count_tokens:n #2 } + \prop_get:NVNTF #1 \l_tmpa_int \l_tmpa_tl + { + \tl_put_right:Nn \l_tmpa_tl { , } + \tl_put_right:NV \l_tmpa_tl #2 + \prop_put:NVV #1 \l_tmpa_int \l_tmpa_tl + } + { + \prop_put:NVV #1 \l_tmpa_int #2 + } + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% evals numerical expr. in #1, which receives the +% evaluated expr. +\cs_new:Npn \__diffco_eval_nos:N #1 + { + \int_set:Nn \l_tmpa_int { #1 } + \tl_set:Nx #1 { \int_use:N \l_tmpa_int } + \int_compare:nNnT { \l_tmpa_int } > { \c_zero } + { \tl_put_left:Nn #1 { + } } + } + +% #1 (tl) has algebraic part & receives eval. expression +% #2 (prop) contains the variables +% The idea is to remove each var but leave coeffs, split into +% num & alg parts, eval. the num part for the net coeff of var +\cs_new:Npn \__diffco_eval_vars:NN #1#2 + { + \bool_set_true:N \l__diffco_vars_noted_bool + \prop_set_eq:NN \l_tmpa_prop #2 + \int_set_eq:NN \l_tmpa_int \c_one + \bool_until_do:nn { \prop_if_empty_p:N #2 } + { + \prop_pop:NVNT #2 \l_tmpa_int \l_tmpa_tl + { + \tl_put_left:Nn \l_tmpa_tl { , } + \tl_put_left:NV \l__diffco_vars_tl \l_tmpa_tl + } + \int_incr:N \l_tmpa_int + } + + \clist_set:NV \l__diffco_vars_clist \l__diffco_vars_tl + \clist_remove_duplicates:N \l__diffco_vars_clist + + \clist_map_inline:Nn \l__diffco_vars_clist + { + \tl_replace_all:Nnn #1 { ##1 } { } + \__diffco_digest_expr:NNNN #1 \l__diffco_nos_aux_tl \l__diffco_alg_aux_tl + \l_tmpa_prop + + \__diffco_eval_nos:N \l__diffco_nos_aux_tl + + \__diffco_cumulate_expr:NnN \l__diffco_nos_aux_tl { ##1 } + \l__diffco_alg_expr_tl + + \tl_set_eq:NN #1 \l__diffco_alg_aux_tl + \tl_clear:N \l__diffco_nos_aux_tl + \tl_clear:N \l__diffco_alg_aux_tl + } + + \tl_set_eq:NN #1 \l__diffco_alg_expr_tl + } + +% #1 is num coeff; #2 is var; #3 is cumulating expr. (All tl.) +\cs_new:Npn \__diffco_cumulate_expr:NnN #1#2#3 + { + \int_case:nnF { #1 } + { + { 0 } { \prg_do_nothing: } + { +1 } { \tl_put_right:Nn #3 { + #2 } } + { -1 } { \tl_put_right:Nn #3 { - #2 } } + } + { + \tl_put_right:NV #3 { #1 } + \tl_put_right:Nn #3 { #2 } + } + } + +% alg. part precedes num. part unless alg. starts +% with - and nums > 0. #1 nums, #2 alg, #3 output +\cs_new:Npn \__diffco_manicure_result:NNN #1#2#3 + { % alg < 0 ? + \exp_args:NV \tl_if_head_eq_charcode:nNTF #2 - + { + \int_compare:nNnTF { #1 } > { \c_zero } + { % nums > 0 + \tl_concat:NNN #3 #1 #2 + } + { % nums =< 0 + \seq_set_split:NnV \l_tmpa_seq { + } #2 + \seq_pop:NN \l_tmpa_seq \l_tmpa_tl + \tl_set:Nx \l_tmpb_tl { \seq_use:Nn \l_tmpa_seq { + } } + \int_compare:nNnF { #1 } = { \c_zero } + { \tl_concat:NNN \l_tmpa_tl \l_tmpa_tl #1 } + \tl_concat:NNN #3 \l_tmpb_tl \l_tmpa_tl + } + } + { % alg >= 0 + \int_compare:nNnTF { #1 } = { \c_zero } + { \tl_set_eq:NN #3 #2 } + { \tl_concat:NNN #3 #2 #1 } + } + + % if everything cancels, output 0 + \tl_if_empty:NTF #3 { \tl_set:Nn #3 { 0 } } + { + % trim initial + sign + \exp_args:NV \tl_if_head_eq_charcode:nNT #3 + + { \tl_set:Nx #3 { \tl_tail:N #3 } } + } + } +%%%%%%%%%%% +% messages +\cs_gset:Npn \msg_error_text:n #1 + { Unimplemented~in~#1.sty: } +\cs_gset:Npn \msg_info_text:n #1 + { Use~the~order-override~option~to~specify~the~overall~order~of~ + differentiation~(or~use~diffcoeffx.sty). } + +\msg_new:nnnn { diffcoeff } { unknown_token } + { \msg_error_text:n { diffcoeff }~#2~in~the~order~specification~#1~ + \msg_line_context:. } + { \msg_info_text:n { diffcoeff } } + +\msg_new:nnnn { diffcoeff } { variable_digit_sequence } + { \msg_error_text:n { diffcoeff }~a~digit~follows~a~variable~in~the~order~ + specification~#1~\msg_line_context:. } + { \msg_info_text:n { diffcoeff } } + +
\ No newline at end of file diff --git a/Master/texmf-dist/tex/latex/diffcoeff/diffcoeffx.sty b/Master/texmf-dist/tex/latex/diffcoeff/diffcoeffx.sty new file mode 100644 index 00000000000..727175176bc --- /dev/null +++ b/Master/texmf-dist/tex/latex/diffcoeff/diffcoeffx.sty @@ -0,0 +1,972 @@ +% This is file `diffcoeffx.sty'. +% +% This program is free software; you can redistribute it and/or modify it +% under the terms of the GNU General Public License as published by the +% Free Software Foundation; either version 2 of the License, or (at your +% option) any later version. +% +% Andrew Parsloe aparsloe@clear.net.nz +% +\RequirePackage{expl3} +\RequirePackage{xparse} +\ProvidesExplPackage{diffcoeffx} {2015/06/27} {1.0} + {Write differential coefficients easily.} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\cs_generate_variant:Nn \tl_if_eq:nnF { no } +\cs_generate_variant:Nn \tl_if_eq:nnT { no } +\cs_generate_variant:Nn \tl_if_in:NnTF { NV } +\cs_generate_variant:Nn \prop_pop:NnNT { NV } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% variables +\tl_new:N \g_diffco_d_ldelim_tl +\tl_new:N \g_diffco_d_rdelim_tl +\tl_new:N \g_diffco_p_ldelim_tl +\tl_new:N \g_diffco_p_rdelim_tl + +\tl_new:N \l__diffco_entiand_tl +\tl_new:N \l__diffco_phantom_tl +\tl_new:N \g_diffco_dop_tl +\tl_new:N \l__diffco_denom_tl +\tl_new:N \l__diffco_coeff_tl + +\clist_new:N \l__diffco_order_clist +\tl_new:N \l__diffco_order_spec_tl +\tl_new:N \l__diffco_nudge_tl + +\bool_new:N \g__diffco_erent_shape_bool +\bool_new:N \l__diffco_no_parens_bool +\bool_new:N \l__diffco_override_bool +\bool_new:N \l__diffco_ordinary_bool +\bool_new:N \l__diffco_fp_bool +\bool_new:N \l__diffco_vars_noted_bool + +\tl_new:N \l__diffco_orders_in_tl +\tl_new:N \l__diffco_order_tl + +\tl_new:N \l__diffco_curr_tok_tl +\tl_new:N \l__diffco_curr_term_tl + +\int_new:N \l__diffco_curr_tok_int +\int_new:N \l__diffco_curr_state_int +\int_new:N \l__diffco_prev_state_int + +\prop_new:N \l__diffco_vars_prop +\tl_new:N \l__diffco_vars_tl +\clist_new:N \l__diffco_vars_clist + +\tl_new:N \l__diffco_curr_var_tl +\tl_new:N \l__diffco_curr_qvar_tl +\tl_new:N \l__diffco_nos_tl +\tl_new:N \l__diffco_nos_aux_tl + +\tl_new:N \l__diffco_alg_tl +\tl_new:N \l__diffco_alg_aux_tl +\tl_new:N \l__diffco_alg_expr_tl + +\tl_const:Nn \c__diffco_digits_tl { 1234567890 } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\keys_define:nn { diffcoeffx } + { + roman .bool_gset:N = \g_diffco_roman_bool, + roman .default:n = { false }, + d-delims .code:n = { + \tl_gset:Nx \g_diffco_d_ldelim_tl { \left \use_i:nn #1 } + \tl_gset:Nx \g_diffco_d_rdelim_tl + { \right \use_ii:nn #1 } + }, + d-delims .default:n = { .| }, + p-delims .code:n = { + \tl_gset:Nx \g_diffco_p_ldelim_tl { \left \use_i:nn #1 } + \tl_gset:Nx \g_diffco_p_rdelim_tl + { \right \use_ii:nn #1 } + }, + p-delims .default:n = { () }, + d-nudge .code:n = { \tl_gset:Nn \g_diffco_d_nudge_tl { \mskip #1 mu } }, + d-nudge .default:n = { 0 }, + p-nudge .code:n = { \tl_gset:Nn \g_diffco_p_nudge_tl { \mskip #1 mu } }, + p-nudge .default:n = { -6 }, + sep .code:n = { \tl_gset:Nn \g_diffco_sep_tl { \mskip #1 mu } }, + sep .default:n = { 2 }, + d-sep .code:n = { \tl_gset:Nn \g_diffco_d_sep_tl { \mskip #1 mu } }, + d-sep .default:n = { 1 }, + p-sep .code:n = { \tl_gset:Nn \g_diffco_p_sep_tl { \mskip #1 mu } }, + p-sep .default:n = { 1 } + } + +\DeclareDocumentCommand \diffset { o } + { + \IfValueTF { #1 } + { + \clist_if_empty:nTF { #1 } + { + \keys_set:nn { diffcoeffx } { roman,d-delims,p-delims, + d-nudge,p-nudge,sep,d-sep,p-sep } + } + { + \keys_set:nn { diffcoeffx } { #1 } + } + } + { + \keys_set:nn { diffcoeffx } { roman,d-delims,p-delims, + d-nudge,p-nudge,sep,d-sep,p-sep } + } + } + +% now set the defaults +\diffset + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Derivatives +% Ordinary differential coefficient. #1 (star) is the append boolean; +% #2 is the order; #3 the differentiand; #4 slash / boolean; +% #5 the variable of differentiation; #6 the point of evaluation. + +\NewDocumentCommand \diff{ s o m t/ m g } + { + \group_begin: + % roman or italic? + \bool_if:NF \g__diffco_erent_shape_bool + { \__diffco_shape:N d } + + \IfValueTF { #2 } + { + \tl_set:Nn \l__diffco_order_tl { \g_diffco_d_sep_tl ^{ #2 } } + \int_compare:nNnTF { \tl_count:n { #5 } } > { \c_one } + { \tl_set:Nn \l__diffco_denom_tl { (#5)^{ #2 } } } + { \tl_set:Nn \l__diffco_denom_tl { #5^{ #2 } } } + } + { + \tl_set:Nn \l__diffco_order_tl { } + \tl_if_in:nnTF { #5 } { ^ } + { \tl_set:Nn \l__diffco_denom_tl { (#5) } } + { \tl_set:Nn \l__diffco_denom_tl { #5 } } + } + + % differentiand appended (star) or numerator (no star); + % slash or fraction? + \IfBooleanTF { #1 } + { % differentiand follows diff coeff + \__diffco_phantom:n { #3 } + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #4 } + { % slash (therefore no phantom) + (\g_diffco_dop_tl \l__diffco_order_tl + / + \g_diffco_dop_tl \l__diffco_denom_tl ) + \l__diffco_entiand_tl + } + { % frac + \frac { \g_diffco_dop_tl \l__diffco_order_tl \l__diffco_phantom_tl } + { \g_diffco_dop_tl \l__diffco_denom_tl } \l__diffco_entiand_tl + } + } + } + { % differentiand in numerator + \tl_set:Nn \l__diffco_coeff_tl + { + \IfBooleanTF { #4 } + { % slash + \g_diffco_dop_tl \l__diffco_order_tl #3 + / + \g_diffco_dop_tl \l__diffco_denom_tl + } + { % frac + \frac { \g_diffco_dop_tl \l__diffco_order_tl #3 } + { \g_diffco_dop_tl \l__diffco_denom_tl } + } + } + } + + % point of eval? + \IfNoValueTF { #6 } + { % no + \l__diffco_coeff_tl + } + { % yes + \bool_set_true:N \l__diffco_ordinary_bool + \__diffco_delims:nnnnN #6;;;;\q_stop \l__diffco_ordinary_bool + } + \group_end: + } + +\NewDocumentCommand \Diff { o m m g } + { + \group_begin: + \__diffco_shape:N D + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\NewDocumentCommand \diffdelta { o m m g } + { + \group_begin: + \__diffco_shape:N \delta + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\NewDocumentCommand \Diffdelta { o m m g } + { + \group_begin: + \__diffco_shape:N \Delta + \bool_gset_true:N \g__diffco_erent_shape_bool + \diff[#1]{#2}{#3}{#4} + \bool_gset_false:N \g__diffco_erent_shape_bool + \group_end: + } + +\cs_new_nopar:Npn \__diffco_shape:N #1 + { + \bool_if:NTF \g_diffco_roman_bool + { \tl_gset:Nn \g_diffco_dop_tl { \mathrm{ #1} } } + { \tl_gset:Nn \g_diffco_dop_tl { #1 } } + } +%%%%% +\cs_new:Npn \__diffco_phantom:n #1 + { + \bool_if:nTF + { + \tl_if_head_eq_meaning_p:nN { #1 } \hfill || + \tl_if_head_eq_meaning_p:nN { #1 } \hfil + } + { + \l__diffco_phantom_aux:nn #1\q_stop + } + { + \bool_if:nTF + { + \tl_if_head_eq_meaning_p:nN { #1 } \hspace || + \tl_if_head_eq_meaning_p:nN { #1 } \hphantom + } + { + \l__diffco_phantom_aux:nnn #1\q_stop + } + { + \tl_set:Nn \l__diffco_entiand_tl { #1 } + } + } + } + +\cs_new:Npn \l__diffco_phantom_aux:nn #1#2\q_stop + { + \tl_set:Nn \l__diffco_phantom_tl { #1 } + \tl_set:Nn \l__diffco_entiand_tl { #2 } + } + +\cs_new:Npn \l__diffco_phantom_aux:nnn #1#2#3\q_stop + { + \tl_set:Nn \l__diffco_phantom_tl { #1 { #2 } } + \tl_set:Nn \l__diffco_entiand_tl { #3 } + } +%%%%%%%%%%%%%%%%%%%% +% Partial differential coefficient. #1 (star) = append differentiand +% to diff. coeff.(displace from numerator); #2 = comma list of the +% orders of diff. in each var.; #3 = order override for the +% numerator; #4 = differentiand; #5 = clist of vars of diff. in +% order; #6 = list of vars held constant +\NewDocumentCommand \diffp{ s o o t/ m m g } + { + \group_begin: + + % diff. order list + \tl_clear:N \l__diffco_order_clist + \IfValueT { #2 } + { + \clist_set:Nn \l__diffco_order_clist { #2 } + \tl_set:Nx \l__diffco_order_spec_tl + { [\clist_use:Nn \l__diffco_order_clist { , }]} + } + + % order list override + \IfValueT { #3 } + { + \bool_set_true:N \l__diffco_override_bool + \tl_set:Nn \l__diffco_order_tl { #3 } + } + + \__diffco_orders:Nn \l__diffco_order_clist { #6 } + + % position of differentiand + \IfBooleanTF { #1 } + { % appended + \__diffco_phantom:n { #5 } + \tl_set:Nn \l__diffco_coeff_tl + { + \frac { \partial + \__diffco_omit_order_if_i:N \l__diffco_order_tl + \l__diffco_phantom_tl } { \l__diffco_denom_tl } + \l__diffco_entiand_tl + } + } + { % in numerator + \tl_set:Nn \l__diffco_coeff_tl + { + \frac { \partial + \__diffco_omit_order_if_i:N \l__diffco_order_tl + #5 } { \l__diffco_denom_tl } + } + } + + \IfNoValueTF { #7 } + { + \l__diffco_coeff_tl + } + { % hold some vars const./ pt of eval. + \bool_set_false:N \l__diffco_ordinary_bool + \__diffco_delims:nnnnN #7;;;;\q_stop \l__diffco_ordinary_bool + } + + \group_end: + } % end of \diffp + + +\NewDocumentCommand \jacob { m m } + { % need extra braces in denom for clist manipulations + \diffp{ (#1) }{ { {{(#2)}} } } + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% #1 subscript; #2 superscript; #3 delims; #4 nudge override +% #5 detritus; #6 boolean (true = ordinary deriv.; false = partial deriv.) +\cs_new:Npn \__diffco_delims:nnnnN #1;#2;#3;#4;#5\q_stop#6 + { + \tl_set:Nn \l__diffco_sub_tl { #1 } + \tl_set:Nn \l__diffco_super_tl { #2 } + \tl_if_empty:nTF { #3 } + { % default delims + \bool_if:NTF #6 + { % ordinary + \tl_set_eq:NN \l__diffco_ldelim_tl \g_diffco_d_ldelim_tl + \tl_set_eq:NN \l__diffco_rdelim_tl \g_diffco_d_rdelim_tl + \tl_if_empty:nTF { #4 } + { \tl_set_eq:NN \l__diffco_nudge_tl \g_diffco_d_nudge_tl } + { \tl_set:Nn \l__diffco_nudge_tl { \mskip #4 mu } } + } + { % partial + \tl_set_eq:NN \l__diffco_ldelim_tl \g_diffco_p_ldelim_tl + \tl_set_eq:NN \l__diffco_rdelim_tl \g_diffco_p_rdelim_tl + \tl_if_empty:nTF { #4 } + { \tl_set_eq:NN \l__diffco_nudge_tl \g_diffco_p_nudge_tl } + { \tl_set:Nn \l__diffco_nudge_tl { \mskip #4 mu } } + } + } + { % delims specified + \tl_set:Nn \l__diffco_ldelim_tl { \left \use_i:nn #3 } + \tl_set:Nn \l__diffco_rdelim_tl { \right \use_ii:nn #3 } + \tl_if_empty:nTF { #4 } + { + \tl_set:NV \l_tmpa_tl { \use_ii:nn #3 } + \__diffco_get_nudge:N \l_tmpa_tl + } + { + \tl_set:Nx \l__diffco_nudge_tl { \mskip #4 mu } + } + } + \l__diffco_ldelim_tl \l__diffco_coeff_tl \l__diffco_rdelim_tl + \tl_if_empty:NF \l__diffco_sub_tl + { \c_math_subscript_token { \l__diffco_nudge_tl \l__diffco_sub_tl } } + \tl_if_empty:NF \l__diffco_super_tl + { \c_math_superscript_token { \l__diffco_nudge_tl \l__diffco_super_tl } } + } + +\cs_new:Npn \__diffco_get_nudge:N #1 + { + \exp_args:NV\tl_case:NnF #1 + { + ) { \tl_set:Nn \l__diffco_nudge_tl { \mskip -6mu } } + > { \tl_set:Nn \l__diffco_nudge_tl { \mskip -6mu } } + \} { \tl_set:Nn \l__diffco_nudge_tl { \mskip -4mu } } + \rangle { \tl_set:Nn \l__diffco_nudge_tl { \mskip -6mu } } + } + { \tl_set:Nn \l__diffco_nudge_tl { \mskip 0mu } } + } + +% #1 is the clist of diff. orders +% #2 is the clist of variables +\cs_new:Npn \__diffco_orders:Nn #1#2 + { + \int_zero:N \l_tmpa_int + \tl_clear:N \l_tmpb_tl + \clist_clear:N \l_tmpa_clist + \clist_clear:N \l_tmpb_clist + \clist_clear:N \l__diffco_vars_clist + + \clist_set:Nn \l__diffco_vars_clist { #2 } + \clist_map_inline:Nn \l__diffco_vars_clist + { + \bool_set_false:N \l__diffco_no_parens_bool + + % get order of each variable + \clist_pop:NNF #1 \l_tmpa_tl + { \tl_set:Nn \l_tmpa_tl { 1 } } + + \tl_if_eq:noT { 1 } \l_tmpa_tl + { \bool_set_true:N \l__diffco_no_parens_bool } + + \clist_put_right:NV \l_tmpa_clist \l_tmpa_tl + + % create the denominator + \tl_clear:N \l_tmpb_tl + \bool_if:nTF + { + \int_compare_p:nNn { \tl_count:n { ##1 } } = { \c_one } + || + \l__diffco_no_parens_bool + } + { \tl_put_right:Nn \l_tmpb_tl { \partial ##1 } } + { \tl_put_right:Nn \l_tmpb_tl { \partial (##1) } } + + \tl_if_eq:noF { 1 } \l_tmpa_tl + { + \tl_put_right:Nn \l_tmpb_tl { ^ } + \tl_put_right:Nx \l_tmpb_tl { { \l_tmpa_tl } } + } + + \clist_put_right:No \l_tmpb_clist \l_tmpb_tl + } % end of inline map + + % numerator order + \bool_if:NF \l__diffco_override_bool + { + \tl_set:Nx \l__diffco_orders_in_tl { \clist_use:Nn \l_tmpa_clist {+} } + \diff_simplify_orders:NN \l__diffco_orders_in_tl \l__diffco_order_tl + } + + % denominator + \tl_set:Nx \l__diffco_denom_tl { \clist_use:Nn + \l_tmpb_clist { \g_diffco_sep_tl } } + } + +% don't display order 1 superscript +\cs_new:Npn \__diffco_omit_order_if_i:N #1 + { + \tl_if_eq:noF { 1 } #1 + { ^{ \g_diffco_p_sep_tl \l__diffco_order_tl } } + } +%%%%%%%%%%%%%%%%%%%%%% +% #1 is expr. in; #2 is expr. out +\cs_new:Npn \diff_simplify_orders:NN #1 #2 + { + \tl_clear:N \l__diffco_nos_tl + + \__diffco_digest_expr:NNNN #1 \l__diffco_nos_tl + \l__diffco_alg_tl \l__diffco_vars_prop + + \__diffco_eval_nos:N \l__diffco_nos_tl + + \tl_if_empty:NF \l__diffco_alg_tl + { \__diffco_eval_vars:NN \l__diffco_alg_tl \l__diffco_vars_prop } + + \__diffco_manicure_result:NNN \l__diffco_nos_tl \l__diffco_alg_tl #2 + } + +% #1 is the expression; #2 (tl) receives the numerical part; +% #3 (tl) receives the algebraic part; #4 (prop) receives the vars +\cs_new:Npn \__diffco_digest_expr:NNNN #1#2#3#4 + { + \tl_put_right:Nn #1 { + } + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set:Nn \l__diffco_curr_state_int { \c_zero } + \int_set:Nn \l__diffco_prev_state_int { \c_zero } + \int_set:Nn \l__diffco_curr_tok_int { \c_zero } + \tl_clear:N #3 + \prop_clear:N #4 + + \tl_map_inline:Nn #1 + { + \tl_set:Nn \l__diffco_curr_tok_tl { ##1 } + \__diffco_get_curr_index:NN ##1 \l__diffco_curr_tok_int + \__diffco_compare_states:NNNNN \l__diffco_curr_state_int + \l__diffco_curr_tok_int #2#3#4 + } + \tl_if_empty:NT #2 { \tl_set:Nn #2 { 0 } } + } + +% #1 contains current token (tl); #2 receives current token index (int) +\cs_new:Npn \__diffco_get_curr_index:NN #1#2 + { + \tl_if_in:NnTF \c__diffco_digits_tl { #1 } + { % digit + \int_set:Nn #2 { \c_one } + } + { + \tl_case:NnF #1 + { + + { \int_set_eq:NN #2 \c_zero } + - { \int_set_eq:NN #2 \c_zero } + ^ { \int_set_eq:NN #2 \c_three } + * { \int_set_eq:NN #2 \c_five } + \times + { + \int_set_eq:NN #2 \c_five + \tl_set:Nn \l__diffco_curr_tok_tl { * } + } + ( { \int_set_eq:NN #2 \c_six } + ) { \int_set_eq:NN #2 \c_seven } + } + { + \token_if_math_subscript:NTF #1 + { \int_set_eq:NN #2 { \c_four } } + { + \int_set_eq:NN #2 \c_two } % var + } + } + } + +% #1 (int) is curr. state; #2 (int) is curr token index; +% #3 (tl) receives the numerical part; +% #4 (tl) receives the algebraic part; +% #5 (prop) receives the vars +\cs_new:Npn \__diffco_compare_states:NNNNN #1#2#3#4#5 + { + \int_case:nn { #1 } + { + { \c_zero } % sgn + { \__diffco_sgn_transitions:NNNN #1#2#3#4 } + { \c_one } % num + { \__diffco_num_transitions:NNNN #1#2#3#4 } + { \c_two } % alg + { \__diffco_alg_transitions:NNNNN #1#2#3#4#5 } + { \c_three } % exp + { \__diffco_exp_transitions:NN #1#2 } + { \c_four } % sub (_) + { \__diffco_sub_transitions:NN #1#2 } + { \c_five } % mul (*,X) + { \__diffco_mul_transitions:NNNN #1#2#3#4 } + } + } + +% transitions from the signed state +% #1 = 0, current state; #2 current token index +% #3 store num. part; #4 store alg. part +\cs_new:Npn \__diffco_sgn_transitions:NNNN #1#2#3#4 + { + \int_case:nnF { #2 } + { + { \c_zero } % tok = s + { + \tl_if_eq:NNTF \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + { \tl_set:Nn \l__diffco_curr_term_tl { + } } + { \tl_set:Nn \l__diffco_curr_term_tl { - } } + \int_set_eq:NN \l__diffco_prev_state_int #1 + % \int_set:Nn #1 { \c_zero } + } + { \c_one } % tok = d + { \__diffco_term_qvar_append:NN #1 #2 } + { \c_two } % tok = v + { + \tl_put_right:Nn \l__diffco_curr_term_tl { 1 } + \__diffco_term_var_append:NN #1 #2 + } + { \c_six } % tok = ( + { + \tl_put_right:Nn \l__diffco_curr_term_tl { 1*( } + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + { \c_seven } % tok = ) + { + \tl_put_right:Nn #3 { ) } + \tl_put_right:Nn #4 { ) } + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { sign } + } + } + +% transitions from the numeric state +% #1 = 1, current state; #2 current token index +% #3 store num. part; #4 store alg. part +\cs_new:Npn \__diffco_num_transitions:NNNN #1#2#3#4 + { + \int_case:nnF { #2 } + { + { \c_zero } % tok = s + { + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_clear:N \l__diffco_curr_qvar_tl + \tl_clear:N \l__diffco_curr_term_tl + \__diffco_term_append:NN #1#2 + } + { \c_one } % tok = d + { \__diffco_term_qvar_append:NN #1 #2 } + { \c_two } % tok = v + { + \tl_clear:N \l__diffco_curr_qvar_tl + \__diffco_term_var_append:NN #1#2 + } + { \c_three } % tok = ^ + { \__diffco_term_qvar_append:NN #1#2 } + { \c_five } % tok = * + { + \tl_clear:N \l__diffco_curr_qvar_tl + \__diffco_term_append:NN #1#2 + } + { \c_six } % tok = ( + { + \tl_put_right:Nn \l__diffco_curr_term_tl { *( } + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + { \c_seven } % tok = ) + { + \tl_put_right:Nn \l__diffco_curr_term_tl { ) } + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_put_right:Nn #4 { +0) } + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { number } + } + } + +% transitions from the algebraic state +% #1 = 2, current state; #2 current token index +% #3 store num. part; #4 store alg. part; #5 var store +\cs_new:Npn \__diffco_alg_transitions:NNNNN #1#2#3#4#5 + { + \int_case:nnF { #2 } + { + { \c_zero } % tok = s + { + \bool_if:NF \l__diffco_vars_noted_bool + { \__diffco_store_var:NN #5 \l__diffco_curr_var_tl } + \tl_clear:N \l__diffco_curr_var_tl + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_clear:N \l__diffco_curr_term_tl + \__diffco_term_append:NN #1 #2 + } + { \c_one } % tok = d + { + \int_compare:nNnTF { \l__diffco_prev_state_int } = { \c_three } + { \__diffco_term_var_append:NN #1 \c_two } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } + { variable } + } + } + { \c_two } % tok = v + { \__diffco_term_var_append:NN #1 #2 } + + { \c_three } % tok = ^ + { \__diffco_term_var_append:NN #1#2 } + { \c_four } % tok = _ + { \__diffco_term_var_append:NN #1#2 } + { \c_seven } % tok = ) + { + \bool_if:NF \l__diffco_vars_noted_bool + { \__diffco_store_var:NN #5 \l__diffco_curr_var_tl } + \tl_clear:N \l__diffco_curr_var_tl + \tl_put_right:Nn \l__diffco_curr_term_tl { ) } + \tl_put_right:Nn #3 { +0) } + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { variable } + } + } + +% transitions from the exponent state +% #1 = 3, current state; #2 current token index +\cs_new:Npn \__diffco_exp_transitions:NN #1#2 + { + \int_case:nn { \l__diffco_prev_state_int } + { + { \c_one } % prev = num + { + \int_case:nnF { #2 } + { + { \c_one } % tok = d + { + \tl_clear:N \l__diffco_curr_qvar_tl + \tl_put_right:Nn \l__diffco_curr_tok_tl { * } + \__diffco_term_append:NN #1 \c_five + \bool_set_true:N \l__diffco_fp_bool + } + { \c_two } % tok = v + { + \__diffco_term_qvar_append:NN #1 \c_two + \tl_set_eq:NN \l__diffco_curr_var_tl \l__diffco_curr_qvar_tl + \tl_clear:N \l__diffco_curr_qvar_tl + } + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { ^ } + } + } + { \c_two } % prev = alg + { + \int_compare:nNnTF { #2 } < { \c_three } + { \__diffco_term_var_append:NN #1 \c_two } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { ^ } + } + } + } + } + +% transitions from the subscript state +% #1 = 4, current state; #2 current token index +\cs_new:Npn \__diffco_sub_transitions:NN #1#2 + { + \bool_if:nTF + {( + \int_compare_p:nNn { \l__diffco_prev_state_int } = { \c_two } && + \int_compare_p:nNn { #2 } < { \c_three } + )} + { + \__diffco_term_var_append:NN #1 \c_two + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } { _ } + } + } + +% transitions from the multiplicative state +% #1 = 5, current state; #2 current token index +% #3 store num. part; #4 store alg. part +\cs_new:Npn \__diffco_mul_transitions:NNNN #1#2#3#4 + { + \int_case:nnF { #2 } + { + { \c_zero } % tok = s + { + \int_compare:nNnTF {\l__diffco_prev_state_int } = { \c_three } + { + \tl_put_right:Nn \l__diffco_curr_term_tl { 1 } + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_clear:N \l__diffco_curr_term_tl + \__diffco_term_append:NN #1 #2 + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } + { multiplicative~token } + } + } + { \c_one } % tok = d + { \__diffco_term_qvar_append:NN #1 #2 } + { \c_two } % tok = v + { + \tl_put_right:Nn \l__diffco_curr_term_tl {1 } + \__diffco_term_var_append:NN #1#2 + } + { \c_five } % tok = * + { \prg_do_nothing: } + { \c_six } % tok = ( + { + \tl_put_right:Nn \l__diffco_curr_term_tl { ( } + \tl_put_right:NV #3 \l__diffco_curr_term_tl + \tl_put_right:NV #4 \l__diffco_curr_term_tl + \tl_set:Nn \l__diffco_curr_term_tl { + } + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set:Nn #1 { \c_zero } + } + } + { + \msg_error:nnxxx { diffcoeffx } { order_spec_construct } + { \l__diffco_order_spec_tl } { \l__diffco_curr_tok_tl } + { multiplicative~token } + } + } + +%%%%%%%%%%%%%%%%%%% +% term/var/qvar appending routines +% #1 current state; #2 current token index +\cs_new:Npn \__diffco_term_var_append:NN #1#2 + { + \tl_put_right:NV \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + \tl_put_right:NV \l__diffco_curr_var_tl \l__diffco_curr_tok_tl + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set_eq:NN #1 #2 + } + +% #1 current state; #2 current token index +\cs_new:Npn \__diffco_term_qvar_append:NN #1#2 + { + \tl_put_right:NV \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + \tl_put_right:NV \l__diffco_curr_qvar_tl \l__diffco_curr_tok_tl + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set_eq:NN #1 #2 + } + +% #1 current state; #2 current token index +\cs_new:Npn \__diffco_term_append:NN #1#2 + { + \tl_put_right:NV \l__diffco_curr_term_tl \l__diffco_curr_tok_tl + \int_set_eq:NN \l__diffco_prev_state_int #1 + \int_set_eq:NN #1 #2 + } + +% #1 is prop list; #2 is tl containing var +\cs_new:Npn \__diffco_store_var:NN #1 #2 + { + \int_set:Nn \l_tmpa_int { \exp_args:NV \tl_count_tokens:n #2 } + \prop_get:NVNTF #1 \l_tmpa_int \l_tmpa_tl + { + \tl_put_right:Nn \l_tmpa_tl { , } + \tl_put_right:NV \l_tmpa_tl #2 + \prop_put:NVV #1 \l_tmpa_int \l_tmpa_tl + } + { + \prop_put:NVV #1 \l_tmpa_int #2 + } + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% evals numerical expr. in #1, which receives the +% evaluated expr. +\cs_new:Npn \__diffco_eval_nos:N #1 + { + \bool_if:NTF \l__diffco_fp_bool + { % exponent; use l3fp + \fp_set:Nn \l_tmpa_fp { #1 } + \int_set:Nn \l_tmpa_int { \fp_use:N \l_tmpa_fp } + \bool_set_false:N \l__diffco_fp_bool + } + { % normal mode; use l3int + \int_set:Nn \l_tmpa_int { #1 } + } + \tl_set:Nx #1 { \int_use:N \l_tmpa_int } + + \int_compare:nNnT { \l_tmpa_int } > { \c_zero } + { \tl_put_left:Nn #1 { + } } + } + +% #1 (tl) has algebraic part & receives eval. expression +% #2 (prop) contains the variables +% The idea is to remove each var but leave coeffs, split into +% num & alg parts, eval. the num part for the net coeff of var +\cs_new:Npn \__diffco_eval_vars:NN #1#2 + { + \bool_set_true:N \l__diffco_vars_noted_bool + \prop_set_eq:NN \l_tmpa_prop #2 + \int_set_eq:NN \l_tmpa_int \c_one + \bool_until_do:nn { \prop_if_empty_p:N #2 } + { + \prop_pop:NVNT #2 \l_tmpa_int \l_tmpa_tl + { + \tl_put_left:Nn \l_tmpa_tl { , } + \tl_put_left:NV \l__diffco_vars_tl \l_tmpa_tl + } + \int_incr:N \l_tmpa_int + } + + \clist_set:NV \l__diffco_vars_clist \l__diffco_vars_tl + \clist_remove_duplicates:N \l__diffco_vars_clist + \bool_if:NF \l__diffco_override_bool + { + \clist_map_inline:Nn \l__diffco_vars_clist + { + \tl_replace_all:Nnn #1 { ##1 } { } + \__diffco_digest_expr:NNNN #1 \l__diffco_nos_aux_tl + \l__diffco_alg_aux_tl \l_tmpa_prop + \__diffco_eval_nos:N \l__diffco_nos_aux_tl + + \__diffco_cumulate_expr:NnN \l__diffco_nos_aux_tl { ##1 } + \l__diffco_alg_expr_tl + + \tl_set_eq:NN #1 \l__diffco_alg_aux_tl + \tl_clear:N \l__diffco_nos_aux_tl + \tl_clear:N \l__diffco_alg_aux_tl + } + } + \tl_set_eq:NN #1 \l__diffco_alg_expr_tl + } + +% #1 is num coeff; #2 is var; #3 is cumulating expr. (All tl.) +\cs_new:Npn \__diffco_cumulate_expr:NnN #1#2#3 + { + \int_case:nnF { #1 } + { + { 0 } { \prg_do_nothing: } + { +1 } { \tl_put_right:Nn #3 { + #2 } } + { -1 } { \tl_put_right:Nn #3 { - #2 } } + } + { + \tl_put_right:NV #3 { #1 } + \exp_args:NNx \tl_if_in:NnT \c__diffco_digits_tl { \tl_head:n { #2 } } + { \tl_put_right:Nn #3 { \times} } + \tl_put_right:Nn #3 { #2 } + } + } + +% alg. part precedes num. part unless alg. starts +% with - and nums > 0. #1 nums, #2 alg, #3 output +\cs_new:Npn \__diffco_manicure_result:NNN #1#2#3 + { % alg < 0 ? + \exp_args:NV \tl_if_head_eq_charcode:nNTF #2 - + { + \int_compare:nNnTF { #1 } > { \c_zero } + { % nums > 0 + \tl_concat:NNN #3 #1 #2 + } + { % nums =< 0 + \seq_set_split:NnV \l_tmpa_seq { + } #2 + \seq_pop:NN \l_tmpa_seq \l_tmpa_tl + \tl_set:Nx \l_tmpb_tl { \seq_use:Nn \l_tmpa_seq { + } } + \int_compare:nNnF { #1 } = { \c_zero } + { \tl_concat:NNN \l_tmpa_tl \l_tmpa_tl #1 } + \tl_concat:NNN #3 \l_tmpb_tl \l_tmpa_tl + } + } + { % alg >= 0 + \int_compare:nNnTF { #1 } = { \c_zero } + { \tl_set_eq:NN #3 #2 } + { \tl_concat:NNN #3 #2 #1 } + } + + % if everything cancels, output 0 + \tl_if_empty:NTF #3 { \tl_set:Nn #3 { 0 } } + { + % trim initial + sign + \exp_args:NV \tl_if_head_eq_charcode:nNT #3 + + { \tl_set:Nx #3 { \tl_tail:N #3 } } + } + } + +%%%%%%%%%%%%%%%%%%%% +% messages +\cs_gset:Npn \msg_error_text:n #1 + { #1.sty.~Is~this~construction~intended } +\cs_gset:Npn \msg_info_text:n #1 + { If~intended~then~use~the~order-override~option~to~specify~the~overall~ + order~of~differentiation. } + +\msg_new:nnnn { diffcoeffx } { order_spec_construct } + { \msg_error_text:n { diffcoeffx }:~#3~followed~by~#2~in~the~order~ + specification~#1~\msg_line_context:? } + { \msg_info_text:n { diffcoeffx } }
\ No newline at end of file diff --git a/Master/tlpkg/bin/tlpkg-ctan-check b/Master/tlpkg/bin/tlpkg-ctan-check index dae3339697c..742e26dd68e 100755 --- a/Master/tlpkg/bin/tlpkg-ctan-check +++ b/Master/tlpkg/bin/tlpkg-ctan-check @@ -197,7 +197,7 @@ my @TLP_working = qw( dejavu delim delimseasy delimtxt denisbdoc dhua diadia diagbox diagmac2 dialogl diagnose dice dichokey - dickimaw dictsym digiconfigs din1505 + dickimaw dictsym diffcoeff digiconfigs din1505 dinat dinbrief dingbat directory dirtree dirtytalk disser dithesis dk-bib dlfltxb dnaseq dnp doc-pictex docbytex doclicense diff --git a/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc b/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc index a5fe0d50f40..439b6962bfa 100644 --- a/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc +++ b/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc @@ -21,6 +21,7 @@ depend concrete depend conteq depend delim depend delimseasy +depend diffcoeff depend drawmatrix depend ebproof depend econometrics diff --git a/Master/tlpkg/tlpsrc/diffcoeff.tlpsrc b/Master/tlpkg/tlpsrc/diffcoeff.tlpsrc new file mode 100644 index 00000000000..e69de29bb2d --- /dev/null +++ b/Master/tlpkg/tlpsrc/diffcoeff.tlpsrc |