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diff --git a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex index 9ade962edd3..ffe2645c6a0 100644 --- a/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex +++ b/Master/texmf-dist/doc/latex/diffcoeff/diffcoeff.tex @@ -1,10 +1,11 @@ -%% LyX 2.3.3 created this file. For more info, see http://www.lyx.org/. +%% LyX 2.4.0-alpha3 created this file. For more info, see https://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{article} \usepackage{lmodern} \renewcommand{\sfdefault}{lmss} \renewcommand{\ttdefault}{lmtt} \usepackage[T1]{fontenc} +\usepackage{textcomp} \usepackage[latin9]{inputenc} \synctex=-1 \usepackage{color} @@ -29,6 +30,8 @@ \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands. +\newenvironment{centred}% + {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}} \newenvironment{example}% {\begin{center}\ttfamily}{\end{center}} \newenvironment{lyxcode} @@ -52,117 +55,97 @@ \title{\texttt{diffcoeff}~\\ a \LaTeX{} package to ease\texttt{ }the~\\ writing of differential coefficients \\ - in all their variety\\ - Version 3.2} + Version 4.0} \author{Andrew Parsloe\\ -{\small{}(ajparsloe@gmail.com)}} +{\small (ajparsloe@gmail.com)}} \maketitle \begin{abstract} -\noindent This package allows the easy and consistent writing of ordinary, -partial and other derivatives of arbitrary (algebraic or numeric) order. -For mixed partial derivatives, the total order of differentiation is calculated -by the package. Optional arguments allow for points of evaluation (ordinary -derivatives), or variables held constant (partial derivatives), and the -placement of the differentiand (numerator or appended). The package uses -\texttt{xtemplate,} allowing systematic fine-tuning of the display and -generation and use of variant forms, including derivatives built from $D$, -$\Delta$ or $\delta$. A command for differentials ensures the $\dl x$ -(or $\dl.up.x$) used in integrals is consistent with the form used in -derivatives. The package requires the \LaTeX 3 bundles \texttt{l3kernel} -and \texttt{l3packages}. +\noindent This package eases the consistent writing of ordinary, partial +and other derivatives of arbitrary (algebraic or numeric) order. For +mixed partial derivatives, the total order of differentiation is calculated +by the package. Optional arguments allow for points of evaluation +(ordinary derivatives), or variables held constant (partial derivatives), +and the placement of the differentiand in numerator or appended. The +package uses \texttt{xtemplate}, allowing systematic fine-tuning of +the display and generation and use of variant forms, including derivatives +built from $D$, $\Delta$ or $\delta$. A command for differentials +ensures the $\dl x$ (or $\dl.up.x$) used in integrals is consistent +with the form used in derivatives. \end{abstract} \tableofcontents{} \section{Introduction} -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 \LaTeX{} package \texttt{diffcoeff.sty} is written in the expl3 +language of \LaTeX 3\texttt{ }and requires the bundles \texttt{l3kernel} +which is now is part of standard \LaTeX{} (since early 2020) and \texttt{l3packages} (the latter for the \texttt{xparse}, \texttt{l3keys2e} and \texttt{xtemplate} packages\texttt{)}. The package is invoked in the usual way by entering \begin{lyxcode} \textbackslash usepackage\{diffcoeff\} \end{lyxcode} -in the preamble of your document. There are two package options. The first -is a switch, \texttt{ISO}, which turns on formatting conforming to ISO -recommendations: +in the preamble of your document. + +\subsubsection*{Note on terminology} + +I refer throughout to the quantity or function being differentiated +as the \emph{differentiand} or \emph{derivand }(in line with \emph{integrand}, +\emph{operand}, etc.). + +\subsubsection*{New in version 4.0} + +(See §\ref{sec:Version-comparison} for a comparison of versions.) +Version 4.0 +\begin{enumerate} +\item enables the insertion of a small space before the differentiand, either +as the default behaviour (package option \verb`spaced`) or at explicit +request (argument \verb`!` of the \verb`\diff` command); see §\ref{subsec:Spacing-before-derivand}; +\item prevents the ligature $df$ that marred previous versions; this is +now rendered $\dl f$; +\item offers the document command \verb`\dlp` for the \emph{partial} differential; +see §\ref{subsec:Partial-differential}; +\item no longer accepts \emph{the braced form} of the\emph{ }trailing optional +argument specifying a point of evaluation or (for partial derivatives) +variables held constant. This was a relic from version 1 of \texttt{diffcoeff}, +and has been deprecated since version 2. +\end{enumerate} + +\subsection{Package options} + +\label{subsec:Package-options}There are three package options, which +are entered in a comma-separated list in the optional argument of +the \texttt{\textbackslash usepackage} command. +\begin{enumerate} +\item The first is a switch, \texttt{ISO}, which turns on formatting conforming +to ISO recommendations, principally upright `d's: \begin{lyxcode} \textbackslash usepackage{[}ISO{]}\{diffcoeff\} \end{lyxcode} -The effect of this is discussed in Section~\ref{sec:Changing-defaults}. -The second is a filename for a file of extension \texttt{.def} containing -definitions of variant forms of derivative: +The full effect of this option is disscussed in §\ref{sec:Changing-defaults}. +\item The second is another switch, \texttt{spaced}, and \emph{is new with +version 4 }of \texttt{diffcoeff}. This switch ensures a small space +is inserted before the differentiand: \begin{lyxcode} -\textbackslash usepackage{[}def-file=<filename>{]}\{diffcoeff\} +\textbackslash usepackage{[}spaced{]}\{diffcoeff\} \end{lyxcode} -This is discussed in Subsection~\ref{subsec:The-.def-file}. Of course -both options can be used in the same call if desired: +This is discussed in §\ref{subsec:Spacing-before-derivand}. +\item The third requires a filename \texttt{<filename>} for a file \texttt{<filename>.def} +containing definitions of variant forms of derivative: \begin{lyxcode} -\textbackslash usepackage{[}ISO,def-file=<filename>{]}\{diffcoeff\} +\textbackslash usepackage{[}def-file=<filename>{]}\{diffcoeff\} \end{lyxcode} -For the present document, the call is +This is discussed in §\ref{subsec:The-.def-file}. For the present +document, the call is \begin{lyxcode} \textbackslash usepackage{[}def-file=diffcoeff-doc{]}\{diffcoeff\} \end{lyxcode} -with the file \texttt{diffcoeff-doc.def} in the same directory as \texttt{diffcoeff.tex}. - -\subsection{Version comparison} - -Unlike version 1, version 2 and later are built on the the \texttt{xtemplate} -package (included in the \texttt{l3packages} bundle) which makes certain -facilities available which it would be silly not to exploit. Hence the -coding in the later versions is completely different and there are consequences. -\begin{enumerate} -\item The \texttt{\textbackslash diffset} command, formerly used to tweak the -display of derivatives, has been superseded by the \texttt{\textbackslash diffdef} -command. \texttt{\textbackslash diffset} now sends a message warning of -its obsolescence to the terminal and \LaTeX{} log but is otherwise functionless. -It should not interfere with the compilation of a document but any intended -fine-tuning of the display by means of the \texttt{\textbackslash diffset} -command\texttt{ }will not eventuate. The warning message is: \texttt{Obsolete -command: \textbackslash diffset has been superseded by the \textbackslash diffdef -command. See the diffcoeff documentation for further information.} The -\texttt{\textbackslash diffdef} command is discussed in Subsection~\ref{subsec:diffdef}. -\item The optional trailing argument used to indicate a point of evaluation or -variables held constant is now delimited by square brackets, \texttt{{[}} -and \texttt{{]}}, as other optional arguments are. For compatibility with -version 1, braces can still be used but their use to delimit an \emph{optional} -argument is now deprecated in \texttt{xparse} on which \texttt{diffcoeff} -depends. Presumably at some stage this provision will be removed from \texttt{xparse}. -For future-proofing documents use square brackets. -\item The commands \texttt{\textbackslash Diff}, \texttt{\textbackslash diffd} -and \texttt{\textbackslash Diffd} used to construct derivatives from $D$, -$\delta$ and $\Delta$ in version 1, are still available in versions 2 -and 3, but deprecated. A new optional argument in the \texttt{\textbackslash diff} -command offers these and a host of other possibilities and is now the preferred -method of forming such variants; see Subsection~\ref{subsec:D-delta-Delta}. -\item Version 3 adds a command, \texttt{\textbackslash dl} (from \emph{d}ifferentia\emph{l}) -to write differentials like $dx$ that occur in integrals and in other -contexts in a manner consistent with the form used in derivatives. After -all, if one is using upright `d's in derivatives, similarly upright `d's -should occur in these other contexts.\footnote{This rather obvious lack in version 2 was pointed out to me by Sergio Callegari.} -\item Version 3.1 enables the differential command to be used before forms like -\texttt{\textbackslash vec\{x\}} (an overlooked possibility causing an -error in version 3). -\item Version 3 also provides some simple spacing commands that can be useful -for tweaking standard spacing. -\item Version 3.2 fixes a bug in which an ordinary derivative as the differentiand -of a partial derivative displayed as a partial derivative. It now displays, -as it should, as an ordinary derivative. -\item In version 3.2, the differential command \texttt{\textbackslash dl} now -also allows negative spacing before it. \end{enumerate} -\subsubsection*{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{A Rogues' Gallery of derivatives} -\label{sec:Rogues'-gallery}Browsing through texts on statistical mechanics, -relativity and classical mechanics I find the following choice examples -of derivatives `disporting every which way'. +\label{sec:Rogues'-gallery}Browsing through texts on statistical +mechanics, relativity and classical mechanics I find the following +choice examples of derivatives `disporting every which way'. Multi-character variables of differentiation un-parenthesized: @@ -173,29 +156,31 @@ Multi-character variables of differentiation parenthesized: \begin{equation} \diffp H{\displaystyle \diffp S{q_{k}}[]},\quad\diffp\varepsilon{(1/\Theta)}.\label{eq:eg2} \end{equation} -Higher-order derivatives where the parentheses do not or do include the -operator: +Higher-order derivatives where the parentheses do not or sometimes +do include the operator: \begin{equation} \diffp[2]q{\frac{1}{\Theta}},\quad\diffp[2]q{1/\Theta},\quad\diffp[2]\varepsilon{a_{i}},\quad\diff.wrapall.[2]{\phi^{i}(x^{i})}{x^{i}}.\label{eq:eg3} \end{equation} -Should the $d$ or $\partial$ be included within the parentheses, as in -the last of (\ref{eq:eg3}), or not, as in the others? Logic says `yes'; -practice suggests (generally) `no'. +Should the $d$ or $\partial$ be included within the parentheses, +as in the last of (\ref{eq:eg3}), or not, as in the others? Logic +says `yes'; practice suggests (generally) `no'. Indicating a point of evaluation is similarly varied: \begin{equation} \diff.pvrule.\phi\varepsilon[\varepsilon=\varepsilon_{0}],\quad\diff.pvrule.[2]\phi\varepsilon[\varepsilon=\varepsilon_{0}],\quad\diff.psqbra.{b^{\beta}}{a^{\alpha}}[b=0],\quad\diff.paren.uv[v=0].\label{eq:eg4} \end{equation} -ISO 80000-2 (item 2.11.13) favours the last of these -- parentheses -- -for ordinary derivatives. Presumably, partial derivatives should follow -suit, although parentheses are also used to indicate variables held constant: +ISO 80000-2 (item 2.11.13) favours the last of these -- parentheses +-- for ordinary derivatives. Presumably, partial derivatives should +follow suit, although parentheses are also used to indicate variables +held constant: \begin{equation} \diffp*{\frac{P}{T}}U[V],\quad\diffp S{N_{2}}[U,V,N_{1}],\quad\diffp S/T[V].\label{eq:eg5} \end{equation} Other symbols besides $d$ and $\partial$ are used to denote derivative-like -quantities. From introductory calculus, classical mechanics and thermodynamics -come $\delta$ and $\Delta$, from fluid mechanics comes $D$: +quantities. From introductory calculus and from classical mechanics +and thermodynamics come $\delta$ and $\Delta$, from fluid mechanics +comes $D$: \begin{equation} \diff.delta.yx,\quad\diff.D.\rho t,\quad\diff.pDelta.UT[V],\quad\diff.Delta.U/T,\quad\diff.delta.{\mathcal{L}}{\eta^{(r)}}.\label{eq:eg6} \end{equation} @@ -205,465 +190,597 @@ There are those, like the International Organization for Standardization \begin{equation} \diff.up.yx.\label{eq:eg11} \end{equation} -When the differentiand is too big or awkward to sit in the numerator and -is appended to the operator, the $d$ or $\partial$ in the numerator is -generally centred -- but not always. In texts prior to the age of computerised -typesetting one will sometimes find the symbol pushed to the \emph{left}: +When the differentiand is too big or awkward to sit in the numerator +and is appended to the operator, the $d$ or $\partial$ in the numerator +is generally centred -- but not always. In texts prior to the age +of computerised typesetting one will sometimes find the symbol pushed +to the \emph{left}: \begin{equation} \diff.pleft.*{\diffp{x^{i^{*}}}{x^{k^{*}}}{}}{x^{l^{*}}},\quad\diff.left.*{\left(\frac{m\mathbf{q}_{x}}{\sqrt{1-q^{2}}}\right)}{t}.\label{eq:eg12} \end{equation} The observant will note an italic adjustment with the first expression, -so that the $\partial$ in the numerator and the $\partial$ in the denominator -line up in a slanting column, but no such adjustment for the $d$-s in -the second derivative. +so that the $\partial$ in the numerator and the $\partial$ in the +denominator line up in a slanting column, but no such adjustment for +the $d$-s in the second derivative. -And finally, the operator in the numerator may differ from that in the -denominator. For instance, in tensor calculus acceleration is sometimes -written +Then there is the case when the operator in the numerator differs +from that in the denominator. For instance, in tensor calculus acceleration +is sometimes written \[ \diff.nabla.{v^{i}}t=\diff{v^{i}}t+\Gamma_{k\hphantom{i}h}^{\hphantom{k}i}v^{h}\diff{y^{k}}t \] -where $\nabla v^{i}$ is the `absolute differential' of the velocity $v^{i}$. +where $\nabla v^{i}$ is the `absolute differential' of the velocity +$v^{i}$. Version 2 or later of the \texttt{diffcoeff} package has the generative -power to cope with all these variations -- see Section~\ref{sec:Changing-defaults} --- although it is unlikely an author should need to call on this capacity -to anything like the extent required for this Rogues' Gallery. +power to cope with all these variations -- see §\ref{sec:Changing-defaults} +-- although it is unlikely an author should need to call on this +capacity to anything like the extent required for this Rogues' Gallery. + +Finally and new with version 4 of \texttt{diffcoeff}, is the ability +to insert space before the differentiand, something that had to be +done explicitly by the user before. One way of thinking of a derivative +is as an operator $\diff{}x$ applied to a function $F(x)$ producing +another function $F'(x)$, a \emph{derived }function, the derivative. +Although the original function is included in the numerator of the +differential coefficient, a small space between the $d$ and $F$ +feels natural to separate the thing operated on from the thing operating: +\[ +F'(x)=\diff!{F(x)}x. +\] +\texttt{diffcoeff} can now produce such spaced derivatives, either +as the default behaviour or at explicit request. + +\newpage{} -\section{Ordinary derivatives \label{sec:Ordinary-derivatives}} +\section{Ordinary derivatives \protect\label{sec:Ordinary-derivatives}} -Writing\textbf{ }\texttt{\textbackslash diff\{y\}\{x\}} will produce $\diff{y}{x}$ -in text style (i.e., placed between \texttt{\$ \$}) or +Writing\textbf{ }\texttt{\textbackslash diff\{y\}\{x\}} will produce +$\diff{y}{x}$ in text style (i.e., placed between \texttt{\textbackslash ( +\textbackslash )} or \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 an argument -- differentiand, variable of differentiation -- -is more than a single token. +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 an argument -- the differentiand +or the variable of differentiation -- is multi-token. \begin{itemize} \item If you want upright `$\mathrm{d}$'s as default, as ISO 80000-2 recommends, -rather than the math-italic `$d$'s used here, this can easily be done; -see Section~\ref{sec:Changing-defaults} on changing default settings. +rather than the math-italic `$d$'s used here, this can easily be +done with the package option \texttt{ISO}; see §\ref{sec:Changing-defaults} +on changing default settings. \end{itemize} For inclusion in a line of text you might prefer to use a slash-fraction form of derivative. That is achieved by inserting a slash, `/', between numerator and denominator arguments: \texttt{\textbackslash diff\{\textbackslash ln -x\}/x} produces $\diff{\ln x}/x$. (Braces are required for the numerator -in this case since it contains more than one token.) - -\subsection{Order of differentiation} - -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 as is a mix of the two: -\begin{example} -\textbackslash diff{[}2{]}yx $\Longrightarrow\quad{\displaystyle \diff[2]yx,}$\medskip{} - -\textbackslash diff{[}n+1{]}yx $\Longrightarrow\quad{\displaystyle \diff[n+1]yx}.$ -\end{example} - -As mentioned, the braces can be and have been omitted around the $x$ and -$y$; the square brackets around the optional argument, the order of differentiation, -are essential. For a first-order derivative, no optional argument is needed -and entering \texttt{1} as the optional argument has no effect: -\begin{example} -\textbackslash diff{[}1{]}yx $\Longrightarrow\quad{\displaystyle \diff[1]yx.}$ -\end{example} - -\noindent In slash style, \texttt{\textbackslash diff{[}2{]}y/x} produces -$\diff[2]y/x$, and \texttt{\textbackslash diff{[}n+1{]}y/x} produces -$\diff[n+1]y/x$. - -\subsection{Multi-character variables of differentiation} - -Differentiating a function of a function may involve a multi-character -differentiation variable. For instance, to differentiate $\ln\sin x$ in -$x$ means forming the product -\begin{example} -\textbackslash diff\{\textbackslash ln\textbackslash sin x\}\{\textbackslash sin -x\}\textbackslash diff\{\textbackslash sin x\}x $\Longrightarrow\quad{\displaystyle \diff{\ln\sin x}{\sin x}\diff{\sin x}x.}$ -\end{example} - -\noindent Forming the \emph{second} derivative of $\ln\sin x$ will now -involve forming (among other quantities) -\begin{example} -\noindent \textbackslash diff{[}2{]}\{\textbackslash ln\textbackslash sin -x\}\{\textbackslash sin x\} $\Longrightarrow\quad{\displaystyle \diff[2]{\ln\sin x}{\sin x}.}$ -\end{example} - -\noindent Parentheses have been inserted automatically by \texttt{diffcoeff} -around $\sin x$ in the denominator to avoid any visual hint that we are -differentiating in the sine of the square of $x$. - -The question is: are the parentheses in the right place? Logically, no. -They should include the $d$: $(d\sin x)^{2}$ -- it is the differential -$d\sin x$ that is of the second order. But as the examples in the Rogues' -Gallery show -- see particularly (\ref{eq:eg3}) -- the inclination seems -to be to do otherwise. This may be because one wants, in any case, to parenthesise -the variable. A second, outer pair of parentheses then seems too fussy -and detracts from comprehending the symbol `at a glance': +x\}/x} produces $\diff{\ln x}/x$. Braces are required for the numerator +in this case since it contains more than one token. + +\subsection{Spacing before the differentiand} + +\label{subsec:Spacing-before-derivand}There are (at least) two different +ways in which we think of derivatives.\footnote{I thank Hans Schülein for first raising this issue with me and for +subsequent thoughtful comments. } We are all familiar with the argument presented in elementary calculus +books where a curve is shown, and a point on the curve through which +a chord has been drawn. The chord is a side -- the hypotenuse -- +of a small right-angled triangle, the other sides having lengths $\delta x$ +and $\delta y$ and being parallel to the coordinate axes. The slope +of the chord is $\diffd yx$. By drawing smaller and smaller chords +through the point, the ratio $\diffd yx$ approaches the slope of +the tangent to the curve at the point. We write \[ -\diff.wrapall.[2]{f(x)}{(1/x)}. +\diff yx \] +for the limit of $\diffd yx$. It is natural to think of $\dl y$ +and $\dl x$ as tiny lengths, like $\delta y$ and $\delta x$, in +which case it would be quite wrong to insert space between the $d$ +and the $y$ (let alone the $d$ and the $x$). $dy$ is a single +object, called a differential, and we write expressions like +\begin{centred} +\verb`\[ dy=\diff yx dx \]` $\Longrightarrow$ +\[ +\dl y=\diff yx\dl x +\] +\end{centred} +and justly call the `fraction' in this expression a differential +coefficient. + +But there is another way of viewing differentiation: as a process +producing (or \emph{deriving}) one function, $f'(x)$, from another, +$f(x)$. Here the sense is of applying $\diff{}x$ to $f(x)$. Although +we include $f(x)$ in the numerator it is not attached to the $d$ +and should be separated from it by a small space: +\begin{centred} +\verb`\[ f'(x)=\diff!{f(x)}x \]` $\Longrightarrow$ +\[ +f'(x)=\diff!{f(x)}x. +\] +\end{centred} +Here the `fraction' on the right is another name for the derived +function $f'$ and is justly called the derivative of $f$. As you +can see a small space has been inserted between the $d$ and the $f$ +in the numerator. By default the space is \texttt{3 mu} but with the +ability to stretch \texttt{1 mu} or shrink\texttt{ 2 mu}{\ttfamily\footnote{In \TeX -speak, \texttt{3mu plus 1mu minus 2mu}. }} +as \TeX{} adjusts lines to fit on the page. (A `mu' is a `math unit' +and is one eighteenth of a quad.) To achieve this result I have inserted +an exclamation mark \verb`!` before the braces delimiting the differentiand. +\begin{itemize} +\item You may want all or most of your derivatives to have this space and +therefore will not want to be inserting exclamation marks in every +\texttt{\textbackslash diff} command. The \texttt{spaced} package +option switches the default behaviour to spaced derivatives (in which +case the \texttt{!} switch now creates an \emph{un-spaced} derivative). +The size of the space inserted by default can be easily changed; see +§\ref{sec:Changing-defaults}. +\end{itemize} +The present document uses the un-spaced default. For backwards compatibility +(and perhaps because the author has done this for the last 60 years), +this is the \textquoteleft out-of-the-box\textquoteright{} default +that \texttt{diffcoeff} uses. Authors should make a habit of using +one form predominantly. (There is also the possibility which I haven't +emphasized of spacing multi-token differentiands and leaving single-token +differentiands unspaced; see §\ref{subsec:A-final-flourish}.) + +Slash-form derivatives also allow space before the differentiand. +By default this has the same value as applied to the fraction form +of derivative (but can be changed; see §\ref{sec:Changing-defaults}): +\begin{centred} +\verb`$ \diff!{\ln\sin x}/x $` $\Longrightarrow$ $ \diff!{\ln\sin x}/x$. +\end{centred} +The \verb`\diff` command has other optional arguments (e.g. to specify +the order of differentiation) but always the exclamation mark, if +used, immediately precedes the differentiand. It is the positioning +of the differentiand that it affects after all and its placement immediately +before that argument seems natural. + +\subsubsection{Ligatures: } + +Prior to v.4 of \texttt{diffcoeff}, \texttt{\textbackslash diff fx} +produced the tight pairing evident in +\[ +\frac{df}{dx}. +\] +This was a bug. From version 4.0 of \texttt{diffcoeff}, such ligatures +are prevented: \verb`\[ \diff fx \]` $\Longrightarrow$ \[ \diff fx.\] -Customary but illogical notations are familiar in mathematics -- think -of the position of the superscripts in an identity like $\sin^{2}\theta+\cos^{2}\theta=1$. -But, like other features of the derivative, the manner of this wrapping -in parentheses of long variables for \emph{higher order} derivatives is -customisable; see Section~\ref{sec:Changing-defaults}. - -For first order derivatives, parenthesising does not occur. If you want -the variable of differentiation to be parenthesised, you need to insert -them yourself: -\begin{example} -\textbackslash diff \{f(x)\}\{1/x\}, \textbackslash quad\textbackslash diff -\{f(x)\}\{(1/x)\} $\Longrightarrow\quad{\displaystyle \diff{f(x)}{1/x},\quad\diff{f(x)}{(1/x)}.}$ -\end{example} - - -\subsubsection{Minutiae of spacing} +\subsubsection{Spacing commands} -You may find the spacing between the `d' and the `f' in the last example -uncomfortably close. The \texttt{diffcoeff} package offers four simple -spacing commands to fine-tune the display. These are +\label{subsec:Spacing-commands}The \texttt{diffcoeff} package also +offers four simple spacing commands to fine-tune the display of derivatives +and of other quantities. These are \begin{description} \item [{\texttt{\textbackslash negmu}}] insert spacing of $-1$ mu -\item [{\texttt{\textbackslash nilmu}}] insert spacing of $0$ mu +\item [{\texttt{\textbackslash nilmu}}] insert spacing of $0$ mu (cf. +use of an empty brace pair \texttt{\textbf{\{\}}} ) \item [{\texttt{\textbackslash onemu}}] insert spacing of $1$ mu \item [{\texttt{\textbackslash twomu}}] insert spacing of $2$ mu \end{description} -Thus for the last example, inserting \texttt{\textbackslash nilmu} and -\texttt{\textbackslash onemu} in the appropriate places produces -\begin{example} -\textbackslash diff \{\textbackslash nilmu f(x)\}\{\textbackslash onemu -1/x\}, \textbackslash quad \textbackslash diff \{\textbackslash nilmu -f(x)\}\{(1/x)\} $\Longrightarrow\quad{\displaystyle \diff{\nilmu f(x)}{\onemu1/x},\quad\diff{\nilmu f(x)}{(1/x)}.}$ -\end{example} - \subsection{Appending the differentiand: \texttt{\textbackslash diff{*}}} -Some differentiands are too big or awkward to be placed neatly in the numerator -of a derivative and it is natural to append them to a preceding differential -operator. One way to do this is to leave the numerator argument empty in -the \texttt{\textbackslash diff} command and follow the command with the -differentiand. A better way is to star the \texttt{\textbackslash diff} -command. This tells \texttt{diffcoeff} to append the differentiand. Thus -suppose the differentiand is a polynomial, say $ax^{2}+bx+c$. Add a star -(an asterisk) to the \texttt{\textbackslash diff} command: -\begin{example} -\textbackslash diff{*}\{(ax\textasciicircum 2+bx+c)\}x $\Longrightarrow\quad{\displaystyle \diff*{(ax^{2}+bx+c)}x.}$ -\end{example} - -A virtue of using an asterisk is that if one isn't sure whether a differentiand -should be appended or not, it is an easy matter to simply insert or delete -the asterisk to compare the results. For example, a second derivative is -an iterated derivative -- one in which a derivative forms the differentiand -of another derivative: -\begin{example} -\textbackslash diff{[}2{]}yx = \textbackslash diff{*}\{\textbackslash diff -yx\}x $\Longrightarrow\quad{\displaystyle \diff[2]yx=\diff*{\diff yx}x},$ -\end{example} - +Some differentiands are too big or awkward to be placed neatly in +the numerator of a derivative and it is natural to append them to +a preceding differential operator. One way to do this is to leave +the numerator argument empty in the \texttt{\textbackslash diff} +command and follow the command with the differentiand. A better way +is to star the \texttt{\textbackslash diff} command. This tells \texttt{diffcoeff} +to append the differentiand. Thus suppose the differentiand is a polynomial, +say $ax^{2}+bx+c$. Add a star (an asterisk) to the \texttt{\textbackslash diff} +command: +\begin{centred} +\verb`\[ \diff*{(ax^2+bx+c)}x \]` $\Longrightarrow$ \[\diff*{(ax^{2}+bx+c)}x\] +\end{centred} +With the \texttt{!} switch or \texttt{spaced} package option, additional +space (by default \texttt{3mu plus 1mu minus 2mu}) is inserted between +the operator and the differentiand: +\begin{centred} +\verb`\[ \diff*!{(ax^2+bx+c)}x \]` $\Longrightarrow$ \[\diff*!{(ax^{2}+bx+c)}x.\] +\end{centred} +A virtue of using an asterisk to append the differentiand is that +if one isn't sure whether a differentiand should be appended or not, +it is an easy matter to simply insert or delete the asterisk to compare +the results. For example, a second derivative is an iterated derivative +-- one in which a derivative forms the differentiand of another derivative: +\begin{centred} +\verb`\[ \diff[2]yx = \diff*{\diff yx}x \]` $\Longrightarrow$ \[ \diff[2]yx=\diff*{\diff yx}x \] +\end{centred} \noindent which is more elegant to my eye than -\begin{example} -\noindent \textbackslash diff{[}2{]}yx = \textbackslash diff\{\textbackslash diff -yx\}x $\Longrightarrow\quad{\displaystyle \diff[2]yx=\diff{\diff yx}x},$ -\end{example} - +\begin{centred} +\noindent \verb`\[ \diff[2]yx = \diff!{\diff yx}x \]` $\Longrightarrow$ +\[ \diff[2]yx=\diff!{\diff yx}x \] +\end{centred} \noindent although whether the \emph{meaning} is clearer is moot.\emph{ -}It is easy to switch between the two forms on the right, simply by inserting -or removing the asterisk. +}It is easy to switch between the two forms on the right, simply by +inserting or removing the asterisk. In slash style with the star option, the polynomial example becomes -\begin{example} -\textbackslash diff{*}\{(ax\textasciicircum 2+bx+c)\}/x $\Longrightarrow\quad\text{\ensuremath{{\displaystyle (\diff{}/{x})(ax^{2}+bx+c)}}, }$ -\end{example} - -\noindent where the parentheses around the differential operator are automatically -inserted by \texttt{diffcoeff}. Like other elements of automatic formatting, -this is user-adjustable; see Section~\ref{sec:Changing-defaults}. - -\subsection{Point of evaluation\label{subsec:Point-of-evaluation}} - -If you want to specify a point at which the derivative is evaluated, append -a final optional argument. Note that there \emph{must be no space} before -the left square bracket of the argument: -\begin{example} -\textbackslash diff{[}2{]}yx{[}0{]} $\Longrightarrow\quad{\displaystyle \diff[2]yx[0]}$ -\end{example} - -\noindent If a space does slip in before the final optional argument, it -will not cause a \LaTeX{} error. Instead, the argument will be treated as -a square-bracketed mathematical expression following the derivative, and -typeset as such. +\begin{centred} +\verb`\[ \diff*{(ax^2+bx+c)}/x \]` $\Longrightarrow$ \[ \diff*{(ax^2+bx+c)}/x \] +\end{centred} +\noindent where the parentheses around the differential operator are +automatically inserted by \texttt{diffcoeff}. Like other elements +of automatic formatting, this is user-adjustable; see §\ref{sec:Changing-defaults}. + +With the \texttt{!} switch or \texttt{spaced} package option, this +becomes +\begin{centred} +\verb`\[ \diff*!{(ax^2+bx+c)}/x \]` $\Longrightarrow$ \[ \diff*!{(ax^2+bx+c)}/x \] +\end{centred} + +\subsection{Point of evaluation\protect\label{subsec:Point-of-evaluation}} + +If you want to specify a point at which the derivative is evaluated, +append a final optional argument. Note that there \emph{must be no +space} before the left square bracket of the argument:\footnote{In v.1 of \texttt{diffcoeff}, this was brace-delimited. From v.2, +square-brackets have been used and braces deprecated. From v.4, braces +are no longer accepted.} +\begin{centred} +\verb`\[ \diff[2]yx[0] \]` $\Longrightarrow\quad{\displaystyle \diff[2]yx[0]}$ +\end{centred} +\noindent If a space does slip in before the final optional argument, +it will not cause a \LaTeX{} error. Instead, the argument will be treated +as a square-bracketed mathematical expression following the derivative, +and typeset as such. \begin{itemize} \item If you prefer to use subscripted \emph{parentheses} around the derivative -to indicate a point of evaluation -- as ISO 80000-2 recommends -- then -this can easily be done; see Section~\ref{sec:Changing-defaults} on changing -default settings. Or use the \texttt{ISO} package option; see the introduction. +to indicate a point of evaluation -- as ISO 80000-2 recommends -- +then this can easily be done; see §\ref{sec:Changing-defaults} on +changing default settings. Or, more simply, use the \texttt{ISO} package +option. \end{itemize} Because the slash form spreads the derivative out horizontally, parentheses are preferred here to indicate a point of evaluation: -\begin{example} -\textbackslash diff\{\textbackslash ln sin x\}/\{sin x\}{[}x=\textbackslash pi/6{]} -$\Longrightarrow\quad{\displaystyle \diff{\ln\sin x}/{\sin x}[x=\pi/6]}$ -\end{example} - -\noindent A vertical rule (or `pipe') can become too remote from the opening -$d$ of the differential coefficient: $\diff.svrule.{\ln\sin x}/{\sin x}[x=\pi/6]$; +\begin{centred} +\verb`$ \diff{\ln sin x}/{sin x}[x=\pi/6] $` $\Longrightarrow$ $ \diff{\ln\sin x}/{\sin x}[x=\pi/6] $. +\end{centred} +\noindent A vertical rule (or `pipe') can easily become too remote +from the opening $d$ of the differential coefficient: $\diff.svrule.{\ln\sin x}/{\sin x}[x=\pi/6]$; parentheses tie the whole cluster of symbols together. \subsubsection{Superscripts} It is easy to add a superscript to a derivative to indicate evaluation at two points and the difference between the values: -\begin{example} -\textbackslash diff \{\textbackslash sin x\}x{[}0{]}\textasciicircum\{\textbackslash pi/2\} -${\displaystyle \Longrightarrow\quad\diff{\sin x}x[0]^{\pi/2}}$ -\end{example} +\begin{centred} +\verb`\[ \diff {\sin x}x[0]^{\pi/2} \]` ${\displaystyle \Longrightarrow}$ +\[ \diff{\sin x}x[0]^{\pi/2} \] +\end{centred} +\noindent If you want only the superscript, no subscript, include +the final optional argument but leave it empty. Thus, for a particle +of mass $m$ moving along a line, distance $x$ at time $t$, the +kinetic energy is: +\begin{centred} +\noindent \verb`$ \tfrac 12 m \diff x/t[]^2 $` $\Longrightarrow$ +$\tfrac{1}{2}m\diff x/t[]^{2}$. +\end{centred} -\noindent If you want only the superscript, no subscript, include the final -optional argument but leave it empty. Thus, for a particle of mass $m$ -moving along a line, distance $x$ at time $t$, the kinetic energy is: -\begin{example} -\noindent \textbackslash tfrac 12 m \textbackslash diff x/t{[}{]}\textasciicircum 2 -$\Longrightarrow\quad{\displaystyle \tfrac{1}{2}m\diff x/t[]^{2}}.$ -\end{example} +\subsection{Order of differentiation} + +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 as is a mix of the two: +\begin{centred} +\verb`\[ \diff[2]yx \]` $\Longrightarrow$ \[ \diff[2]yx,\]\verb`\[ \diff[n+1]yx \]` +$\Longrightarrow$ \[ \diff[n+1]yx.\] +\end{centred} +As mentioned, the braces can be and have been omitted around the $x$ +and $y$; the square brackets around the optional argument, the order +of differentiation, are essential. For a first-order derivative, no +optional argument is needed and entering \texttt{1} as the optional +argument has no effect: +\begin{centred} +\verb`$ \diff[1]yx $` $\Longrightarrow$$\diff[1]yx$. +\end{centred} +In slash style, \texttt{\$\textbackslash diff{[}2{]}y/x\$} produces +$\diff[2]y/x$, and \texttt{\$\textbackslash diff{[}n+1{]}y/x\$} +produces $\diff[n+1]y/x$. +\subsection{Multi-character variables of differentiation} + +Differentiating a function of a function may involve a multi-character +differentiation variable. For instance, to differentiate $\ln\sin x$ +in $x$ means forming the product +\begin{centred} +\verb`\[ \diff!{\ln\sin x}{\sin x}\diff{\sin x}x \]` $\Longrightarrow$ +\[\diff!{\ln\sin x}{\sin x}\diff{\sin x}x.\] +\end{centred} +\noindent (Although I am mainly using un-spaced differentiands in +this document, a space before the differentiand in the first of these +derivatives -- the \verb`!` switch -- improves the appearance of +the expression to my eye.) + +Forming the \emph{second} derivative of $\ln\sin x$ will now involve +forming (among other quantities) +\begin{centred} +\noindent \verb`\[\diff[2]{\ln\sin x}{\sin x}\]` $\Longrightarrow$ +\[\diff[2]{\ln\sin x}{\sin x}\] +\end{centred} +\noindent Parentheses have been inserted automatically by \texttt{diffcoeff} +around $\sin x$ in the denominator to avoid any visual hint that +we are differentiating in the sine of $x^{2}$. + +The question is: are the parentheses in the right place? Logically, +no. They should include the $d$: $(d\sin x)^{2}$ -- it is the differential +$d\sin x$ that is of the second order. But as the examples in the +Rogues' Gallery show -- see particularly (\ref{eq:eg3}) -- the +inclination seems to be to do otherwise. This may be because one wants, +in any case, to parenthesise the variable. A second, outer pair of +parentheses then seems too fussy and detracts from comprehending the +symbol `at a glance': +\[ +\diff.wrapall.[2]{f(x)}{(1/x)}. +\] -\section{Partial derivatives\label{sec:Partial-derivatives}} -\noindent \begin{flushleft} -Partial derivatives follow the same pattern as ordinary derivatives, with -some extensions. The command this time is \texttt{\textbackslash diffp}. -Thus \texttt{\textbackslash diffp\{F\}\{x\}}, or, with a saving on keystrokes,\texttt{ -\textbackslash diffp Fx}, produces $\diffp Fx$ in text style and +Customary but illogical notations are familiar in mathematics -- +think of the position of the superscripts in an identity like $\sin^{2}\theta+\cos^{2}\theta=1$. +But, like other features of the derivative, the manner of this wrapping +in parentheses of long variables for \emph{higher order} derivatives +is customisable; see §\ref{sec:Changing-defaults}. + +For first order derivatives, parenthesising does not occur. If you +want the variable of differentiation to be parenthesised, you need +to insert them yourself: +\begin{centred} +\verb`\[\diff {f(x)}{1/x}, \quad \diff {f(x)}{(1/x)}.\]` $\Longrightarrow$ +\[ \diff {f(x)}{1/x}, \quad\diff {f(x)}{(1/x)}.\] +\end{centred} + +\section{Partial derivatives\protect\label{sec:Partial-derivatives}} + +\noindent Partial derivatives follow the same pattern as ordinary +derivatives, with some extensions. The command this time is \texttt{\textbackslash diffp}. +Thus \texttt{\textbackslash diffp\{F\}\{x\}}, or, with a saving on +keystrokes,\texttt{ \textbackslash diffp Fx}, produces $\diffp Fx$ +in text style and \[ \diffp{F}{x} \] -in display style. (As for \texttt{\textbackslash diff}, the omission of -braces is possible when the differentiand or the differentiation variable -are single tokens.) As for \texttt{\textbackslash diff}, there is a slash -form, generally preferred for inline use, \texttt{\textbackslash diffp +in display style. (As for \texttt{\textbackslash diff}, the omission +of braces is possible when the differentiand or the differentiation +variable are single tokens.) As for \texttt{\textbackslash diff}, +there is a slash form, generally preferred for inline 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: -\par\end{flushleft} -\begin{example} -\textbackslash diffp{[}3{]}F/x , \textbackslash quad \textbackslash diffp{[}n{]}F/x -$\Longrightarrow\quad{\displaystyle {\displaystyle {\displaystyle \diffp[3]F/x}},\quad{\displaystyle \diffp[n]F/x.}}$\medskip{} - \textbackslash diffp{[}n+1{]}Fx $\Longrightarrow\quad{\displaystyle {\displaystyle \diffp[n+1]Fx,}}$ -\end{example} - +takes 8 keystrokes to type, the slash form \emph{does }economise on +keystrokes for a partial derivative. + +With either the \verb`spaced` package option (see §\ref{subsec:Package-options}) +or the \verb`!` switch a space, defaulting to \verb`3 mu` with some +stretch and shrink, can be inserted before the differentiand: \verb`\[ \diffp!{F(x,y)}x \]` +$\Longrightarrow$ \[ \diffp!{F(x,y)}x. \] + +\noindent (But note that if the \verb`spaced` package option is used, +the \verb`!` switch \emph{removes} any extra space.) + +Again an optional argument allows the specification of the order of +differentiation which may be numeric or algebraic or a mix of the +two: +\begin{centred} +\verb`\[ \diffp[3]F/x , \quad \diffp[n]F/x \]` $\Longrightarrow$ +\[\diffp[3]F/x , \quad \diffp[n]F/x\] \verb`\[ \diffp[n+1]Fx. \]` +$\Longrightarrow$ \[ \diffp[n+1]Fx. \] +\end{centred} \subsection{Variables held constant} -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 square-bracketed optional argument and ensure that -it follows \emph{immediately} on the preceding mandatory argument. A space -here will detach the argument from the derivative and result in it being -treated as a mathematical expression following the derivative. Thus to -differentiate the entropy $S$ in temperature $T$ while holding the volume -$V$ constant, write -\begin{example} -\textbackslash diffp ST{[}V{]} $\Longrightarrow\quad{\displaystyle \diffp ST[V]}$ -\end{example} - +In a subject like thermodynamics, there is a need to indicate which +variables are held constant when the differentiation occurs. To show +this, append a final square-bracketed optional argument and ensure +that it follows \emph{immediately} on the preceding mandatory argument. +A space here will detach the argument from the derivative and result +in it being treated as a mathematical expression following the derivative. +Thus to differentiate the entropy $S$ in temperature $T$ while holding +the volume $V$ constant, write +\begin{centred} +\verb`\[ \diffp ST[V] \]` $\Longrightarrow$ \[ \diffp ST[V]. \] +\end{centred} \noindent In slash form the same expression looks like -\begin{example} -\noindent \textbackslash diffp S/T{[}V{]} $\Longrightarrow\quad{\displaystyle \diffp{S}/{T}[V]}.$ -\end{example} - -This use of a parenthesised, subscripted form to indicate a variable or -variables held constant, leaves open the question: how do we represent +\begin{centred} +\noindent \verb`$ \diffp S/T[V] $` $\Longrightarrow$ $ \diffp S/T[V] $. +\end{centred} +This use of a parenthesised, subscripted form to indicate a variable +or variables held constant, leaves open the question: how do we represent a point of evaluation? ISO 80000-2 makes no recommendation for \emph{partial} derivatives; presumably we follow the same practice as their recommendation for ordinary derivatives: \begin{example} -\textbackslash diffp \{F(x,y)\}x{[}(0,0){]} $\Longrightarrow\quad{\displaystyle \diffp{F(x,y)}x[(0,0)]}$ +\verb`\[ \diffp {F(x,y)}x[(0,0)] \]` $\Longrightarrow$ \[ \diffp {F(x,y)}x[(0,0)] \] \end{example} -However, you may prefer (as many do) to use a vertical rule for this purpose: +However, you may prefer (as many do) to use a vertical rule for this +purpose: \[ \diff.pvrule.{F(x,y)}x[(0,0)] \] -Making this possibility available is discussed in Section~\ref{sec:Changing-defaults}. - -An empty final argument produces a parenthesised derivative with no subscript: -\begin{example} -\textbackslash diffp yx{[}{]}${\displaystyle \Longrightarrow\quad\diffp yx[]}.$ -\end{example} - -\noindent This can be useful sometimes, e.g. for writing Lagrange's equations -of motion in analytic mechanics: -\begin{example} -\noindent \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} - +Making this possibility available is discussed in §\ref{sec:Changing-defaults}. + +An empty final argument produces a parenthesised derivative with no +subscript, +\begin{centred} +\verb`\[ \diffp yx[] \]` ${\displaystyle \Longrightarrow}$ \[ \diffp yx[] \] +\end{centred} +\noindent which can be useful sometimes. An instance is the writing +of Lagrange's equations of motion in analytic mechanics: +\begin{centred} +\noindent \verb`\[ \diffp L{q_k}-\diff*{\diffp L{\dot{q}_k}[]}t = 0 \]` +$\Longrightarrow$ \[ \diffp L{q_k}-\diff*{\diffp L{\dot{q}_k}[]}t = 0. \] +\end{centred} \subsubsection{Text-style derivatives} -The \texttt{diffcoeff} package assumes that derivatives formed as `numerator -over denominator' will be used in display-style expressions, and that the -slash form will be used for inline use (text style). This is the familiar -practice in the literature. If one \emph{does} use the first form in an -inline expression where a variable is held constant, say \texttt{\textbackslash diffp -ST{[}V{]}} as here $\diffp ST[V]$, the result is unsatisfactory, the subscript -too tight on the closing parenthesis and too much space between parentheses -and derivative. The matter is easily resolved using `variant forms' -- -see Subsection~\ref{subsec:Text-and-script-style} below -- giving, for -our example, $\diff.ptxt.ST[V]$. +The \texttt{diffcoeff} package assumes that derivatives formed as +`numerator over denominator' will be used in display-style expressions, +and that the slash form will be used for inline use (text style). +This is the familiar practice in the literature. If one \emph{does} +use the first form in an inline expression where a variable is held +constant, say \texttt{\textbackslash diffp ST{[}V{]}} as here $\diffp ST[V]$, +the result is unsatisfactory, the subscript too tight on the closing +parenthesis and too much space between parentheses and derivative. +The matter is easily resolved using `variant forms' -- see §\ref{subsec:Text-and-script-style} +below -- giving, for our example, $\diff.ptxt.ST[V]$. -\subsection{Appending the differentiand\label{subsec:Partial-appending}} +\subsection{Appending the differentiand\protect\label{subsec:Partial-appending}} For a long or awkward differentiand, it is generally better to \emph{append} it to a preceding differential operator, rather than create a fractional form with the long expression in the numerator. As with ordinary derivatives, this is achieved by adding an asterisk to (i.e. by starring) the \texttt{\textbackslash diffp} -command. -\begin{example} -\textbackslash diffp{*}{[}2{]}\{\textbackslash Phi(x,y,z)\}x $\Longrightarrow\quad{\displaystyle \diffp*[2]{\Phi(x,y,z)}x.}$ -\end{example} - -\noindent Alternatively you could leave the first mandatory argument empty -and manually append the differentiand, but by deleting or inserting an +command. +\begin{centred} +\verb`\[ \diffp*[2]{\Phi(x,y,z)}x \]` $\Longrightarrow$ \[ \diffp*[2]{\Phi(x,y,z)}x. \] +\end{centred} +With the \verb`spaced` package option, or with the \verb`!` switch +if the package option is not used (which is the case for this document), +one gets +\begin{centred} +\verb`\[ \diffp*[2]!{\Phi(x,y,z)}x \]` $\Longrightarrow$ \[ \diffp*[2]!{\Phi(x,y,z)}x \] +\end{centred} +which is an improvement (to my eye). + +Alternatively you could leave the first mandatory argument empty and +manually append the differentiand, but by deleting or inserting an asterisk, it is easy to compare the two forms, differentiand-in-the-numerator, differentiand-appended, and see which is preferable. In slash form, parentheses are automatically inserted around the differential operator when the differentiand is appended, -\begin{example} -\textbackslash diffp{*}{[}n{]}\{f(x)\}/x $\Longrightarrow\quad{\displaystyle \diffp*[n]{f(x)}/x,}$ -\end{example} - -\noindent although this behaviour can be changed (Section~\ref{sec:Changing-defaults} +\begin{centred} +\verb`\[ \diffp*[n]{f(x)}/x \]` $\Longrightarrow$ \[ \diffp*[n]{f(x)}/x, \] +\end{centred} +\noindent although this behaviour can be changed (§\ref{sec:Changing-defaults} again). If you wish to both append the differentiand \emph{and} indicate variables held constant, then the starred form is much the easier way to achieve this. Thus, to express a relation in thermodynamics, -\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} - +\begin{centred} +\verb`\[ \diffp*{\frac PT}U[V] = \diffp*{\frac 1T}V[U] \]` $\Longrightarrow$ +\[ \diffp*{\frac PT}U[V] = \diffp*{\frac 1T}V[U], \] +\end{centred} \noindent where the starring automatically takes care of the parentheses and subscripts. \subsection{Iterated derivatives} -Derivatives can be the differentiands of derivatives, as seen above when -writing Lagrange's equations, -\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} - -\noindent However, in versions of \texttt{diffcoeff} before version 3.2, -an ordinary derivative within a partial derivative rendered as a partial -derivative. That is now corrected, the ordinary derivative rendering correctly: -\begin{example} -\textbackslash diffp{*}\{\textbackslash diff\{x\textasciicircum\textbackslash mu\}\{\textbackslash lambda\}\}\{x\textasciicircum\textbackslash sigma\} -$\Longrightarrow\quad{\displaystyle \diffp*{\diff{x^{\mu}}{\lambda}}{x^{\sigma}}}$ -\end{example} - +Derivatives can be the differentiands of derivatives, as seen above +when writing Lagrange's equations, +\begin{centred} +\noindent \verb`\[ \diffp L{q_k}-\diff*{\diffp L{\dot{q}_k}[]}t = 0 \]` +$\Longrightarrow$ \[ \diffp L{q_k}-\diff*{\diffp L{\dot{q}_k}[]}t = 0. \] +\end{centred} +\noindent However, in versions of \texttt{diffcoeff} before version +3.2, an ordinary derivative within a partial derivative rendered as +a partial derivative. That is now corrected, the ordinary derivative +rendering correctly: +\begin{centred} +\verb`\[ \diffp*{\diff{x^\mu}{\lambda}}{x^\sigma} \]` $\Longrightarrow$ +\[ \diffp*{\diff{x^\mu}{\lambda}}{x^\sigma}. \] +\end{centred} \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. Suppose $F$ is a function -of three variables, $x$, $y$ and $z$. Then -\begin{example} -\textbackslash diffp F\{x,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 -- \texttt{diffcoeff} does the -calculation. There is also a slash form: -\begin{example} -\textbackslash diffp F/\{x,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 for the 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 -- 6 -- 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 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 differentiation of $z$ is 1 by default. But you \emph{cannot} -use an order specification like \texttt{{[} , ,2{]}}. Instead write \texttt{{[}1,1,2{]}}.\textbf{ -}It is only the \emph{tail} of an order specification which can be omitted. - -\subsubsection{Minutiae of spacing (again)\label{subsec:Minutiae-of-spacing}} - -In Chapter 18 of the \emph{The\TeX book}, Knuth suggests inserting a thin -space, \texttt{\textbackslash ,} (or 3 mu), between differentials in appropriate -contexts, giving as an example $dx\,dy=r\,dr\,d\theta$. In the denominator -of a derivative, however, that degree of extra spacing seems too great -(to my eye), interfering with seeing the derivative `as a whole', +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. +Suppose $F$ is a function of three variables, $x$, $y$ and $z$. +Then +\begin{centred} +\verb`\[ \diffp F{x,y,z} \]` $\Longrightarrow$ \[ \diffp F{x,y,z}, \] + +\verb`\[ \diffp !{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$ \[ \diffp !{F(x,y,z)}{x,y,z}. \] +\end{centred} +In the second of these I have inserted space before the differentiand +by using the \verb`!` switch. If the \verb`spaced` package option +was being used, this would have been done automatically. + +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 -- \texttt{diffcoeff} +does the calculation. There is also a slash form: +\begin{centred} +\verb`$ \diffp F/{x,y,z} $` $\Longrightarrow$ $ \diffp F/{x,y,z}$. +\end{centred} +If we want to differentiate variables to higher order, then their +orders need to be specified explicitly. To do so use a comma list +for the optional argument: +\begin{centred} +\verb`\[ \diffp[2,3]F{x,y,z} \]` $\Longrightarrow$ \[ \diffp[2,3]F{x,y,z}. \] +\end{centred} +\noindent Notice that the overall order of the derivative -- 6 -- +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 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 differentiation +of $z$ is 1 by default. But you \emph{cannot} use an order specification +like \texttt{{[} , ,2{]}}. Instead write \texttt{{[}1,1,2{]}}.\textbf{ +}It is only the \emph{tail} of an order specification which can be +omitted. + +\subsubsection{Minutiae of spacing (again)\protect\label{subsec:Minutiae-of-spacing}} + +In Chapter 18 of the \emph{The\TeX book}, Knuth suggests inserting +a thin space, {\ttfamily\verb`\,`} (or {\ttfamily\verb`3 mu`}), +between differentials in appropriate contexts, giving as an example +$dx\,dy=r\,dr\,d\theta$. In the denominator of a derivative, however, +that degree of extra spacing seems too great (to my eye), interfering +with seeing the derivative `as a whole', \[ \diff.pwide.F{x,y,z}, \] especially for the slash-form of derivative: $\diff.pwide.F/{x,y,z}$. -Some spacing is desirable, but less; by default \texttt{diffcoeff} inserts -2 mu between the differentials in the fraction form of derivative and 1 -mu in the slash form. +Some spacing is desirable, but less; by default \texttt{diffcoeff} +inserts \verb`2 mu` between the differentials in the fraction form +of derivative and \verb`1 mu`\texttt{ }in the slash form. Should a differentiation occur to higher order and so a variable acquire -a superscript, an adjustment is made to the extra spacing. By default 1 -mu is subtracted from the default spacing. Thus in +a superscript, an adjustment is made to the extra spacing. By default +\verb`1 mu` is subtracted from the default spacing. Thus in \[ \diffp[2]F{x,y,z}, \] -2 mu of spacing is inserted between the $\partial y$ and $\partial z$, -but because the superscript already provides some separation between them, -only 1 mu is inserted between $\partial x^{2}$ and $\partial y$. - -The values used for the spacing and its adjustment in the presence of a -superscript can be adjusted by the user; see Section~\ref{sec:Changing-defaults}. -Some other minutiae of spacing are discussed below in Subsection~\ref{subsec:Comma-list-of-vars} -when the variables themselves are super- or subscripted (as happens in -tensor calculus, for instance). +\verb`2 mu` of spacing is inserted between the $\partial y$ and +$\partial z$, but because the superscript already provides some separation +between them, only \verb`1 mu` is inserted between $\partial x^{2}$ +and $\partial y$. + +The values used for the spacing and its adjustment in the presence +of a superscript can be changed by the user; see §\ref{sec:Changing-defaults}. +Some other minutiae of spacing are discussed below in §\ref{subsec:Comma-list-of-vars} +when the variables themselves are super- or subscripted (as happens +in tensor calculus, for instance). + +Note that the \verb`!` switch, if used, is inserted \emph{immediately +before} the differentiand. It is the placement of the differentiand, +after all, that it is affecting: +\begin{centred} +\verb`\[ \diffp[2]!F{x,y,z} \]` $\Longrightarrow$ \[ \diffp[2]!F{x,y,z}. \] +\end{centred} \subsubsection{Algebraic orders of differentiation} -Calculation of the overall order of differentiation still occurs even when -some or all of the individual orders are algebraic variables rather than -numbers. For example, differentiating in three variables to orders \texttt{1, -m+1, m-1}, -\begin{example} -\textbackslash diffp{[}1,m+1,m-1{]}\{F(x,y,z)\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[1,m+1,m-1]{F(x,y,z)}{x,y,z}.}$ -\end{example} - +Calculation of the overall order of differentiation still occurs even +when some or all of the individual orders are algebraic variables +rather than numbers. For example, differentiating in three variables +to orders \texttt{1, m+1, m-1}, +\begin{centred} +\verb`\[ \diffp[1,m+1,m-1]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[1,m+1,m-1]{F(x,y,z)}{x,y,z}. \] +\end{centred} \subsubsection{Error messages} -Should you specify \emph{more} orders in the comma list of the order specification -than there are variables, compilation will fail and an error message will -be sent to the terminal and \LaTeX{} log . For example, if on line 53 (say) -of my document I specify \texttt{{[}1,m-1,m+1,2{]}} for the orders of differentiation -but list only \texttt{\{x,y,z\}} for the variables, the message will be +Should you specify \emph{more} orders in the comma list of the order +specification than there are variables, compilation will fail and +an error message will be sent to the terminal and \LaTeX{} log . For +example, if on (say) line 53 of my document I specify \texttt{{[}1,m-1,m+1,2{]}} +for the orders of differentiation but list only \texttt{\{x,y,z\}} +for the variables, the message will be \begin{lyxcode} !~Package~diffcoeff~Error:~4~orders~of~differentiation~ @@ -671,9 +788,9 @@ specified~for~3~variables;~orders~{[}1,m+1,m-1,2{]} (on~line~53)~for~variables~x,y,z. \end{lyxcode} -Should you try using a \emph{number} raised to a power in an order specification, -it will cause an error. For example, if on line 53 (say) of my document -I try to form \texttt{\textbackslash diffp{[}2\textasciicircum 2{]}f\{x,y,z\}} +Should you try using a \emph{number} raised to a power in an order +specification, it will cause an error. For example, if on line 53 +(say) of my document I try to form \texttt{\textbackslash diffp{[}2\textasciicircum 2{]}f\{x,y,z\}} compilation will fail with the message \begin{lyxcode} !~Package~diffcoeff~Error:~number~followed~by~\textasciicircum ~in~the~ @@ -686,120 +803,118 @@ in~this~case.~Use~the~order-override~option~to~enter~ the~overall~order. \end{lyxcode} -Similarly, you cannot use multiplication (\texttt{\textbackslash times} -or \texttt{{*}})\texttt{ }or division (\texttt{/} or \texttt{\textbackslash div}) -signs after \emph{numbers} in an order specification; only \texttt{+} or -\texttt{-} or a left parenthesis (see below) are allowed. +The order-overide option mentioned here is discussed next. Similarly, +you cannot use multiplication (\texttt{\textbackslash times} or \texttt{{*}})\texttt{ +}or division (\texttt{/} or \texttt{\textbackslash div}) signs after +\emph{numbers} in an order specification; only \texttt{+} or \texttt{-} +or a left parenthesis (see below) are allowed. \subsubsection{The order-override option} \noindent Perhaps the differentiations are to orders \texttt{{[}2km,m-1,m+1{]}}: -\begin{example} -\noindent \textbackslash diffp{[}2km,m-1,m+1{]}\{F(x,y,z)\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[2km,m-1,m+1]{F(x,y,z)}{x,y,z}.}$ -\end{example} - -\noindent Here the overall order is presented as \texttt{2km+2m}. You might -prefer this to be presented as \texttt{2m(k+1)}. Although \texttt{diffcoeff} -takes some steps to present the overall order appropriately, it is not -a computer algebra system and does not factorise expressions. If you want -to present the order in a manner distinct from that presented by \texttt{diffcoeff}, -use the \emph{order-override option}. - -This is a second optional argument immediately following the order specification. -For our last example, filling the override option with \texttt{2m(k+1)} -gives the desired result: -\begin{example} -\textbackslash diffp{[}2km,m-1,m+1{]}{[}2m(k+1){]}\{F(x,y,z)\}\{x,y,z\} -$\Longrightarrow\quad{\displaystyle \diffp[2km,m-1,m+1][2m(k+1)]{F(x,y,z)}{x,y,z}}.$ -\end{example} - +\begin{centred} +\noindent \verb`\[ \diffp[2km,m-1,m+1]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[2km,m-1,m+1]{F(x,y,z)}{x,y,z}. \] +\end{centred} +\noindent Here the overall order is presented as \texttt{2km+2m}. +You might prefer this to be presented as \texttt{2m(k+1)}. Although +\texttt{diffcoeff} takes some steps to present the overall order appropriately, +it is not a computer algebra system and does not factorise expressions. +If you want to present the order in a manner distinct from that presented +by \texttt{diffcoeff}, use the \emph{order-override option}. + +This is a second optional argument immediately following the order +specification. For our last example, filling the override option with +\texttt{2m(k+1)} gives the desired result: +\begin{centred} +\verb`\[ \diffp[2km,m-1,m+1][2m(k+1)]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[2km,m-1,m+1][2m(k+1)]{F(x,y,z)}{x,y,z}. \] +\end{centred} \noindent As another example, left to its own devices, \texttt{diffcoeff} produces -\begin{example} -\noindent \textbackslash diffp{[}m/2+n/2,m/2,n/2{]}F\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[m/2+n/2,m/2,n/2]F{x,y,z},}$ -\end{example} - -\noindent whereas we would like the total order to be presented as $m+n$. -Using the override option, -\begin{example} -\noindent \textbackslash diffp{[}m/2+n/2,m/2,n/2{]}{[}m+n{]}F\{x,y,z\} -$\Longrightarrow\quad{\displaystyle \diffp[m/2+n/2,m/2,n/2][m+n]F{x,y,z}}.$ -\end{example} - +\begin{centred} +\noindent \verb`\[ \diffp[m/2+n/2,m/2,n/2]F{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[m/2+n/2,m/2,n/2]F{x,y,z}, \] +\end{centred} +\noindent whereas we would like the total order to be presented as +$m+n$. Using the override option, +\begin{centred} +\noindent \verb`\[ \diffp[m/2+n/2,m/2,n/2][m+n]F{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[m/2+n/2,m/2,n/2][m+n]F{x,y,z}. \] +\end{centred} The order-override option does exactly that: overrides the presentation of the calculated order with the manually given one. In fact the calculation -algorithm does not get called at all when the override option is used so -that one can in this way present the total order in whatever manner one -wishes or, indeed, add essentially arbitrary material as a superscript +algorithm does not get called at all when the override option is used +so that one can in this way present the total order in whatever manner +one wishes or, indeed, add essentially arbitrary material as a superscript to the $\partial$ symbol in the numerator. -\subsubsection{Comma list of variables of differentiation\label{subsec:Comma-list-of-vars}} - -In tensor calculus the differentiations are almost always in terms of super- -or subscripted coordinates. In many other contexts this is the case too --- the reciprocal of the temperature in thermodynamics or generalized -coordinates in analytical mechanics. This is why a comma list is used in -\texttt{diffcoeff} for specifying variables of differentiation for mixed -partial derivatives. Although it would be nice to write the minimal \texttt{\{xy\}} -rather than \texttt{\{x,y}\} when two variables $x$ and $y$ are involved, -the extra writing is trivial and the comma list allows a simpler handling -of multi-character variables. For instance in tensor calculus we get expressions -like -\begin{example} -\textbackslash diffp\{A\_i\}\{ x\textasciicircum j,x\textasciicircum k -\} $\Longrightarrow\quad{\displaystyle \diffp{A_{i}}{x^{j},x^{k}}.}$ -\end{example} - +\subsubsection{Comma list of variables of differentiation\protect\label{subsec:Comma-list-of-vars}} + +In tensor calculus the differentiations are almost always in terms +of super- or subscripted coordinates. In many other contexts this +is the case too -- the reciprocal of the temperature in thermodynamics +or generalized coordinates in analytical mechanics. This is why a +comma list is used in \texttt{diffcoeff} for specifying variables +of differentiation for mixed partial derivatives. Although it would +be nice to write the minimal \texttt{\{xy\}} rather than \texttt{\{x,y}\} +when two variables $x$ and $y$ are involved, the extra writing is +trivial and the comma list allows a simpler handling of multi-character +variables. For instance in tensor calculus we get expressions like +\begin{centred} +\verb`\[ \diffp{A_i}{ x^j,x^k } \]` $\Longrightarrow$ \[ \diffp{A_i}{ x^j,x^k }. \] +\end{centred} \noindent It is easier to write \texttt{\{x\textasciicircum j,x\textasciicircum k\}} here than, say, \texttt{\{\{x\textasciicircum j\}\{x\textasciicircum k\}\}} -to distinguish the variables. It does mean that should the variable of -differentiation include a comma then that comma needs to be enclosed in -braces. There are plenty of instances of this out in the world (see, e.g., -the last equation of (\ref{eq:eg1})) but it is overall a rare occurrence. +to distinguish the variables. It does mean that should the variable +of differentiation include a comma then that comma needs to be enclosed +in braces. There are plenty of instances of this out in the world +(see, e.g., the last equation of (\ref{eq:eg1})) but it is overall +a rare occurrence. \paragraph*{Minutiae of spacing (yet again):} -In Subsection~\ref{subsec:Minutiae-of-spacing} above, I discussed a slight -reduction in the space inserted between the terms occurring in the denominator -of a mixed partial derivative when a higher order differentiation is involved. -The superscript appearing on a differentiation variable in that case \emph{of -itself} introduced a spacing adjustment between the terms. But the present -discussion involves only first order differentiations and no such reduction -is automatically made by \texttt{diffcoeff}. However it is still possible -to explicitly make such an adjustment with the \texttt{\textbackslash negmu} -command, which inserts \texttt{-1 mu} of spacing. For our example, -\begin{example} -\textbackslash diffp\{A\_i\}\{ x\textasciicircum j\textbackslash negmu,x\textasciicircum k -\} $\Longrightarrow\quad{\displaystyle \diffp{A_{i}}{x^{j}\negmu,x^{k}},}$ -\end{example} - -\noindent the \texttt{\textbackslash negmu} decreasing the spacing between -the terms from the default 2 mu to 1 mu. +In §\ref{subsec:Minutiae-of-spacing} above, I discussed a slight +reduction in the space inserted between the terms occurring in the +denominator of a mixed partial derivative when a higher order differentiation +is involved. The superscript appearing on a differentiation variable +in that case \emph{of itself} introduced a spacing adjustment between +the terms. But the present discussion involves only first order differentiations +and no such reduction is automatically made by \texttt{diffcoeff}. +However it is still possible to explicitly make such an adjustment +with the \texttt{\textbackslash negmu} command introduced earlier +(§\ref{subsec:Spacing-commands}), which inserts \texttt{-1 mu} of +spacing. For our example, in +\begin{centred} +\verb`\[ \diffp{A_i}{ x^j\negmu,x^k } \]` $\Longrightarrow$ \[ \diffp{A_i}{ x^j\negmu,x^k } \] +\end{centred} +\noindent the \texttt{\textbackslash negmu} decreases the spacing +between the terms from the default \verb`2 mu` to \verb`1 mu`. \subsubsection{Overkill territory} -Two previous examples illustrate limitations of the algorithm that calculates -the overall order of differentiation: \texttt{2m/2+2n/2} is not simplified -to \texttt{m+n} and \texttt{2km+2m} is not factorised to \texttt{2m(k+1)}. -But there is much that the algorithm \emph{can} handle -- for instance, -the simple use of parentheses: -\begin{example} -\textbackslash diffp{[}2m-(k+1),2(k+1)-m{]}\{F(x,y,z)\}\{x,y,z\} $\Longrightarrow\quad{\displaystyle \diffp[2m-(k+1),2(k+1)-m]{F(x,y,z)}{x,y,z}}.$ -\end{example} - +Two previous examples illustrate limitations of the algorithm that +calculates the overall order of differentiation: \texttt{2m/2+2n/2} +is not simplified to \texttt{m+n} and \texttt{2km+2m} is not factorised +to \texttt{2m(k+1)}. But there is much that the algorithm \emph{can} +handle -- for instance, the simple use of parentheses: +\begin{centred} +\verb`\[ \diffp[2m-(k+1),2(k+1)-m]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$ +\[ \diffp[2m-(k+1),2(k+1)-m]{F(x,y,z)}{x,y,z}. \] +\end{centred} \paragraph*{Dynamic use of parentheses} \noindent For parenthetic expressions to be evaluated as in this example --- the \emph{dynamic} use of parentheses -- the left parenthesis must -be preceded at most by a sign or a number; the right parenthesis must be -followed at most by a sign. +-- the \emph{dynamic} use of parentheses -- the left parenthesis +must be preceded at most by a sign or a number; the right parenthesis +must be followed at most by a sign. If a right parenthesis is followed by a \emph{variable}, say by \texttt{m} -as in the order spec. \texttt{{[}(2n+1)m,(2n-1)m{]}}, it will throw an -error and halt compilation. A message will be sent to the terminal and -the \LaTeX{} log like the following (which assumes the order spec. was on -line 53 of the document): +as in the order spec. \texttt{{[}(2n+1)m,(2n-1)m{]}}, it will throw +an error and halt compilation. A message will be sent to the terminal +and the \LaTeX{} log like the following (which assumes the order spec. +was on line 53 of the document): \begin{lyxcode} !~Package~diffcoeff~Error:~)~followed~by~m~in~the~ @@ -811,378 +926,461 @@ differentiation~in~this~case.~Use~the~order-override~ option~to~enter~the~overall~order. \end{lyxcode} -This is a limitation on the dynamic use of parentheses -- but they \emph{can} -be nested. +This is a limitation on the dynamic use of parentheses -- but they +\emph{can} be nested. \paragraph*{Static use of parentheses} -If a left parenthesis is preceded by a \emph{variable} (i.e., not a sign -or a number) this signals to \texttt{diffcoeff} the \emph{static} use of -parentheses. No attempt is made to evaluate what is between them and they -are treated simply as an extension of the variable. For example, -\begin{example} -\textbackslash diffp{[}f(k+1)+1,f(k-1)-1{]}\{F(x,y)\}\{x,y\} $\Longrightarrow\quad{\displaystyle \diffp[f(k+1)+1,f(k-1)-1]{F(x,y)}{x,y}}.$ -\end{example} - +If a left parenthesis is preceded by a \emph{variable} (i.e., not +a sign or a number) this signals to \texttt{diffcoeff} the \emph{static} +use of parentheses. No attempt is made to evaluate what is between +them and they are treated simply as an extension of the variable. +For example, +\begin{centred} +\verb`\[ \diffp[f(k+1)+1,f(k-1)-1]{F(x,y)}{x,y} \]` $\Longrightarrow$ +\[ \diffp[f(k+1)+1,f(k-1)-1]{F(x,y)}{x,y}. \] +\end{centred} \noindent In the static case you \emph{can} follow the right parenthesis -by a variable without generating an error.\emph{ }You can nest them, and -you can include static parentheses within a dynamic pair; for example, -\begin{example} -\noindent \textbackslash diffp{[}2(3+f(k))+1,1-3(f(k)-2){]}\{F(x,y)\}\{x,y\} -$\Longrightarrow\quad{\displaystyle \diffp[2(3+f(k))+1,1-3(f(k)-2)]{F(x,y)}{x,y}}.$ -\end{example} - +by a variable without generating an error.\emph{ }You can nest them, +and you can include static parentheses within a dynamic pair; for +example, +\begin{centred} +\noindent \verb`\[ \diffp[2(3+f(k))+1,1-3(f(k)-2)]{F(x,y)}{x,y} \]` +$\Longrightarrow$ \[ \diffp[2(3+f(k))+1,1-3(f(k)-2)]{F(x,y)}{x,y}. \] +\end{centred} \noindent However, the reverse is not possible: you can't have dynamic parentheses within a static pair. \paragraph*{Other refinements} -Exponents and subscripts on a \emph{variable} are fine in an order specification, -so long as the exponent or subscript consists of a \emph{single} token: -\begin{example} -\textbackslash diffp{[}m\textasciicircum 2+2(k-1),m\textasciicircum 2-(k+1){]}F\{x,y,z,w\} -$\Longrightarrow\quad{\displaystyle \diffp[m^{2}+2(k-1),m^{2}-(k+1)]F{x,y,z,w}}.$ -\end{example} - +Exponents and subscripts on a \emph{variable} are fine in an order +specification, so long as the exponent or subscript consists of a +\emph{single} token: +\begin{centred} +\verb`\[ \diffp[m^2+2(k-1),m^2-(k+1)]F{x,y,z,w} \]` $\Longrightarrow$ +\[ \diffp[m^2+2(k-1),m^2-(k+1)]F{x,y,z,w}. \] +\end{centred} \noindent Braced arguments containing \emph{multiple} tokens as exponents -or subscripts to variables will generally not halt compilation but will -usually give nonsensical results, as will \emph{signs} treated as superscripts -or subscripts. Neither circumstance is checked for by \texttt{diffcoeff}. +or subscripts to variables will generally not halt compilation but +will usually give nonsensical results, as will \emph{signs} treated +as superscripts or subscripts. Neither circumstance is checked for +by \texttt{diffcoeff}. \paragraph*{Override} There are limitations on what order specifications the \texttt{diffcoeff} -package can `digest'; equally, it can digest a wide variety of such constructs, -but it is \emph{not} a computer algebra system. In all those cases where -it fails to calculate or present a correct total order, the order-override -option is available. Yes, this is not as convenient as having the overall -order calculated automatically but (let's remind ourselves) we are deep -in overkill territory. Mixed partial derivatives are used far less often -than the pure derivatives, and when they \emph{are} used it is nearly always -to orders 1 or 2 in the variables. Mixed partial derivatives to exotic -orders of differentiation are rarely used, so that the limitations of the -calculational algorithm are of little real moment -- and the override -option is always available for such cases. +package can `digest'; equally, it can digest a wide variety of such +constructs, but it is \emph{not} a computer algebra system. In all +those cases where it fails to calculate or present a correct total +order, the order-override option is available. Yes, this is not as +convenient as having the overall order calculated automatically but +(let's remind ourselves) we are deep in overkill territory. Mixed +partial derivatives are used far less often than the pure derivatives, +and when they \emph{are} used it is nearly always to orders 1 or 2 +in the variables. Mixed partial derivatives to exotic orders of differentiation +are rarely used, so that the limitations of the calculational algorithm +are of little real moment -- and the override option is always available +for such cases. \subsection{Parentheses around multi-character variables} In thermodynamics and statistical mechanics one may want to differentiate in the reciprocal of the temperature, $1/T$ (or $1/\Theta$): -\begin{example} -\textbackslash diffp{[}2{]}q\{\textbackslash frac 1\textbackslash Theta\} -$\Longrightarrow\quad{\displaystyle \diffp[2]q{\frac{1}{\Theta}}.}$ -\end{example} - -\noindent In this case and for other \emph{higher order} derivatives of -multi-character variables of differentiation, the parentheses are inserted -automatically by \texttt{diffcoeff}. Precisely where parentheses should -be placed is moot. The placement in this example is not strictly logical, -although it feels intuitive, but the placement can be customised (Section~\ref{sec:Changing-defaults}). +\begin{centred} +\verb`\[ \diffp[2]q{\frac 1\Theta} \]` $\Longrightarrow$ \[ \diffp[2]q{\frac 1\Theta}. \] +\end{centred} +\noindent In this case and for other \emph{higher order} derivatives +of multi-character variables of differentiation, the parentheses are +inserted automatically by \texttt{diffcoeff}. Precisely where parentheses +should be placed is moot. The placement in this example is not strictly +logical, although it feels intuitive, but the placement can be customised +(§\ref{sec:Changing-defaults}). Parentheses are automatically inserted like this only for higher order derivatives. When the differentiation is to first order, parenthesising is up to the user: -\begin{example} -\textbackslash diffp q\{(\textbackslash frac 1\textbackslash Theta),V\} -$\Longrightarrow{\displaystyle \quad\diffp q{(\frac{1}{\Theta}),V}.}$ -\end{example} - +\begin{centred} +\verb`\[ \diffp q{(\frac 1\Theta),V} \]` $\Longrightarrow$ \[ \diffp q{(\frac 1\Theta),V}. \] +\end{centred} \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\} $\Longrightarrow\quad{\displaystyle \jacob{u,v,w}{x,y,z}.}$ -\end{example} - +\texttt{diffcoeff} provides a command \texttt{\textbackslash jacob} +for constructing Jacobians. For example +\begin{centred} +\verb`\[ \jacob{u,v,w}{x,y,z} \]` $\Longrightarrow$ \[ \jacob{u,v,w}{x,y,z}. \] +\end{centred} 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 \texttt{\textbackslash jacob\{u,v,w\}\{x,y\}} +does \emph{not} check that the two arguments contain the same number +of variables, so it is perfectly possible to form an object like \texttt{\textbackslash jacob\{u,v,w\}\{x,y\}} which as far as I know has no meaning. -\section{Changing defaults; variant forms\label{sec:Changing-defaults}} +\section{Changing defaults; variant forms\protect\label{sec:Changing-defaults}} -To write the range of different examples displayed in the Rogues' Gallery -(Section~\ref{sec:Rogues'-gallery}) I have had to make extensive use -of forms of derivative other than the default. \texttt{Diffcoeff} is built -on the facilities offered by the \texttt{xtemplate} package (included in -the \LaTeX 3 bundle \texttt{l3packages)}. These facilities are harnessed -by means of a command, \texttt{\textbackslash diffdef}, and a further -optional argument of the \texttt{\textbackslash diff} command.\texttt{ } - -How a derivative is displayed in a document is determined by specifying -values in a `key = value' list. This is done with the \texttt{\textbackslash diffdef} -command, which also allows a name to be associated with the list. By using -that name as an argument in the \texttt{\textbackslash diff} command, -a derivative is formed shaped by those values. Examples will make the process -clear. +To write the range of different examples displayed in the Rogues' +Gallery (§\ref{sec:Rogues'-gallery}) I have had to make extensive +use of forms of derivative other than the default. \texttt{diffcoeff} +is built on the facilities offered by the \texttt{xtemplate} package +(included in the \LaTeX 3 bundle \texttt{l3packages)}. These facilities +are harnessed by means of a command, \texttt{\textbackslash diffdef}, +and a further optional argument of the \texttt{\textbackslash diff} +command.\texttt{ } \subsection{Default values: ordinary derivatives} -Table~\ref{tab:Ordinary-derivatives} lists the keys available for forming -derivatives and the default values\footnote{Note that a mu is a `math unit', 1/18 of an em in the math font used.} -they have been assigned. These default values have been chosen to coincide -with those relevant for \emph{ordinary} derivatives -- apart from the -keys \texttt{denom-term-sep}, \texttt{/-denom-term-sep} and \texttt{term-sep-adjust -}which are ignored for ordinary derivatives but apply to the case of mixed -partial derivatives when there is more than one variable of differentiation. -Keys with an opening slash, /, apply only to the slash form of the derivative; -keys with an opening asterisk, {*}, apply only when the differentiand is -appended. - -\begin{wraptable}[22]{o}{0.5\columnwidth}% -\noindent \begin{centering} -\vspace{-4.5ex} - \caption{{\small{}Defaults (ordinary derivatives})\label{tab:Ordinary-derivatives}} -\par\end{centering} -\noindent \centering{}\abovetopsep=.5ex{}% +Table~\ref{tab:Ordinary-derivatives} lists the keys available for +forming derivatives and the default values\footnote{Note that a mu is a `math unit', 1/18 of a quad.} +they have been assigned. These default values have been chosen to +coincide with those relevant for \emph{ordinary} derivatives -- apart +from the keys \texttt{denom-term-sep}, \texttt{/-denom-term-sep}, +\texttt{term-sep-adjust} and \texttt{/-term-sep-adjust} which are +ignored for ordinary derivatives but apply to the case of mixed partial +derivatives when there is more than one variable of differentiation. +Keys with an opening slash, /, apply only to the slash form of the +derivative; keys with an opening asterisk, {*}, apply only when the +differentiand is appended. + +Note that these settings are, in general, font dependent. The values +given are (in the author's opinion) appropriate for the default \LaTeX{} +math fonts. There are also likely to be variations required for whether +a derivative is used in a display-style or text-style or script-style +expression. That matter is discussed below in §\ref{subsec:Text-and-script-style}. +All values specifying a space require the unit (\texttt{mu}) to be +given; a number alone does not suffice. +\noindent \begin{center} +\begin{table} +\centering{}\caption{{\small Defaults (ordinary derivatives})\protect\label{tab:Ordinary-derivatives}} \begin{tabular}{lr} \toprule -{\small{}key} & {\small{}default}\tabularnewline +{\small key} & {\small default}\tabularnewline \midrule -{\small{}op-symbol} & \texttt{\small{}d}\tabularnewline -{\small{}op-symbol-alt} & \texttt{\small{}= op-symbol}\tabularnewline -{\small{}op-order-sep} & \texttt{\small{}1 mu}\tabularnewline -{\small{}{*}-op-left} & \texttt{\small{}false}\tabularnewline -{\small{}{*}-italic-nudge} & \texttt{\small{}3 mu}\tabularnewline -{\small{}{*}/-op-wrap} & \texttt{\small{}true}\tabularnewline -{\small{}long-var-wrap} & \texttt{\small{}d(v)}\tabularnewline -{\small{}denom-term-sep} & \texttt{\small{}2 mu}\tabularnewline -{\small{}/-denom-term-sep} & \texttt{\small{}1 mu}\tabularnewline -term-sep-adjust & \texttt{-1 mu}\tabularnewline -{\small{}left-delim} & \texttt{\small{}\textbackslash left .}\tabularnewline -{\small{}right-delim} & \texttt{\small{}\textbackslash right |}\tabularnewline -{\small{}elbowroom} & \texttt{\small{}0 mu}\tabularnewline -{\small{}subscr-nudge} & \texttt{\small{}0 mu}\tabularnewline -{\small{}/-left-delim} & \texttt{\small{}(}\tabularnewline -{\small{}/-right-delim} & \texttt{\small{})}\tabularnewline -{\small{}/-elbowroom} & \texttt{\small{}0 mu}\tabularnewline -{\small{}/-subscr-nudge} & \texttt{\small{}0 mu}\tabularnewline +{\small op-symbol} & {\small\texttt{d}}\tabularnewline +{\small op-symbol-alt} & {\small\texttt{= op-symbol}}\tabularnewline +{\small op-order-sep} & {\small\texttt{1 mu}}\tabularnewline +{\small derivand-sep} & {\small\texttt{3 mu plus 1 mu minus 2 mu}}\tabularnewline +{\small /-derivand-sep} & {\small\texttt{= derivand-sep}}\tabularnewline +{\small{*}-derivand-sep} & {\small\texttt{= derivand-sep}}\tabularnewline +{\small{*}/-derivand-sep} & {\small\texttt{= /-derivand-sep}}\tabularnewline +{\small denom-term-sep} & {\small\texttt{2 mu}}\tabularnewline +{\small /-denom-term-sep} & {\small\texttt{1 mu}}\tabularnewline +{\small term-sep-adjust} & {\small\texttt{-1 mu}}\tabularnewline +{\small left-delim} & {\small\texttt{\textbackslash left .}}\tabularnewline +{\small right-delim} & {\small\texttt{\textbackslash right |}}\tabularnewline +{\small /-left-delim} & {\small\texttt{(}}\tabularnewline +{\small /-right-delim} & {\small\texttt{)}}\tabularnewline +{\small elbowroom} & {\small\texttt{0 mu}}\tabularnewline +{\small /-elbowroom} & {\small\texttt{0 mu}}\tabularnewline +{\small subscr-nudge} & {\small\texttt{0 mu}}\tabularnewline +{\small /-subscr-nudge} & {\small\texttt{0 mu}}\tabularnewline +{\small long-var-wrap} & {\small\texttt{d(v)}}\tabularnewline +{\small{*}/-op-wrap} & {\small\texttt{true}}\tabularnewline +{\small{*}-op-left} & {\small\texttt{false}}\tabularnewline +{\small{*}-italic-nudge} & {\small\texttt{3 mu}}\tabularnewline \bottomrule -\end{tabular}\end{wraptable}% -Note that these settings are, in general, font dependent. The values given -are (in the author's opinion) appropriate for the default \LaTeX{} math -fonts. There are also likely to be variations required for whether a derivative -is used in a display-style or text-style or script-style expression. That -matter is discussed below in Subsection~\ref{subsec:Text-and-script-style}. -All values specifying a space require the unit (\texttt{mu}) to be given; -a number alone does not suffice. +\end{tabular} +\end{table} +\par\end{center} \begin{description} \item [{op-symbol}] the operator symbol; for ordinary derivatives, generally -one of \texttt{d} or \texttt{\textbackslash mathrm\{d\}}, \texttt{D} or -\texttt{\textbackslash mathrm\{D\}}, \texttt{\textbackslash delta} or -\texttt{\textbackslash Delta}; for partial derivatives \texttt{\textbackslash partial}; +one of \texttt{d} or \texttt{\textbackslash mathrm\{d\}}, \texttt{D} +or \texttt{\textbackslash mathrm\{D\}}, \texttt{\textbackslash delta} +or \texttt{\textbackslash Delta}; for partial derivatives \texttt{\textbackslash partial}; default = \texttt{d} -\item [{op-symbol-alt}] if different from \textbf{op-symbol} then used in the -denominator while \textbf{op-symbol} is used in the numerator; e.g. for -$\diff.nabla.{v^{i}}t$, \texttt{op-symbol = \textbackslash nabla} and -\texttt{op-symbol-alt = d}; otherwise (and usually) defaults to \textbf{op-symbol} -\end{description} -~\vspace{-5ex} - -\begin{description} -\item [{op-order-sep}] extra horizontal space added between the op-symbol and -the superscripted order of differentiation in higher order derivatives; -compare $d^{2}$ with $d^{\mkern1mu 2}$, $\partial^{2}$ with $\partial^{\mkern1mu 2}$ -where the first symbol in each case has no extra space and the second has -an extra 1 mu; default = \texttt{1 mu} -\item [{{*}-op-left}] a choice of \texttt{true} or \texttt{false} indicating -whether the op-symbol is left-aligned or not when the differentiand is -appended; generally it is centred; does not apply to slash forms of the -derivative; default = \texttt{false} -\item [{{*}-italic-nudge}] if \textbf{{*}-op-left} is \texttt{true}, makes an -italic adjustment in the numerator, so that the op-symbols in numerator -and denominator align in the same slanting column; for an upright \texttt{d} -this would be set to \texttt{0 mu}; default = \texttt{3 mu} -\item [{{*}/-op-wrap}] a choice of \texttt{true} or \texttt{false} for slash -forms of the derivative when the differentiand is appended, dictating whether -the derivative is wrapped in parentheses, as here $\diffp*{F(x,y)}/x$, -or not; default = \texttt{true} -\item [{long-var-wrap}] to avoid ambiguity in higher order derivatives it may -be advisable to wrap multi-token variables of differentiation in parentheses; -default = \texttt{d(v)}; the choices are -\begin{description} -\item [{\texttt{dv}}] no wrapping, e.g. $dx_{i}^{2}$ or $d\frac{1}{\Theta}^{2}$, -$\partial x_{i}^{2}$ or $\partial\frac{1}{\Theta}^{2}$, -\item [{\texttt{d(v)}}] wrap the variable only, e.g. $d(x_{i})^{2}$ or $d(\frac{1}{\Theta})^{2}$, -$\partial(x_{i})^{2}$ or $\partial(\frac{1}{\Theta})^{2}$ -\item [{\texttt{(dv)}}] wrap the op-symbol and variable, e.g. $(dx_{i})^{2}$ -or $(d\frac{1}{\Theta})^{2}$, $(\partial x_{i})^{2}$ or $(\partial\frac{1}{\Theta})^{2}$ -\end{description} +\item [{op-symbol-alt}] if different from \textbf{op-symbol} then used +in the denominator while \textbf{op-symbol} is used in the numerator; +e.g. for $\diff.nabla.{v^{i}}t$, \texttt{op-symbol = \textbackslash nabla} +and \texttt{op-symbol-alt = d}; otherwise (and usually) defaults to +\textbf{op-symbol} +\item [{op-order-sep}] extra horizontal space added between the op-symbol +and the superscripted order of differentiation in higher order derivatives; +for the math-italic forms compare $d^{2}$ with $d^{\mkern1mu 2}$, +$\partial^{2}$ with $\partial^{\mkern1mu 2}$ where the first symbol +in each case has no extra space and the second has an extra 1 mu; +default = \texttt{1 mu} +\item [{derivand-sep}] horizontal space added before the differentiand +(derivand) if the \verb`spaced` package option is used, or by the +\verb`!` switch if it is not; the default has some stretch and shrink; +default = \verb`3mu plus 1mu minus 2mu` +\item [{/-derivand-sep}] for the slash form of derivative, horizontal space +added before the differentiand (derivand) if the \verb`spaced` package +option is used, or by the \verb`!` switch if it is not; default = +\verb`derivand-sep` +\item [{{*}-derivand-sep}] when the derivand is appended, horizontal space +added before the differentiand (derivand) if the \verb`spaced` package +option is used, or by the \verb`!` switch if it is not; default = +\verb`derivand-sep` +\item [{{*}/-derivand-sep}] for the slash form of derivative when the derivand +is appended, horizontal space added before the differentiand (derivand) +if the \verb`spaced` package option is used, or by the \verb`!` +switch if it is not; default = \verb`/-derivand-sep` \item [{denom-term-sep}] (mixed partial derivatives only) horizontal spacing inserted between the differentials in the denominator of a mixed partial derivative to avoid a solid cluster like $\partial x\partial y\partial z$; -with the default 2 mu this is $\dl.p.x\dl.p.2y\dl.p.2z$; default = \texttt{2 +with the default 2 mu this is $\dl.p.x\dl.p.2y\dl.p.2z$; default += \texttt{2 mu} +\item [{/-denom-term-sep}] (mixed partial derivatives only) horizontal +spacing inserted between the differentials in the denominator of a +slash-form mixed partial derivative; because a slash-form derivative +is already spread out horizontally, the default spacing is less than +for the \texttt{\textbackslash frac} form derivative; default = \texttt{1 mu} -\item [{/-denom-term-sep}] (mixed partial derivatives only) horizontal spacing -inserted between the differentials in the denominator of a slash-form mixed -partial derivative; because a slash-form derivative is already spread out -horizontally, the default spacing is less than for the \texttt{\textbackslash frac} -form derivative; default = \texttt{1 mu} \item [{term-sep-adjust}] (mixed partial derivatives only) adjustment (i.e. -reduction) to \textbf{(/-)denom-term-sep} when differentiation in a variable -occurs to an order other than 1; if, e.g., $\dl.p.x^{2}\dl.p.1y\dl.p.2z$ -is the denominator of a mixed partial derivative in three variables, because -of the superscript the spacing between $\partial x^{2}$ and $\partial y$ -is reduced by\textbf{ term-sep-adjust} from the spacing between $\partial y$ +reduction) to \textbf{denom-term-sep} or \textbf{/-denom-term-sep} +when differentiation in a variable occurs to an order other than 1; +if, e.g., $\dl.p.x^{2}\dl.p.1y\dl.p.2z$ is the denominator of a mixed +partial derivative in three variables, because of the superscript +the spacing between $\partial x^{2}$ and $\partial y$ is reduced +by\textbf{ term-sep-adjust} from the spacing between $\partial y$ and $\partial z$; default = \texttt{-1 mu} \item [{left-delim}] the left member of a delimiter pair wrapping the derivative, the right member of which is subscripted to indicate a point of evaluation -or variables held constant; default = \texttt{\textbackslash left .} -\item [{right-delim}] the right member of a delimiter pair wrapping the derivative -and subscripted to indicate a point of evaluation or variables held constant; -default = \texttt{\textbackslash right |} +or variables held constant; default = \texttt{\textbackslash left +.} +\item [{right-delim}] the right member of a delimiter pair wrapping the +derivative and subscripted to indicate a point of evaluation or variables +held constant; default = \texttt{\textbackslash right |} +\item [{/-left-delim}] for the slash form of derivative, the left member +of a delimiter pair wrapping the derivative and subscripted to indicate +a point of evaluation or variables held constant; default = \texttt{(} +\item [{/-right-delim}] for the slash form of derivative, the right member +of a delimiter pair wrapping the derivative, the right member of which +is subscripted to indicate a point of evaluation or variables held +constant; default = \texttt{)} \item [{elbowroom}] adjustment to the whitespace between the left and right -delimiters and the enclosed derivative; negative values reduce the space; -default = \texttt{0 mu} -\item [{subscr-nudge}] horizontal adjustment of the subscript's placing relative -to the \textbf{right-delim}iter, e.g., a negative value compensates for -the curving inwards of a large right parenthesis; may be font dependent; -default = \texttt{0 mu} -\item [{/-left-delim}] for the slash form of derivative, the left member of -a delimiter pair wrapping the derivative and subscripted to indicate a -point of evaluation or variables held constant; default = \texttt{(} -\item [{/-right-delim}] for the slash form of derivative, the right member of -a delimiter pair wrapping the derivative, the right member of which is -subscripted to indicate a point of evaluation or variables held constant; -default = \texttt{)} -\item [{/-elbowroom}] adjustment to the whitespace between the left and right -delimiters and the enclosed slash-form derivative; default = \texttt{0 -mu} +delimiters and the enclosed derivative; negative values reduce the +space; default = \texttt{0 mu} +\item [{/-elbowroom}] adjustment to the whitespace between the left and +right delimiters and the enclosed slash-form derivative; default = +\texttt{0 mu} +\item [{subscr-nudge}] horizontal adjustment of the subscript's placing +relative to the \textbf{right-delim}iter, e.g., a negative value compensates +for the curving inwards of a large right parenthesis; may be font +dependent; default = \texttt{0 mu} \item [{/-subscr-nudge}] for the slash form of derivative, horizontal adjustment of the subscript's placing relative to the /-\textbf{right-delim}iter; may be font dependent; default = \texttt{0 mu} +\item [{long-var-wrap}] to avoid ambiguity in higher order derivatives +it may be advisable to wrap multi-token variables of differentiation +in parentheses; default = \texttt{d(v)}; the choices are +\begin{description} +\item [{\texttt{dv}}] no wrapping, e.g. $dx_{i}^{2}$ or $d\frac{1}{\Theta}^{2}$, +$\partial x_{i}^{2}$ or $\partial\frac{1}{\Theta}^{2}$, +\item [{\texttt{d(v)}}] wrap the variable only, e.g. $d(x_{i})^{2}$ or +$d(\frac{1}{\Theta})^{2}$, $\partial(x_{i})^{2}$ or $\partial(\frac{1}{\Theta})^{2}$ +\item [{\texttt{(dv)}}] wrap the op-symbol and variable, e.g. $(dx_{i})^{2}$ +or $(d\frac{1}{\Theta})^{2}$, $(\partial x_{i})^{2}$ or $(\partial\frac{1}{\Theta})^{2}$ +\end{description} +\item [{{*}/-op-wrap}] a choice of \texttt{true} or \texttt{false} for +slash forms of the derivative when the differentiand is appended, +dictating whether the derivative is wrapped in parentheses, as here +$\diffp*{F(x,y)}/x$, or not; default = \texttt{true} +\item [{{*}-op-left}] a choice of \texttt{true} or \texttt{false} indicating +whether the op-symbol is left-aligned or not when the differentiand +is appended; generally it is centred; does not apply to slash forms +of the derivative; default = \texttt{false} +\item [{{*}-italic-nudge}] if \textbf{{*}-op-left} is \texttt{true}, makes +an italic adjustment in the numerator, so that the op-symbols in numerator +and denominator align in the same slanting column; for an upright +\texttt{d} this would be set to \texttt{0 mu}; default = \texttt{3 +mu} \end{description} \subsection{ISO defaults} -You may not like the default settings that come with \texttt{diffcoeff}. -The package does not follow ISO 80000-2 -- it does not use upright `d's -nor does it wrap an ordinary differential coefficient in subscripted parentheses -to indicate a point of evaluation. Both `defects' can be remedied by calling -the package with the option \texttt{ISO}:\footnote{One can also use \texttt{ISO=true} to turn the option on and \texttt{ISO=false +\label{subsec:ISO-defaults}You may not like the default settings +that come with \texttt{diffcoeff}. The package does not follow ISO +80000-2 -- it does not use upright `d's nor does it wrap an ordinary +differential coefficient in subscripted parentheses to indicate a +point of evaluation. Both `defects' can be remedied by calling the +package with the option \texttt{ISO}:\footnote{One can also use \texttt{ISO=true} to turn the option on and \texttt{ISO=false }to turn the option off. } \begin{lyxcode} \textbackslash usepackage{[}ISO{]}\{diffcoeff\} \end{lyxcode} -\begin{wraptable}[9]{o}{0.4\columnwidth}% +\begin{wraptable}[10]{o}{0.4\columnwidth}% \centering{}\vspace{-4ex} - \caption{{\small{}ISO default changes}\label{tab:ISO-setting-changes}} + \caption{{\small ISO default changes}\protect\label{tab:ISO-setting-changes}} \abovetopsep=.5ex % \begin{tabular}{lr} \toprule -{\small{}key} & {\small{}default}\tabularnewline +{\small key} & {\small default}\tabularnewline \midrule -{\small{}op-symbol} & \texttt{\small{}\textbackslash mathrm\{d\}}\tabularnewline -{\small{}op-order-sep} & \texttt{\small{}0 mu}\tabularnewline -{\small{}left-delim} & \texttt{\small{}\textbackslash left (}\tabularnewline -{\small{}right-delim} & \texttt{\small{}\textbackslash right )}\tabularnewline -{\small{}subscr-nudge} & \texttt{\small{}-6 mu}\tabularnewline +{\small op-symbol} & {\small\texttt{\textbackslash mathrm\{d\}}}\tabularnewline +{\small op-order-sep} & {\small\texttt{0 mu}}\tabularnewline +{\small left-delim} & {\small\texttt{\textbackslash left (}}\tabularnewline +{\small right-delim} & {\small\texttt{\textbackslash right )}}\tabularnewline +{\small subscr-nudge} & {\small\texttt{-6 mu}}\tabularnewline \bottomrule \end{tabular}\end{wraptable}% The uppercase is essential -- an option \texttt{iso} is not recognised. -The \texttt{ISO} option results in changes to the default settings of Table~\ref{tab:Ordinary-derivatives} -as listed in Table~\ref{tab:ISO-setting-changes}. Any settings not mentioned -in Table~\ref{tab:ISO-setting-changes} retain the values presented in -Table~\ref{tab:Ordinary-derivatives}. Note that the subscript nudge figure -specified here is \emph{not} part of the standard, which makes no recommendation -about the subscript's positioning. But: the \texttt{-6 mu} figure with -a default or latin modern font gives a better representation of what is -displayed in the standard than a zero figure. - -Because the `d' is upright with the \texttt{ISO} option, no extra space -is required between the symbol and the superscript in a higher order derivative. -Hence the zero value for the \texttt{op-order-sep}. ISO recommends subscripted -parentheses to indicate a point of evaluation. Hence the other entries -in the table. Because a large right parenthesis (penultimate setting) bends -inwards, to the left, a negative value for the last entry ensures the subscript -does not become detached from the derivative, looking lost in a sea of -whitespace. - -Note that the \texttt{ISO} option will also produce upright `D's in derivatives -formed from `D'; see Subsection~\ref{subsec:D-delta-Delta} below. +The \texttt{ISO} option results in changes to the default settings +of Table~\ref{tab:Ordinary-derivatives} as listed in Table~\ref{tab:ISO-setting-changes}. +Any settings not mentioned in Table~\ref{tab:ISO-setting-changes} +retain the values presented in Table~\ref{tab:Ordinary-derivatives}. +Note that the subscript nudge figure specified here is \emph{not} +part of the standard, which makes no recommendation about the subscript's +positioning. But: the \texttt{-6 mu} figure with a default or latin +modern font gives a better representation of what is displayed in +the standard than a zero figure. + +Because the `d' is upright with the \texttt{ISO} option, no extra +space is required between the symbol and the superscript in a higher +order derivative. Hence the zero value for the \texttt{op-order-sep}. +ISO recommends subscripted parentheses to indicate a point of evaluation. +Hence the other entries in the table. Because a large right parenthesis +(penultimate setting) bends inwards, to the left, a negative value +for the last entry ensures the subscript does not become detached +from the derivative, looking lost in a sea of whitespace. + +Note that the \texttt{ISO} option will also produce upright `D's in +derivatives formed from `D'; see §\ref{subsec:D-delta-Delta} below. \subsection{Partial derivatives} \begin{wraptable}{o}{0.4\columnwidth}% -\centering{}\vspace{-4.5ex} - \caption{{\small{}Default changes: partial derivatives}\label{tab:Partial-deriv-defaults}} +\centering{}\vspace{-4ex} + \caption{{\small Default changes for partial derivatives}\protect\label{tab:Partial-deriv-defaults}} \abovetopsep=.5ex % \begin{tabular}{lr} \toprule -{\small{}key} & {\small{}default}\tabularnewline +{\small key} & {\small default}\tabularnewline \midrule -{\small{}op-symbol } & \texttt{\small{}\textbackslash partial}\tabularnewline -{\small{}left-delim} & \texttt{\small{}\textbackslash left (}\tabularnewline -{\small{}right-delim} & \texttt{\small{}\textbackslash right )}\tabularnewline -{\small{}subscr-nudge } & \texttt{\small{}-6 mu}\tabularnewline +{\small op-symbol } & {\small\texttt{\textbackslash partial}}\tabularnewline +{\small left-delim} & {\small\texttt{\textbackslash left (}}\tabularnewline +{\small right-delim} & {\small\texttt{\textbackslash right )}}\tabularnewline +{\small subscr-nudge } & {\small\texttt{-6 mu}}\tabularnewline \bottomrule \end{tabular}\end{wraptable}% -The default values given in Table~\ref{tab:Ordinary-derivatives}, when -they are relevant, apply to \emph{ordinary} derivatives. For partial derivatives, -the defaults are those of Table~\ref{tab:Partial-deriv-defaults}. All -other keys take the default values listed in Table~\ref{tab:Ordinary-derivatives}. -The last three entries here are not an attempt at ISO compatibility but -reflect the use of subscripted parentheses with partial derivatives to -indicate variables held constant, for instance in the Maxwell relations +The default values given in Table~\ref{tab:Ordinary-derivatives}, +when they are relevant, apply to \emph{ordinary} derivatives. For +partial derivatives, the defaults are those of Table~\ref{tab:Partial-deriv-defaults}. +All other keys take the default values listed in Table~\ref{tab:Ordinary-derivatives}. +The last three entries here are not an attempt at ISO compatibility +but reflect the use of subscripted parentheses with partial derivatives +to indicate variables held constant, for instance in the Maxwell relations of thermodynamics, one of which is \[ \diffp SV[T]=\diffp PT[V]. \] -\subsection{Setting your own defaults: \texttt{\textbackslash diffdef\label{subsec:diffdef}}} +\subsection{Setting your own defaults: \texttt{\textbackslash diffdef\protect\label{subsec:diffdef}}} Versions 2 and later of the \texttt{diffcoeff} package provide a command, -\texttt{\textbackslash diffdef}, that enables users to set their own defaults.\texttt{ }For -example, if you wish to use upright `d's but not follow the ISO's use of -subscripted parentheses to indicate a point of evaluation, enter in the -preamble of your document the command\vspace{-2ex} - -\begin{lyxcode} -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% +\texttt{\textbackslash diffdef}, that enables users to set their +own defaults.\texttt{ }For example, if you wish to use upright `d's +but not follow the ISO's use of subscripted parentheses to indicate +a point of evaluation, enter in the preamble of your document the +command \begin{lyxcode} \textbackslash diffdef~\{\}~~~~ -\begin{lyxcode} -\{~~~~~~ -\begin{lyxcode} -op-symbol~~~~=~\textbackslash mathrm\{d\},~~~~ -op-order-sep~=~0~mu -\end{lyxcode} -\} -\end{lyxcode} -\end{lyxcode} -% -\end{minipage} +~~\{~~~~~~ + +~~~~op-symbol~~~~=~\textbackslash mathrm\{d\},~~~~ + +~~~~op-order-sep~=~0~mu + +~~\} \end{lyxcode} \noindent Since a list of settings, like this one, is a comma-\emph{separated} list, no comma is required for the last entry. That entry is a consequence -of the first: upright symbols do not require any extra separation between -the `d' and the superscript in a higher order derivative. +of the first: upright symbols do not require any extra separation +between the `d' and the superscript in a higher order derivative. The other point to note is the empty pair of braces after the \texttt{\textbackslash diffdef} -command. \emph{They matter}. Their emptiness is what determines that it -is the \emph{default} values that are changed. If they contain some content, -then that content provides a \emph{name} for the particular set of values -in the following list. The \texttt{diffcoeff} package exploits this facility -to cope with the wide variety of forms displayed in the Rogues' Gallery -of Section~\ref{sec:Rogues'-gallery}. +command. \emph{They matter}. Their emptiness is what determines that +it is the \emph{default} values that are changed. If they contain +some content, then that content provides a \emph{name} for the particular +set of values in the following list. The \texttt{diffcoeff} package +exploits this facility to cope with the wide variety of forms displayed +in the Rogues' Gallery of §\ref{sec:Rogues'-gallery}. + +\subsubsection{Space before the differentiand} + +\label{subsec:A-final-flourish}\begin{wraptable}{o}{0.6\columnwidth}% +\begin{centering} +\caption{Keys for spacing the derivand} +\medskip{} +\begin{tabular}{lr} +\toprule +{\small key} & {\small default}\tabularnewline +\midrule +{\small derivand-sep} & {\small\texttt{3mu plus 1mu minus 2mu}}\tabularnewline +{\small{*}-derivand-sep} & {\small\texttt{= derivand-sep}}\tabularnewline +{\small /-derivand-sep} & {\small\texttt{= derivand-sep}}\tabularnewline +{\small{*}/-derivand-sep} & {\small\texttt{= /-derivand-sep}}\tabularnewline +\bottomrule +\end{tabular} +\par\end{centering} +\end{wraptable}% +The insertion of a small space before the differentiand is effected +by the \verb`!` key inserted immediately before the differentiand +argument in the \verb`\diff` command. You may wish to make the insertion +of this space the \emph{default} behaviour. This is done by using +the \verb`spaced` package option (which makes the \verb`!` switch +now reverse this new default and put \emph{no} extra space before +the derivand). The amount of space inserted is, by default, \verb`3mu plus 1mu minus 2mu`, +meaning the space is generally $3$mu but can stretch to $4$mu or +shrink to $1$mu as \TeX{} strives to fit content in a line or on the +page. + +Perhaps this doesn't suit. You may want a fixed space here, with no +stretch or shrink. The key to change is \verb`derivand-sep`. By default, +this setting applies not only to the fraction form of derivative, +but also to the slash form and to when the derivand is appended. + +If you feel a little less space should be used for slash derivatives, +then the key to change is \verb`/-deriv-sep`. This changed value +will also be used for an appended derivand in a slash derivative. + +Thus to meet both wishes you might put in the preamble of your document +something like +\begin{lyxcode} +\textbackslash diffdef~\{\}~~~~ + +~~\{~~~~~~ + +~~~~derivand-sep~~~=~3~mu,~~~~ + +~~~~/-derivand-sep~=~2~mu + +~~\} +\end{lyxcode} +This will insert a fixed space of $3$mu before the differentiand +in both the numerator and when appended in the fraction form of derivative, +and a fixed space of $2$mu in the slash form of derivative, both +in the numerator and when appended. + +If you want a different spacing when the derivand is appended, the +keys to change are \verb`*-derivand-sep` and \verb`*/-derivand-sep`. + +\paragraph{Selective spacing} + +I have treated the \verb`spaced` package option thus far as if it +were an \verb`ON/OFF` switch and, indeed, the presence of the package +option \verb`spaced` behaves as \verb`ON` and its absence as \verb`OFF`. +Internally, however, \verb`spaced` is equivalent to \verb`spaced=1` +and its absence to \verb`spaced=0`. Entering \verb`spaced=n` in +the package option where \verb`n` is a positive integer is equivalent +to entering \verb`spaced=1` (and hence to simply entering \verb`spaced`), +but if \verb`n` is a negative integer, a new effect is produced. + +Entering \verb`spaced=-1` (or any negative integer) will insert a +space (by default \verb`3mu plus 1mu minus 2mu`) before the differentiand +provided the differentiand\emph{ is longer than a single token} but +will insert no space before single-token differentiands. The switch +\verb`!` reverses this behaviour. \subsection{Variant forms} -For this package I needed to have a number of variant forms available to -illustrate the wide variety of ways in which derivatives are displayed. +For this package I needed to have a number of variant forms available +to illustrate the wide variety of ways in which derivatives are displayed. The \texttt{\textbackslash diffdef} command in which the first argument -is \emph{filled} provides one half of the means of doing this. I've given -the single-letter name \texttt{p} to the following settings: - -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% +is \emph{filled} provides one half of the means of doing this. I've +given the single-letter name \texttt{p} to the following settings: \begin{lyxcode} \textbackslash diffdef~\{~p~\} @@ -1198,34 +1396,28 @@ the single-letter name \texttt{p} to the following settings: ~~\} \end{lyxcode} -% -\end{minipage} - -The second half of providing variant forms is to insert this name, \texttt{p}, -between dots (periods, full stops) as the \emph{first} argument of the -\texttt{\textbackslash diff} command. Thus, repeating an example at the -end of Subsection~\ref{subsec:Partial-appending}, -\begin{example} -\textbackslash diff.p.{*}\{\textbackslash frac PT\}U{[}V{]} = \textbackslash diff.p.{*}\{\textbackslash frac -1T\}V{[}U{]} $\Longrightarrow\quad{\displaystyle \diff.p.*{\frac{P}{T}}U[V]=\diff.p.*{\frac{1}{T}}V[U]}$ -\end{example} - +The second half of providing variant forms is to insert this name, +\texttt{p}, between dots (periods, full stops) as the \emph{first} +argument of the \texttt{\textbackslash diff} command. Thus, repeating +an example at the end of §\ref{subsec:Partial-appending}, +\begin{centred} +\verb`\[ \diff.p.*{\frac PT}U[V] = \diff.p.*{\frac 1T}V[U] \]` $\Longrightarrow$ +\[ \diff.p.*{\frac PT}U[V] = \diff.p.*{\frac 1T}V[U] \] +\end{centred} \noindent The effect is exactly the same as previously, when the \texttt{\textbackslash diffp} command was used. Indeed, \texttt{diffcoeff} identifies \texttt{\textbackslash diffp} -with \texttt{\textbackslash diff.p.}, saving a few keystrokes and maintaining -compatibility with version 1 of the package. In \LaTeXe{} synatx, +with \texttt{\textbackslash diff.p.}: \begin{lyxcode} -\textbackslash newcommand~\{~\textbackslash diffp~\}~\{~\textbackslash diff.p.~\} +\textbackslash NewDocumentCommand~\textbackslash diffp~\{~\}~\{~\textbackslash diff.p.~\} \end{lyxcode} -Note that this identification of \texttt{\textbackslash diffp} with \texttt{\textbackslash diff.p. }means -there is no equivalent dot-delimited argument available for \texttt{\textbackslash diffp}. -The dot-delimited argument applies only to \texttt{\textbackslash diff}. - -For example, to illustrate the upright-d form of derivative, without changing -the default math-italic form (which I prefer), one might enter in the preamble -\begin{lyxcode} -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% +Note that this identification of \texttt{\textbackslash diffp} with +\texttt{\textbackslash diff.p. }means there is no equivalent dot-delimited +argument available for \texttt{\textbackslash diffp}. The \emph{dot-delimited +argument applies only to} \texttt{\textbackslash diff}. + +For example, to illustrate the upright-d form of derivative, without +changing the default math-italic form (which I prefer), one might +enter in the preamble \begin{lyxcode} \textbackslash diffdef~\{~up~\}~ @@ -1237,83 +1429,69 @@ the default math-italic form (which I prefer), one might enter in the preamble ~~\} \end{lyxcode} -% -\end{minipage} -\end{lyxcode} -Apart from the key = value settings, the critical feature here is the name, -\texttt{up} (which is at your discretion and could equally be \texttt{upright} -or \texttt{roman} or even \texttt{Fred} if you so fancied). This ensures -that the changed settings apply only to this particular variant and do -not `infect' the overall defaults. To use this variant, all that is needed -is to add the name, between dots, to the \texttt{\textbackslash diff} -command: -\begin{example} -\textbackslash diff.up.yx $\Longrightarrow\quad{\displaystyle \diff.up.yx}.$ -\end{example} - -\noindent Each variant derivative inherits all the default values that -it does not explicitly countermand. Thus a point of evaluation is indicated -by a vertical rule which is the \texttt{diffcoeff} default\footnote{\noindent But not the ISO recommendation.}: -\begin{example} -\noindent \textbackslash diff.up.{*}\{\textbackslash frac\{F(x)\}\{G(x)\}\}x{[}x=1{]} -$\Longrightarrow\quad{\displaystyle \diff.up.*{\frac{F(x)}{G(x)}}x[x=1]}$ -\end{example} - -\noindent Dot-delimited arguments must always be the \emph{first} argument -of the \texttt{\textbackslash diff} command, even preceding an asterisk -(star) as in this example. +Apart from the \emph{key = value} settings, the critical feature here +is the name, \texttt{up} (which is at your discretion and could equally +be \texttt{upright} or \texttt{roman} or even \texttt{Fred} if you +so fancied). This ensures that the changed settings apply only to +this particular variant and do not `infect' the overall defaults. +To use this variant, all that is needed is to add the name, between +dots, to the \texttt{\textbackslash diff} command: +\begin{centred} +\verb`\[ \diff.up.yx \]` $\Longrightarrow$ \[ \diff.up.yx. \] +\end{centred} +\noindent Each variant derivative inherits all the default values +that it does not explicitly countermand. Thus a point of evaluation +is indicated by a vertical rule which is the \texttt{diffcoeff} default\footnote{\noindent But not the ISO recommendation.}: +\begin{centred} +\noindent \verb`\[ \diff.up.*{\frac{F(x)}{G(x)}}x[x=1] \]` $\Longrightarrow$ +\[ \diff.up.*{\frac{F(x)}{G(x)}}x[x=1] \] +\end{centred} +\noindent Dot-delimited arguments must always be the \emph{first} +argument of the \texttt{\textbackslash diff} command, even preceding +an asterisk (star) as in this example. As another example, suppose for the subscripted indication of variables held constant in a partial derivative that you want to see what things -look like if the subscript is \emph{not }nudged in towards the right parenthesis. -In that case define a variant form +look like if the subscript is \emph{not }nudged in towards the right +parenthesis. In that case define a variant form \begin{lyxcode} \textbackslash diffdef~\{~padrift~\}~\{~subscr-nudge~=~0~mu~\} \end{lyxcode} I have attached a name, \texttt{padrift},\texttt{ }to this setting, -\begin{example} -\textbackslash diff.padrift.Fx{[}y{]} $\Longrightarrow{\displaystyle \diff.padrift.Fx[y]}$ -\end{example} - -\noindent since, to my eye, the subscript seems detached from the expression -it qualifies -- is it perhaps a typo? -- and `adrift in a sea of whitespace'. -This is to be compared with the default -\begin{example} -\noindent \textbackslash diffp Fx{[}y{]} $\Longrightarrow{\displaystyle \diffp Fx[y]}$ -\end{example} +\begin{centred} +\verb`\[ \diff.padrift.Fx[y] \]` $\Longrightarrow$ \[ \diff.padrift.Fx[y] \] +\end{centred} +\noindent since, to my eye, the subscript seems detached from the +expression it qualifies -- is it perhaps a typo? -- and `adrift +in a sea of whitespace'. This is to be compared with the default \verb`\[ \diffp Fx[y] \]` +$\Longrightarrow$ \[ \diffp Fx[y] \] \noindent where the subscript is tucked in close to the right parenthesis and is clearly connected to it and the expression it delimits. -Some might want to distinguish notationally a point of evaluation for a -partial derivative from variables held constant, perhaps using a vertical -rule for the former and (the default) parentheses for the latter. It would -suffice then to add to the preamble -\begin{lyxcode} -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% +Some might want to distinguish notationally a point of evaluation +for a partial derivative from variables held constant, perhaps using +a vertical rule for the former and (the default) parentheses for the +latter. It would suffice then to add to the preamble \begin{lyxcode} \textbackslash diffdef~\{~pvrule~\}~\{~op-symbol~=~\textbackslash partial~\}~ \end{lyxcode} -% -\end{minipage} -\end{lyxcode} (or some other name of your choosing). This gives -\begin{example} -\textbackslash diff.pvrule.\{F(x,y)\}x{[}x=1{]}$\Longrightarrow{\displaystyle \diff.pvrule.{F(x,y)}x[x=1]}$ -\end{example} - +\begin{centred} +\verb`\[ \diff.pvrule.{F(x,y)}x[x=1] \]`$\Longrightarrow$ \[ \diff.pvrule.{F(x,y)}x[x=1] \] +\end{centred} -\subsubsection{Text-style and script-style derivatives\label{subsec:Text-and-script-style}} +\subsubsection{Text-style and script-style derivatives\protect\label{subsec:Text-and-script-style}} As noted earlier, the \texttt{diffcoeff} package assumes that derivatives -of fraction-like form will be used in display-style expressions and that -the slash form will be used for inline use (text style). This is the usual -practice. But if one does want to use the fraction form in an inline expression, -say \texttt{\textbackslash diffp ST{[}V{]}} displaying as $\diffp ST[V]$, -then some tweaking of settings is necessary: the subscript is obviously -too close to the right parenthesis and, to my eye, there is too much `elbowroom' -between the derivative and the enclosing parentheses. Hence define +of fraction-like form will be used in display-style expressions and +that the slash form will be used for inline use (text style). This +is the usual practice. But if one does want to use the fraction form +in an inline expression, say \texttt{\textbackslash diffp ST{[}V{]}} +displaying as $\diffp ST[V]$, then some tweaking of settings is necessary: +the subscript is obviously too close to the right parenthesis and, +to my eye, there is too much `elbowroom' between the derivative and +the enclosing parentheses. Hence define \noindent % \noindent\begin{minipage}[t]{1\columnwidth}% @@ -1339,34 +1517,28 @@ between the derivative and the enclosing parentheses. Hence define % \end{minipage} -This gives, for the same example, \texttt{\textbackslash diff.ptxt.ST{[}V{]}} -displaying as $\diff.ptxt.ST[V]$, where the subscript is better positioned -and there is a better fit between parentheses and derivative. +We can now write, for the same example, \texttt{\textbackslash diff.ptxt.ST{[}V{]}} +which displays as $\diff.ptxt.ST[V]$, where the subscript is better +positioned and there is a better fit between parentheses and derivative. -\subsubsection{Derivatives from D, \textbackslash delta, \textbackslash Delta\label{subsec:D-delta-Delta}} +\subsubsection{Derivatives from D, \textbackslash delta, \textbackslash Delta\protect\label{subsec:D-delta-Delta}} -In addition to \texttt{\textbackslash diff.p.},\texttt{ diffcoeff} has -three other \emph{built-in} variant forms that are commonly used: \texttt{\textbackslash diff.D.}, -\texttt{\textbackslash diff.delta.}, and \texttt{\textbackslash diff.Delta.}, -corresponding to derivatives formed from $D$, $\delta$ and $\Delta$ -respectively. +In addition to \texttt{\textbackslash diff.p.},\texttt{ diffcoeff} +has three other \emph{built-in} variant forms that are commonly used: +\texttt{\textbackslash diff.D.}, \texttt{\textbackslash diff.delta.}, +and \texttt{\textbackslash diff.Delta.}, corresponding to derivatives +formed from $D$, $\delta$ and $\Delta$ respectively. In fluid dynamics the \emph{material }or \emph{substantive} derivative -uses an uppercase $D$ in place of $d$. For example, the continuity equation -is, -\begin{example} -\textbackslash diff.D.\{\textbackslash rho\}t=\textbackslash diffp\textbackslash rho -t + \textbackslash mathbf\{u\textbackslash cdot\}\textbackslash nabla\textbackslash rho -$\Longrightarrow{\displaystyle \diff.D.\rho t=\diffp\rho t+\mathbf{u\cdot}\nabla\rho}$ -\end{example} - +uses an uppercase $D$ in place of $d$. For example, the continuity +equation is, +\begin{centred} +\verb`\[ \diff.D.{\rho}t=\diffp\rho t + \mathbf{u\cdot}\nabla\rho \]`$\Longrightarrow$ +\[ \diff.D.{\rho}t=\diffp\rho t + \mathbf{u\cdot}\nabla\rho \] +\end{centred} \noindent where \texttt{\textbackslash diff.D.} produces the D-derivative. -If you want upright `D's, then the \texttt{ISO} package option will produce -that effect. Alternatively, \vspace{-2ex} - -\begin{lyxcode} -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% +If you want upright `D's, then the \texttt{ISO} package option will +produce that effect. Alternatively, \begin{lyxcode} \textbackslash diffdef~\{~Up~\}~ @@ -1378,84 +1550,78 @@ that effect. Alternatively, \vspace{-2ex} ~~\} \end{lyxcode} -% -\end{minipage} -\end{lyxcode} \noindent provides a variant with upright `D's. -In introductory calculus texts the simple $\delta$-derivative is used. -This is achieved with the \texttt{\textbackslash diff.delta.} command -\begin{example} -\textbackslash diff.delta.yx $\Longrightarrow{\displaystyle \diff.delta.yx}.$ -\end{example} +In introductory calculus texts the simple $\delta$-derivative is +used. This is achieved with the \texttt{\textbackslash diff.delta.} +command: \verb`\[ \diff.delta.yx \]` $\Longrightarrow$ \[ \diff.delta.yx. \] -\noindent This form also features in analytical mechanics (in the Rogues' -Gallery, the final example at (\ref{eq:eg6})). +\noindent This form also features in analytical mechanics (in the +Rogues' Gallery, the final example at (\ref{eq:eg6})). -Similarly, \texttt{\textbackslash diff.Delta.} forms a derivative from -$\Delta$: -\begin{example} -\textbackslash diff.Delta.y/x $\Longrightarrow{\displaystyle \diff.Delta.y/x,}$ -\end{example} - -\noindent where the slash form of the derivative is shown in this instance. +Similarly, \texttt{\textbackslash diff.Delta.} forms a derivative +from $\Delta$: +\begin{centred} +\verb`$ \diff.Delta.y/x $` $\Longrightarrow$ $ \diff.Delta.y/x $ +\end{centred} +\noindent where the slash form of the derivative is shown in this +instance. Higher order forms of these derivatives, points of evaluation, appending -the differentiand with a star argument, all follow exactly as for the `pure' -\texttt{\textbackslash diff} command. +the differentiand with a star argument, all follow exactly as for +the `pure' \texttt{\textbackslash diff} command. \paragraph{The commands \textbackslash Diff, \textbackslash diffd, \textbackslash Diffd} -For compatibility with version 1 of \texttt{diffcoeff}, the commands \texttt{\textbackslash Diff}, -\texttt{\textbackslash diffd} and \texttt{\textbackslash Diffd} are available -and also produce the $D$, $\delta$ and $\Delta$ derivatives. Just as -\texttt{\textbackslash diffp} is identified with \texttt{\textbackslash diff.p.} -for partial derivatives, these commands are identified with \texttt{\textbackslash diff.D.}, +For compatibility with version 1 of \texttt{diffcoeff}, the commands +\texttt{\textbackslash Diff}, \texttt{\textbackslash diffd} and +\texttt{\textbackslash Diffd} are available and also produce the +$D$, $\delta$ and $\Delta$ derivatives. Just as \texttt{\textbackslash diffp} +is identified with \texttt{\textbackslash diff.p.} for partial derivatives, +these commands are identified with \texttt{\textbackslash diff.D.}, \texttt{\textbackslash diff.delta.}, and \texttt{\textbackslash diff.Delta} -through commands equivalent to\footnote{In fact the actual commands in \texttt{diffcoeff.sty} use the syntax of -the \texttt{xparse} package, e.g. \texttt{\textbackslash NewDocumentCommand -\{ \textbackslash Diff \} \{ \} \{ }\textbackslash diff.D. \}, and similarly -for the others.} +through the commands \begin{lyxcode} -\textbackslash newcommand\{\textbackslash Diff\}\{\textbackslash diff.D.\} +\textbackslash NewDocumentCommand~\textbackslash Diff~\{\}~~\{\textbackslash diff.D.\} -\textbackslash newcommand\{\textbackslash diffd\}\{\textbackslash diff.delta.\} +\textbackslash NewDocumentCommand~\textbackslash diffd~\{\}~\{\textbackslash diff.delta.\} -\textbackslash newcommand\{\textbackslash Diffd\}\{\textbackslash diff.Delta.\} +\textbackslash NewDocumentCommand~\textbackslash Diffd~\{\}~\{\textbackslash diff.Delta.\} \end{lyxcode} -Unless one is using such variant forms frequently, it seems simpler to -remember that they are available as dot-delimited arguments to the \texttt{\textbackslash diff} -command, using the obvious name in each case, rather than having to remember -the precise camel-case form of name of the \texttt{\textbackslash Diff}, -\texttt{\textbackslash diffd} and \texttt{\textbackslash Diffd} commands. +Unless one is using such variant forms frequently, it seems simpler +to remember that they are available as dot-delimited arguments to +the \texttt{\textbackslash diff} command, using the obvious name +in each case, rather than having to remember the precise camel-case +form of name of the \texttt{\textbackslash Diff}, \texttt{\textbackslash diffd} +and \texttt{\textbackslash Diffd} commands. -\subsection{The \texttt{.def} file\label{subsec:The-.def-file}} +\subsection{The \texttt{.def} file\protect\label{subsec:The-.def-file}} This mechanism of variant formation is how I have been able to illustrate -in the Rogues' Gallery, Section~\ref{sec:Rogues'-gallery}, the wide variety +in the Rogues' Gallery, §\ref{sec:Rogues'-gallery}, the wide variety of different usages culled from the literature. But the thought arises: -if a variant is to be used only once or twice, isn't this a lot of bother? -Why not just construct the variant derivative `by hand' out of \texttt{\textbackslash frac} -and \texttt{\textbackslash mkern} for example? The reason for making such -definitions is that they can be transferred from document to document. -For instance, definitions placed in the preamble can be copied to the preamble -of another document. - -But that is hardly optimal. Instead, \texttt{diffcoeff} allows such definitions -to be placed in a text file with the the extension \texttt{.def} and a -name of your choosing. For the present document the file is called \texttt{diffcoeff-doc.def} -and has been placed in the same directory as \texttt{diffcoeff.tex}. To -use these definitions, the \texttt{diffcoeff} package is called with the -command +if a variant is to be used only once or twice, isn't this a lot of +bother? Why not just construct the variant derivative `by hand' out +of \texttt{\textbackslash frac} and \texttt{\textbackslash mskip} +for example? The reason for making such definitions is that they can +be transferred from document to document. For instance, definitions +placed in the preamble can be copied to the preamble of another document. + +But that is hardly optimal. Instead, \texttt{diffcoeff} allows such +definitions to be placed in a text file with the the extension \texttt{.def} +and a name of your choosing. For the present document the file is +called \texttt{diffcoeff-doc.def} and has been placed in the same +directory as \texttt{diffcoeff.tex}. To use these definitions, the +\texttt{diffcoeff} package is called with the command \begin{lyxcode} \textbackslash usepackage{[}def-file=diffcoeff-doc{]}\{diffcoeff\} \end{lyxcode} But even this process still means copying a definition file from directory -to directory as one works on different documents. The solution is to make -a definition file available for \emph{all} documents and the way to do -that is by placing it in the texmf tree, preferably not the one created -by your \TeX{} distribution (perhaps MiKTeX or TexLive), but your own \emph{personal} -texmf tree. +to directory as one works on different documents. The solution is +to make a definition file available for \emph{all} documents and the +way to do that is by placing it in the texmf tree, preferably not +the one created by your \TeX{} distribution (perhaps MiKTeX or TexLive), +but your own \emph{personal} texmf tree. \texttt{\vspace{2ex} } @@ -1464,19 +1630,20 @@ texmf tree. \noindent\fbox{\begin{minipage}[t]{1\linewidth - 2\fboxsep - 2\fboxrule}% \textbf{Personal texmf tree? } -This is a directory for `waifs and strays' of the \TeX{} system that are -not included in standard distributions like MiK\TeX{} or \TeX Live. For -instance, it is the place for personal packages designed for your own particular -circumstances or preferences, and is structured like the standard MiK\TeX{} -or \TeX Live hierarchy but placed in another location so that there is -no chance of its being overwritten when MiK\TeX{} or \TeX Live are updated. -However, those distributions need to be alerted to its existence. For MiK\TeX , -open the MiK\TeX{} console, click on \textsf{Settings} and then the \textsf{Directories} -tab. Click the \textsf{+} button and navigate to your personal texmf tree -to add it to the MiK\TeX{} search path. Having added it, you will then need -to refresh the filename database by clicking on the \textsf{Tasks} menu -and selecting the obvious entry. I am not familiar with \TeX Live but presume -a similar process will apply there.% +This is a directory for `waifs and strays' of the \TeX{} system that +are not included in standard distributions like MiK\TeX{} or \TeX Live. +For instance, it is the place for personal packages designed for your +own particular circumstances or preferences, and is structured like +the standard MiK\TeX{} or \TeX Live hierarchy but placed in another +location so that there is no chance of its being overwritten when +MiK\TeX{} or \TeX Live are updated. However, those distributions need +to be alerted to its existence. For MiK\TeX , open the MiK\TeX{} console, +click on \textsf{Settings} and then the \textsf{Directories} tab. +Click the \textsf{+} button and navigate to your personal texmf tree +to add it to the MiK\TeX{} search path. Having added it, you will then +need to refresh the filename database by clicking on the \textsf{Tasks} +menu and selecting the obvious entry. I am not familiar with \TeX Live +but presume an analogous process will apply there.% \end{minipage}} \medskip{} @@ -1487,68 +1654,61 @@ to all documents. \subsubsection{Structure of the \texttt{.def} file} -The best way to see what a \texttt{.def} file looks like is to view \texttt{diffcoeff-doc.def -}in a text editor.\footnote{This file should be in the same directory as \texttt{diffcoeff.pdf} and -\texttt{diffcoeff.tex} in your \LaTeX{} distribution.} +The best way to see what a \texttt{.def} file looks like is to view +\texttt{diffcoeff-doc.def }in a text editor.\footnote{This file should be in the same directory as \texttt{diffcoeff.pdf} +and \texttt{diffcoeff.tex} in your \LaTeX{} distribution.} -If you want your variant definitions to use defaults different from those -supplied with the \texttt{diffcoeff} package, then the first definition -in the \texttt{.def} file should be the one setting the new defaults, with -an \emph{empty} first argument to the \texttt{\textbackslash diffdef} +If you want your variant definitions to use defaults different from +those supplied with the \texttt{diffcoeff} package, then the first +definition in the \texttt{.def} file should be the one setting the +new defaults, with an \emph{empty} first argument to the \texttt{\textbackslash diffdef} command: \begin{lyxcode} -\noindent % -\noindent\begin{minipage}[t]{1\columnwidth}% -\begin{lyxcode} \textbackslash diffdef~\{\}~~~~ -\begin{lyxcode} -\{~~~~~~ -\begin{lyxcode} -key-1~=~value-1,~~~~ -key-2~=~value-2, +~~\{~~~~~~ -... +~~~~key-1~=~value-1,~~~~ -key-n~=~value-n -\end{lyxcode} -\} -\end{lyxcode} -\end{lyxcode} -% -\end{minipage} +~~~~key-2~=~value-2, + +~~~~... + +~~~~key-n~=~value-n + +~~\} \end{lyxcode} -The key-value list is a comma-separated list; hence the last entry doesn't -need to end with a comma. Nudge and separation values need to include the -unit, \texttt{mu}; a numerical value alone will result in error.\texttt{ }Because -a \texttt{.def} file\texttt{ }is a \LaTeX{} file, comments need to start -with a \texttt{\%} character. +The key-value list is a comma-separated list; hence the last entry +doesn't need to end with a comma. Nudge and separation values need +to include the unit, \texttt{mu}; a numerical value alone will result +in error.\texttt{ }Because a \texttt{.def} file\texttt{ }is a \LaTeX{} +file, comments need to start with a \texttt{\%} character. \subsubsection{\texttt{diffcoeff.def}} -Note that if the \texttt{diffcoeff} package is invoked without an explicit -\texttt{def-file= <filename>} option statement, as here, +Note that if the \texttt{diffcoeff} package is invoked without an +explicit \texttt{def-file= <filename>} option statement, as here, \begin{lyxcode} \textbackslash usepackage\{diffcoeff\} \end{lyxcode} -then it will search in the texmf tree (the \LaTeX{} distribution's and your -personal one) and the document directory for a file \texttt{diffcoeff.def} -and if found will load that. This file should contain definitions of those -variants you are likely to use in multiple documents. In my personal texmf -tree (which I've put at \texttt{D:\textbackslash texmf\textbackslash} -on a Windows machine) the file \texttt{diffcoeff.def} is located in the -directory \texttt{D:\textbackslash texmf\textbackslash tex\textbackslash latex\textbackslash diffcoeff\textbackslash}. +then it will search in the texmf tree (the \LaTeX{} distribution's +and your personal one) and the document directory for a file \texttt{diffcoeff.def} +and if found will load that. This file should contain definitions +of those variants you are likely to use in multiple documents. In +my personal texmf tree (which I've put at \texttt{E:\textbackslash texmf\textbackslash} +on a Windows machine) the file \texttt{diffcoeff.def} is located in +the directory \texttt{E:\textbackslash texmf\textbackslash tex\textbackslash latex\textbackslash diffcoeff\textbackslash}. (The backslashes are replaced by forward slashes on linux machines.) -Variants likely to be of value only to a specific document should be added -to the preamble of that document. Alternatively, they could be added to -\texttt{diffcoeff.def} but that added-to file saved to the document directory -under a \emph{different} name -- e.g. I've saved the variants required -for the present document under the name \texttt{diffcoeff-doc.def}. Many -of these variants were created solely to illustrate points in the present -document and I have no intention of using them in my own work. Consequently -my \texttt{diffcoeff.def} file is smaller, containing only a selection -from \texttt{diffcoeff-doc.def}. +Variants likely to be of value only to a specific document should +be added to the preamble of that document. Alternatively, they could +be added to \texttt{diffcoeff.def} but that added-to file saved to +the document directory under a \emph{different} name -- e.g. I've +saved the variants required for the present document under the name +\texttt{diffcoeff-doc.def}. Many of these variants were created solely +to illustrate points in the present document and I have no intention +of using them in my own work. Consequently my \texttt{diffcoeff.def} +file is smaller, containing only a selection from \texttt{diffcoeff-doc.def}. \section{Differentials in integrals, etc.} @@ -1557,261 +1717,359 @@ point, and its omission from previous versions of \texttt{diffcoeff}.} for examp \[ \dl P=\diffp Px\dl x+\diffp Py\dl y+\diffp Pz\dl z, \] -or an integral like $\int\sin x\dl2x$, or a multi-variable integral like +or an integral like $\int\sin x\dl2x$, or a multi-variable integral +like \[ \iiintop_{-\infty}^{\infty}V(x,y,z)\dl3x\dl2y\dl2z. \] -Surely we want the `d's in these expressions to correspond to their form -(upright or math italic) in derivatives? If, for instance, the \texttt{ISO} -package option has been set, one doesn't want to be writing \texttt{\textbackslash mathrm\{d\}} -in every (or indeed any) integral. To this end, \texttt{diffcoeff} provides -a command \texttt{\textbackslash dl} to write the `d' in a differential -in a manner consistent with the default form used in derivatives. In the -present document, the default form is math-italic and so -\begin{example} -\textbackslash dl x $\Longrightarrow\dl x.$ -\end{example} - -\noindent It is also possible\footnote{\noindent From version 3.1 of \texttt{diffcoeff}; version 3.0 of the package -produced a \LaTeX{} error.} to use the command before, for instance, \texttt{\textbackslash vec\{x\}} -or \texttt{\textbackslash mathbf\{x\}}: -\begin{example} -\textbackslash dl \textbackslash vec\{x\} $\Longrightarrow\dl\vec{x}$~~~~\textbackslash dl -\textbackslash mathbf\{x\} $\Longrightarrow\dl0\mathbf{x}.$ -\end{example} - +Surely we want the `d's in these expressions to correspond to their +form (upright or math italic) in derivatives? If, for instance, the +\texttt{ISO} package option has been set, one doesn't want to be writing +\texttt{\textbackslash mathrm\{d\}} in every (or indeed any) integral. +To this end, \texttt{diffcoeff} provides a command \texttt{\textbackslash dl} +to write the `d' in a differential in a manner consistent with the +default form used in derivatives. In the present document, the default +form is math-italic and so +\begin{centred} +\verb`$ \dl x $` $\Longrightarrow$ $ \dl x. $ +\end{centred} +\noindent It is also possible\footnote{\noindent From version 3.1 of \texttt{diffcoeff}; version 3.0 of the +package produced a \LaTeX{} error.} to use the command before, for instance, \texttt{\textbackslash mathbf\{x\}} +or \texttt{\textbackslash vec\{x\}}: +\begin{centred} +\verb`$ \dl \vec{x} $` $\Longrightarrow$ $ \dl \vec{x} $,~~~~\verb`$ \dl \mathbf{x} $` +$\Longrightarrow$$ \dl \mathbf{x}. $ +\end{centred} \subsection{Options} There are two options available with the differential command. -The first is the dotted name option discussed in Section~\ref{sec:Changing-defaults}. +The first is the dotted name option discussed in §\ref{sec:Changing-defaults}. Thus, to illustrate a differential with an upright `d', write -\begin{example} -\textbackslash dl.up.x $\Longrightarrow\dl.up.x.$ -\end{example} - -\noindent This presumes that a variant derivative with upright `d's has -been defined and the definition is available, either in the preamble or -in an accessible \texttt{.def} file, as discussed in the previous section. -Alternatively, if you have defined your default derivative to use upright -`d's (perhaps with the \texttt{ISO} option) then \texttt{\textbackslash dl} -alone will suffice to produce an upright `d'. - -Since the variant \texttt{\textbackslash diff.p.} is defined in \texttt{diffcoeff.sty} -itself, \texttt{\textbackslash dl.p.} is always available and at 6 keystrokes -offers a slightly shorter way of writing \texttt{\textbackslash partial} -(8 keystrokes). But the only reason one might use this variant `in earnest' -is because of the convenience offered by the second option to the \texttt{\textbackslash dl} -command. +\begin{centred} +\verb`$ \dl.up.x $` $\Longrightarrow$ $ \dl.up.x. $ +\end{centred} +\noindent This presumes that a variant derivative with upright `d's +has been defined and the definition is available, either in the preamble +or in an accessible \texttt{.def} file, as discussed in the previous +section. Alternatively, if you have defined your default derivative +to use upright `d's (perhaps with the \texttt{ISO} option) then +\texttt{\textbackslash dl} alone will suffice to produce an upright +`d'. + +\subsubsection{Partial differential} + +\label{subsec:Partial-differential}Since the variant \texttt{\textbackslash diff.p.} +is defined in \texttt{diffcoeff.sty} itself, \texttt{\textbackslash dl.p.} +is always available and at 6 keystrokes offers a slightly shorter +way of writing \texttt{\textbackslash partial} (8 keystrokes). However, +there are sufficient contexts where expressions like \verb`\partial_x` +are used, perhaps as a shortcut for a partial derivative, for it to +be worthwhile to define an appropriate command for this variant. From +version 4.0, \texttt{diffcoeff} therefore provides \verb`\dlp`, defined +by +\begin{lyxcode} +\textbackslash NewDocumentCommand~\textbackslash dlp~\{\}~\{~\textbackslash dl.p.~\} +\end{lyxcode} +Thus, for instance, \verb`$ \dlp_x $` $\Longrightarrow$ $ \dlp_x $. \subsubsection{Spacing} This second option inserts spacing before the `d'. If \texttt{\textbackslash dl} -is followed by a digit (0, 1, 2, ..., 9) it will insert a horizontal space -of that number of mu before the `d'; (\texttt{\textbackslash dl0x} has -the same effect as \texttt{\textbackslash dl x}.)\texttt{ }Thus, an alternative -way of writing an example in Chapter~18 of \emph{The \TeX book} is -\begin{example} -\textbackslash dl x\textbackslash dl3y=r\textbackslash dl3r\textbackslash dl3\textbackslash theta -$\Longrightarrow\dl x\dl3y=r\dl3r\dl3\theta.$ -\end{example} - +is followed by a digit (0, 1, 2, ..., 9) it will insert a horizontal +space of that number of mu before the `d'; (\texttt{\textbackslash dl0x} +has the same effect as \texttt{\textbackslash dl x}.)\texttt{ }Thus, +an alternative way of writing an example in Chapter~18 of \emph{The +\TeX book} is +\begin{centred} +\verb`$ \dl x\dl3y=r\dl3r\dl3\theta $` $\Longrightarrow$ $ \dl x\dl3y=r\dl3r\dl3\theta. $ +\end{centred} \noindent To my eye this is too much space; I prefer -\begin{example} -\noindent \textbackslash dl x\textbackslash dl2y=r\textbackslash dl2r\textbackslash dl2\textbackslash theta -$\Longrightarrow\dl x\dl2y=r\dl2r\dl2\theta$ -\end{example} - +\begin{centred} +\noindent \verb`$ \dl x\dl2y=r\dl2r\dl2\theta $` $\Longrightarrow$ +$ \dl x\dl2y=r\dl2r\dl2\theta. $ +\end{centred} \noindent I used \texttt{\textbackslash dl3x\textbackslash dl2y\textbackslash dl2z} -when writing the triple integral above, \emph{no} extra spacing when writing -the total differential expression, since the differentials are already -distinct from the preceding fraction forms $\diff.ptxt.Px$ etc., and \texttt{\textbackslash dl2x} -when writing $\int\sin x\dl2x$. +when writing the triple integral above, \emph{no} extra spacing when +writing the total differential expression, since the differentials +are already distinct from the preceding fraction forms $\diff.ptxt.Px$ +etc., and \texttt{\textbackslash dl2x} when writing $\int\sin x\dl2x$. Note that only \emph{one} digit can be used. If two are used, as here, -\begin{example} -\textbackslash dl20x $\Longrightarrow\dl20x$ -\end{example} - -\noindent the effect is unlikely to be what is wanted. - -From version 3.2 of \texttt{diffcoeff} it is possible to also add \emph{negative} -space before the differential, which might be useful in special contexts, -perhaps to construct a symbol: -\begin{example} -/\textbackslash dl-9x $\Longrightarrow/\dl-9x.$ -\end{example} - -\noindent For negative space before the differential, add a minus sign -before the (single) digit. +\verb`$ \dl20x $` $\Longrightarrow$ $ \dl20x $, the effect is unlikely +to be what is wanted. + +From version 3.2 of \texttt{diffcoeff} it is possible to also add +\emph{negative} space before the differential, which might be useful +in special contexts, perhaps to construct a symbol: +\begin{centred} +\verb`$ /\dl-9x $` $\Longrightarrow$ $ /\dl-9x. $ +\end{centred} +\noindent For negative space before the differential, add a minus +sign before the (single) digit. The spacing digit option \emph{follows} the dot-delimited name option. -For example, to illustrate spacing in the denominator of a mixed partial -derivative, I have earlier used \texttt{\textbackslash dl.p.x\textbackslash dl.p.2y\textbackslash dl.p.2z}, -and the variation to that spacing when a higher-order differentiation occurs: -\begin{example} -\textbackslash dl.p.x\textasciicircum 2\textbackslash dl.p.1y\textbackslash dl.p.2z -$\Longrightarrow\dl.p.x^{2}\dl.p.1y\dl.p.2z.$ -\end{example} - +For example, earlier, in the denominator of a mixed partial derivative, +I have used what is effectively \texttt{\textbackslash dlp x\textbackslash dlp2y\textbackslash dlp2z}, +and the variation to that spacing when a higher-order differentiation +occurs: +\begin{centred} +\verb`$ \dlp x^2\dlp1y\dlp2z $` $\Longrightarrow$ $ \dlp x^2\dlp1y\dlp2z. $ +\end{centred} \section{Rationale} Version 1 of the \texttt{diffcoeff} package arose from a need to simplify -the parsing of differential coefficients for another program I was working -on which was struggling to `read' all the possible permutations of \texttt{\textbackslash frac} -or \texttt{\textbackslash tfrac} or \texttt{\textbackslash dfrac} or -slash forms of the derivative, of \texttt{d} or \texttt{\textbackslash mathrm\{d\}} -or \texttt{\textbackslash partial} or \texttt{D} or \texttt{\textbackslash mathrm\{D\}} -or \texttt{\textbackslash nabla},\texttt{ }and of points of evaluation -delimited by vertical rules or parentheses.\texttt{ }Although regular expressions -coped with most of these cases, it was \emph{messy}. +the parsing of differential coefficients for another program I was +working on which was struggling to `read' all the possible permutations +of \texttt{\textbackslash frac} or \texttt{\textbackslash tfrac} +or \texttt{\textbackslash dfrac} or slash forms of the derivative, +of \texttt{d} or \texttt{\textbackslash mathrm\{d\}} or \texttt{\textbackslash partial} +or \texttt{D} or \texttt{\textbackslash mathrm\{D\}} or \texttt{\textbackslash nabla},\texttt{ +}and of points of evaluation delimited by vertical rules or parentheses.\texttt{ }Although +regular expressions coped with most of these cases, it was \emph{messy}. There are other packages which have commands for the derivative (e.g., -\texttt{bropd}, \texttt{commath},\texttt{ esdiff}, \texttt{physymb}) but -none quite gave what I wanted -- although they probably cope with most -users' needs. \texttt{esdiff} came closest to what I was seeking but failed -when it came to combining algebraic and numeric orders of differentation -in a mixed partial derivative (and made heavier use of braces than I would -like in that case too). +\texttt{bropd}, \texttt{commath},\texttt{ esdiff}, \texttt{physymb}) +but none quite gave what I wanted -- although they probably cope +with most users' needs. \texttt{esdiff} came closest to what I was +seeking but failed when it came to combining algebraic and numeric +orders of differentation in a mixed partial derivative (and made heavier +use of braces than I liked in that case too). \subsection{\texttt{diffcoeff.sty}} -I have tried to make using \texttt{diffcoeff} intuitive. Looking at the -other packages mentioned, writing something like \texttt{\textbackslash diff{[}n{]}\{f\}\{x\}} -(which can be trimmed to \texttt{\textbackslash diff{[}n{]}fx} for single-token -arguments) seems `natural' -- only \texttt{physymb} deviates from the -pattern. +I have tried to make using \texttt{diffcoeff} intuitive. Looking at +the other packages mentioned, writing something like \texttt{\textbackslash diff{[}n{]}\{f\}\{x\}} +(which can be trimmed to \texttt{\textbackslash diff{[}n{]}fx} for +single-token arguments) seems `natural' -- only \texttt{physymb} +deviates from the pattern. \begin{itemize} \item It seems consistent with this pattern to use a comma list for specifying the orders of differentiation of the variables in a higher order mixed partial derivative (and its suppression when all are of order 1) \item Having specified the orders, surely the program itself should calculate -the overall order? \texttt{esdiff} does this for numerical orders; \texttt{diffcoeff} -does this for both numeric and algebraic orders, +the overall order? \texttt{esdiff} does this for numerical orders; +\texttt{diffcoeff} does this for both numeric and algebraic orders, \end{itemize} +\begin{centred} +\verb`\[ \diffp[m-(k+1),m+(k-1)]{F(x,y,z)}{x,y,z} \]` +\end{centred} \begin{example} -\textbackslash diffp{[}m-(k+1),m+(k-1){]}\{F(x,y,z)\}\{x,y,z\} - -$\Longrightarrow{\displaystyle \diffp[m-(k+1),m+(k-1)]{F(x,y,z)}{x,y,z}},$ +$\Longrightarrow$ \[ \diffp[m-(k+1),m+(k-1)]{F(x,y,z)}{x,y,z} \] \end{example} \begin{itemize} -\item and where it fails, either to calculate at all or to present the result +\item and where it fails either to calculate at all or to present the result in a preferred form, offers the order-override option: \end{itemize} +\begin{centred} +\verb`\[ \diffp[m+(k+1),m+(k-1)][2(m+k+1)]{F(x,y,z,w)}{x,y,z,w} \]` +\end{centred} \begin{example} -\textbackslash diffp{[}m+(k+1),m+(k-1){]}{[}2(m+k+1){]}\{F(x,y,z,w)\}\{x,y,z,w\} - -$\Longrightarrow{\displaystyle \diffp[m+(k+1),m+(k-1)][2(m+k+1)]{F(x,y,z,w)}{x,y,z,w}}.$ +$\Longrightarrow$ \[ \diffp[m+(k+1),m+(k-1)][2(m+k+1)]{F(x,y,z,w)}{x,y,z,w} \] \end{example} \begin{itemize} -\item I wished to avoid the unnecessary writing of superscripts, subscripts and -brace pairs. In the examples just given, no superscript tokens \texttt{\textasciicircum} -are written by the user despite the higher-order differentiation in $x$ -and $y$, and only the two inescapable brace pairs are required. +\item I wished to avoid the unnecessary writing of superscripts, subscripts +and brace pairs. In the examples just given, no superscript tokens +\texttt{\textasciicircum} are written by the user despite the higher-order +differentiation in $x$ and $y$, and only the two inescapable brace +pairs are required. \item The use of a comma list for the second mandatory argument in a partial -derivative -- the list of variables -- makes differentiations in super- -or subscripted symbols (as occurs prolifically in tensor calculus) easier -to both write and read by avoiding unnecessary brace pairs. +derivative -- the list of variables -- makes differentiations in +super- or subscripted symbols (as occurs prolifically in tensor calculus) +easier to both write and read by avoiding unnecessary brace pairs. \end{itemize} -\begin{example} -\textbackslash diffp\{A\_i\}\{ x\textasciicircum j,x\textasciicircum k -\} $\Longrightarrow\quad{\displaystyle \diffp{A_{i}}{x^{j},x^{k}}.}$ -\end{example} - +\begin{centred} +\verb`\[ \diffp{A_i}{ x^j,x^k } \]` $\Longrightarrow$ \[ \diffp{A_i}{ x^j,x^k } \] +\end{centred} \begin{itemize} -\item Should a point of evaluation or variables held constant be considered part -of the derivative? Thermodynamic usage was decisive here. The partial derivative -alone is ambiguous -- the parentheses and subscript are essential to understand -what is being stated: +\item Should a point of evaluation or variables held constant be considered +part of the derivative? Thermodynamic usage was decisive here. The +partial derivative alone is ambiguous -- the parentheses and subscript +are essential to understand what is being stated: \[ \diffp ST[V] \] Hence provision for these extra elements was included in the derivative commands. \item Given the position of the subscripted symbol in the displayed derivative, -it's positioning as the \emph{final} argument in the derivative commands +it's positioning as the \emph{final} argument in the derivative command feels inevitable. \item Version 1 of \texttt{diffcoeff} used braces for this argument to avoid -any possible confusion with a following mathematical expression. That use -of braces is now deprecated in \texttt{xparse}. Consequently later versions -of the package use square brackets, conforming with familiar \LaTeX{} practice. -The only special remembering needed is avoidance of a space before the -argument -- and if it does slip in, it won't cause a \LaTeX{} error. It -will be treated as part of a following mathematical expression and displayed -as such. -\item The star option also prompted the question: is it needed? After all, 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 derivative commands, the star option -became the simplest way of handling appended differentiands since the parentheses -for a variable held constant must wrap around the differential operator -\emph{and} differentiand. Once available, it provides a simple way of switching -between (and comparing) the appearance of differentiand-in-the-numerator -and differentiand-appended. -\item The slash option was added to the derivative commands after seeing how -widely such forms are used in texts at all levels. The placement of the -slash, between the two mandatory arguments, seems more-or-less self-evident. -\item The final option added to \texttt{\textbackslash diff} (and not present +any possible confusion with a following mathematical expression. That +use of braces is now deprecated in \texttt{xparse}, has been deprecated +in \texttt{diffcoeff} since version 2, and is no longer compatible +with version 4. Later versions of \texttt{diffcoeff} use square brackets, +conforming with familiar \LaTeX{} practice. The only special remembering +needed is avoidance of a space before the argument -- and if it does +slip in, it won't cause a \LaTeX{} error. It will be treated as part +of a following mathematical expression and displayed as such. +\item The star option also prompted the question: is it needed? After all, +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 derivative +commands, the star option became the simplest way of handling appended +differentiands since the parentheses for a variable held constant +must wrap around the differential operator \emph{and} differentiand. +Once available, it provides a simple way of switching between (and +comparing) the appearance of differentiand-in-the-numerator and differentiand-appended. +\item The slash option was added to the derivative commands after seeing +how widely such forms are used in texts at all levels. The placement +of the slash, between the two mandatory arguments, seems more-or-less +self-evident. +\item A later option added to \texttt{\textbackslash diff} (and not present in version 1) was the dot-delimited name option. Once \texttt{xtemplate} was used as the basis of the package this seemed the most straightforward -way of making available, ready to hand, the wealth of variants that \texttt{xtemplate} -makes possible. (It's just a pity that the second dot is needed, and a -single-dot naming scheme can't be used, but \texttt{xparse} forces my hand -here.) -\item Having added the dot-delimited name option, the use of a \texttt{.def} -file to store variants or preferred defaults is more-or-less forced, otherwise -one is faced with making these definitions anew for each new document (or -locating a previous document and copying from that to the new one). -\item To handle possible differences between display-style and text-style (and -script-style) derivatives (see Subsection~\ref{subsec:Text-and-script-style}) +way of making available, ready to hand, the wealth of variants that +\texttt{xtemplate} makes possible. (It's just a pity that the second +dot is needed, and a single-dot naming scheme can't be used, but \texttt{xparse} +forces my hand here.) +\item Having added the dot-delimited name option, the use of a \texttt{def} +file to store variants or preferred defaults is more-or-less forced, +otherwise one is faced with making these definitions anew for each +new document (or locating a previous document and copying from that +to the new one). +\item To handle possible differences between display-style and text-style +(and script-style) derivatives (see §\ref{subsec:Text-and-script-style}) I considered using \TeX 's \texttt{\textbackslash mathchoice} command. -This command takes four arguments, corresponding to display-, text-, script- -and scriptscript-styles and would require forming four derivatives each -time a derivative is used, `just in case'. In fact fraction-form derivatives -are used overwhelmingly in display-style expressions, the slash form being -used for inline use. Given the ease of defining a fraction-form variant -for text-style use, and the rareness of such use, employing variants seemed -the way to go. It is the one adopted and avoids the computational burden -associated with the use of \texttt{\textbackslash mathchoice}. -\item After version 2 of the package appeared on CTAN, it was pointed out to -me that there was an issue of consistency between the form of `d' used -in a derivative (upright or math-italic) and the form used in an integral. -I had overlooked this matter completely and in version 3 of the package -remedied the omission with the differential command \texttt{\textbackslash dl}. -A spacing option for \texttt{\textbackslash dl} was almost inevitable. -The demands of writing the present document forced another, the dot-delimited -name option. +This command takes four arguments, corresponding to display-, text-, +script- and scriptscript-styles and would require forming four derivatives +each time a derivative is used, `just in case'. In fact fraction-form +derivatives are used overwhelmingly in display-style expressions, +the slash form being used for inline use. Given the ease of defining +a fraction-form variant for text-style use, and the rareness of such +use, employing variants seemed the way to go. It is the one adopted +and avoids the computational burden associated with the use of \texttt{\textbackslash mathchoice}. +\item After version 2 of the package appeared on CTAN, it was pointed out +to me that there was an issue of consistency between the form of `d' +used in a derivative (upright or math-italic) and the form used in +an integral. I had overlooked this matter completely and in version +3 of the package remedied the omission with the differential command +\texttt{\textbackslash dl}. A spacing option for \texttt{\textbackslash dl} +was almost inevitable. From version 4 the partial analogue \texttt{\textbackslash dlp} +has been added. +\item Space before the differentiand was requested by a user. Once considered +it became clear that there are (at least) two ways of thinking of +a derivative: as $\diff y/x$, a ratio of differentials where $\dl y$ +is a unit and it makes no sense to insert space between the `d' +and the `y', and as +\[ +\diff!{F(x)}x +\] +where the function $F(x)$ is being operated on by $\diff{}/x$ and +it is natural to insert space between the `d' and the `F'. With +that realisation came the need for a simple switch-like package option +(\verb`spaced`) to turn spacing on or off, and a switch-like argument +(\verb`!`) to countermand the package option in exceptional cases +(version 4). \end{itemize} +\section{Version comparison} + +\label{sec:Version-comparison}Unlike version 1, version 2 and later +are built on the the \texttt{xtemplate} package which makes certain +facilities available which it would be silly not to exploit. Hence +the coding in the later versions is completely different and there +are consequences. +\begin{enumerate} +\item From version 2.0 +\begin{enumerate} +\item The \texttt{\textbackslash diffset} command, formerly used to tweak +the display of derivatives, has been superseded by the \texttt{\textbackslash diffdef} +command. \texttt{\textbackslash diffset} now sends a message warning +of its obsolescence to the terminal and \LaTeX{} log but is otherwise +functionless. It should not interfere with the compilation of a document +but any intended fine-tuning of the display by means of the \texttt{\textbackslash diffset} +command\texttt{ }will not eventuate. The warning message is: \texttt{Obsolete +command: \textbackslash diffset has been superseded by the \textbackslash diffdef +command.} \texttt{See the diffcoeff} \texttt{doc\-umentation for +further information.} The \texttt{\textbackslash diffdef} command +is discussed in §\ref{subsec:diffdef}; +\item The optional trailing argument used to indicate a point of evaluation +or variables held constant is now delimited by square brackets, \texttt{{[}} +and \texttt{{]}}, as other optional arguments are. For compatibility +with version 1, versions 2 and 3 still accepted braces to delimit +this argument but from version 4 of \texttt{diffcoeff} only the square-bracket +delimited argument is accepted. (The use of braces around \emph{optional} +arguments while once accepted is now deprecated in \texttt{xparse} +on which \texttt{diffcoeff} depends); +\item The commands \texttt{\textbackslash Diff}, \texttt{\textbackslash diffd} +and \texttt{\textbackslash Diffd} used to construct derivatives from +$D$, $\delta$ and $\Delta$ in version 1, are still available but +deprecated. A new optional argument in the \texttt{\textbackslash diff} +command offers these and a host of other possibilities and is now +the preferred method of forming such variants; see §\ref{subsec:D-delta-Delta}. +\end{enumerate} +\item Version 3.0 +\begin{enumerate} +\item adds a command, \texttt{\textbackslash dl} (from \emph{d}ifferentia\emph{l}) +to write differentials like $dx$ that occur in integrals and in other +contexts in a manner consistent with the form used in derivatives. +After all, if one is using upright `d's in derivatives, similarly +upright `d's should occur in these other contexts;\footnote{This rather obvious lack in version 2 was pointed out to me by Sergio +Callegari.} +\item provides some simple spacing commands that can be useful for tweaking +standard spacing. +\end{enumerate} +\item Version 3.1 enables the differential command to be used before forms +like \texttt{\textbackslash vec\{x\}} (an overlooked possibility +causing an error in earlier versions). +\item Version 3.2 +\begin{enumerate} +\item allows negative spacing before the differential command \texttt{\textbackslash dl}; +\item fixes a bug in which an ordinary derivative as the differentiand of +a partial derivative displayed as a partial derivative. It now displays, +as it should, as an ordinary derivative. +\end{enumerate} +\item Version 4.0 +\begin{enumerate} +\item enables the insertion of a small space before the differentiand, either +as the default behaviour (package option \verb`spaced`) or at explicit +request (argument \verb`!` of the \verb`\diff` command); see §\ref{subsec:Spacing-before-derivand}. +\item prevents the ligature $df$ that marred previous versions; this is +now rendered $\dl f$. +\item offers the document command \verb`\dlp` for the \emph{partial} differential; +see §\ref{subsec:Partial-differential}; +\item no longer accepts \emph{the braced form} of the\emph{ }trailing optional +argument specifying a point of evaluation or (for partial derivatives) +variables held constant. This was a relic from version 1 of \texttt{diffcoeff}, +and has been deprecated since version 2. +\end{enumerate} +\end{enumerate} + \section{Commands} \begin{description} -\item [{\texttt{\textbackslash diff}}] options: +\item [{\texttt{\textbackslash diff}}] arguments (all optional unless +otherwise indicated): \begin{enumerate} \item .\emph{name}. for the given settings \item {*} append-differentiand switch \item {[}\emph{order}{]} or {[}\emph{comma-list of orders}{]} of differentiation \item {[}\emph{order-override}{]} +\item ! countermand before-differentiand spacing of the \verb`spaced` package +option \item \{\emph{differentiand}\} (mandatory) \item / slash-form switch \item \{\emph{comma list of differentiation variables}\} (mandatory) \item {[}\emph{point of evaluation/variables held constant}{]} -\item \{as for previous, but using braces\} (deprecated) \end{enumerate} -\item [{\texttt{\textbackslash diffdef}}] options: +\item [{\texttt{\textbackslash diffdef}}] arguments (all mandatory): \begin{enumerate} \item \{\emph{name}\} \item \{\emph{key=value comma list}\} \end{enumerate} -\item [{\texttt{\textbackslash diffp}}] \texttt{= \textbackslash diff.p.} -\item [{\texttt{\textbackslash dl}}] options: +\item [{\texttt{\textbackslash diffp}}] \texttt{= \textbackslash diff.p.} +\end{description} +\newpage{} +\begin{description} +\item [{\texttt{\textbackslash dl}}] arguments (all optional): \begin{enumerate} \item .\emph{name}. (as for \texttt{\textbackslash diff}) -\item \emph{minus sign} (optional, use only if negative spacing before the `d' -is wanted) +\item \emph{minus sign} (optional, use only if negative spacing before the +`d' is wanted) \item \emph{digit} (insert spacing of this number of mu before the `d') \end{enumerate} \item [{\texttt{\textbackslash negmu}}] insert a $-1$ mu space @@ -1819,7 +2077,8 @@ is wanted) \item [{\texttt{\textbackslash onemu}}] insert a $1$ mu space \item [{\texttt{\textbackslash twomu}}] insert a $2$ mu space \end{description} -Deprecated: +Deprecated (why try remembering the special names when the forms on +the right are self-explanatory): \begin{description} \item [{\texttt{\textbackslash Diff}}] preferred form: \texttt{\textbackslash diff.D.} \item [{\texttt{\textbackslash diffd}}] preferred form \texttt{\textbackslash diff.delta.} |