%% file: TXSeqns.doc - Equations - TeXsis version 2.18 %% @(#) $Id: TXSeqns.doc,v 18.2 2000/05/17 00:19:46 myers Exp $ %======================================================================* % (C) Copyright 1989, 1992 by Eric Myers and Frank E. Paige % This file is a part of TeXsis. Distribution and/or modifications % are allowed under the terms of the LaTeX Project Public License (LPPL). % See the file COPYING or ftp://ftp.texsis.org/texsis/LPPL %======================================================================* \ifx\undefined\meta\texsis\input TXSdocM.doc\draft\fi \section{Equations \label{sect.eqns}} Plain \TeX\ was designed to make it easy to typeset equations and mathematical formulae. \TeXsis\ provides additional features that make the job even easier. Among these are the automatic numbering of equations (with the ability to refer to numbered equations by symbolic labels), automatic sizing of parentheses, and a collection of special symbols which are often used by physicists. \subsection{Equation Numbering \label{sect.eqno}}% In Plain \TeX\ equation numbers are added to displayed equations with the \cs{eqno} command, as described in Chapter~19 of \TeXbook. For example, typing \index{equations!numbering} \TeXexample $$ E=mc^2. \eqno (47) $$ |endTeXexample will produce \offparens $$ E = mc^2 \eqno (47) $$ With \TeXsis\ the procedure for getting automatically numbered equations is similar, except that you use \CS{EQN} instead of \cs{eqno}, and what follows this (up to the ``|$$|'') is not used as the number of the equation, but as an internal {\it label}\, for the equation. For example, if you had already used |\EQN| to create three numbered equations, then typing \eqnum=3 \TeXexample $$ \vec F = -{G_N Mm \over r^2} \, \hat r \EQN Newton $$ |endTeXexample would produce $$ \vec F = -{G_N Mm \over r^2} \, \hat r \EQN Newton $$ \TeXsis\ automatically assigns the next available equation number to the equation and prints this number on the right, in parentheses. The equations that follow the one above would be numbered 4.5, 4.6, and so on. The purpose of the "label" after the |\EQN| is to let you refer to the equation in the text without knowing the equation number that has been assigned to it. For example, to refer to \Eq{Newton} above, which has the label ``Newton'', you might type \index{equations!labeling} \TeXexample The dependence of the force on the inverse square of the distance can be seen in \Eq{Newton}. |endTeXexample This would be printed as: \example The dependence of the force on the inverse square of the distance can be seen in \Eq{Newton}. \endexample The control sequence \CS{Eq}\arg{label} gives the same result as typing "|Eq.~(|$nn$|)|", where $nn$ is the appropriate equation number for the given label. Two similar control words allow you to refer to a range of equations, or to refer to an equation simply by number, without the preceding "|Eq.~|." Typing \CS{Eqs}\arg{label} produces "Eqs.~($nn$)", while \CS{Ep}\arg{label} gives just the equation number associated with the label, enclosed in parentheses. Thus typing \TeXexample The class of metrics defined by \Eqs{cond-1} through \Ep{cond-5} can all be shown to satisfy inequality \Ep{triangle}, as well as the condition in \Eq{Killing}. |endTeXexample would produce output something like \example The class of metrics defined by Eqs.~(14) through (19) can all be shown to satisfy inequality (23), as well as the condition in Eq.~(7). \endexample Almost any sequence of characters can be used as a "label" for an equation. This includes letters, numbers, and punctuation marks, but it is best to avoid \TeX's special characters, such as |\|, |&|, |}|, |{|, |$|, |_|, and |^|. Spaces may be included but will be ignored, and there {\sl is} a difference between uppercase and lowercase letters. Also, the semicolon (|;|) is reserved for a special purpose, which is described below. You may want to give equations "names," as in the example called "Newton" above, or you can give the equations labels that are numbers. This is the best thing to do if the equations in the original manuscript already have numbers, but if you do this then you should be sure to remember that the internal label and the number which is printed (the ``external'' number) are not necessarily the same. If the equation {\it labeled} ``15'' is the fifteenth equation in the document then it will also be equation number 15, but if you move the equation around in the file so that it is the sixth equation in the document it will be printed as equation 6, even though it has the label ``15''. If you have a heavily edited document and the arrangement of the numbers and labels gets confusing there is a useful aid to help straighten things out. If you say \example \CS{eqnotracetrue} \endexample then the equation numbers will be printed as usual, but with the label appearing next to it in the right margin in square brackets. For example, the equation \TeXexample $$ (i\gamma^\mu \partial_\mu - m)\psi (x) = 0 \EQN Dirac $$ |endTeXexample when printed with |\eqnotracetrue| would appear as: \begingroup\eqnotracetrue $$ (i\gamma^\mu \partial_\mu - m)\psi (x) = 0 \EQN Dirac$$ \endgroup Since the label appears in the margin the spacing of your document remains exactly the same. Obviously you would not print the final copy of the document this way, but it can be very useful for editing. You can turn off this equation tracing feature anytime by saying \cs{eqnotracefalse}. If you say \cs{draft} at the beginning of your document then the equation tracing will be turned on automatically (along with some other useful features). \index{equations!labeling} \bigskip Sometimes when several equations are closely related you will want to give the whole set the same equation number. To distinguish separate formulae in such a group you then generally add a letter to the number. You might, for example, have the equation \index{equations!multiple} $$ J^2\ket{j,m} = j(j+1) \ket{j,m} \EQN eigen;a $$ and then a short while later you might have the related equation $$ J_z\ket{j,m} = m \ket{j,m} \EQN eigen;b $$ These may be referred to together as \Eqs{eigen} or may be referred to separately as \Eqs{eigen;a} and \Ep{eigen;b}. To get an equation number with a letter in it you still use \CS{EQN}, but you add the appropriate letter to the end of the label, separated {from} the label by a semicolon. The equations above were created by typing: \TeXexample $$ J^2\ket{j,m} = j(j+1) \ket{j,m} \EQN eigen;a $$ |endTeXexample and \TeXexample $$ J_z\ket{j,m} = m \ket{j,m} \EQN eigen;b $$ |endTeXexample The semicolon is not printed --- it is just used to separate the letter from the label. If the letter is ``a'' then the next available equation number is used, as usual, but if the letter is anything else then the equation number is not incremented and instead the number from the previous equation is used. You can use any sequence of letters you like, although it is usually best to use ``a,b,c$\ldots$'' for consecutive equations. To refer to equations that have both numbers and letters you can either use the letter to pick out a specific equation, or simply leave out the letter and use the label to refer to the whole set together. You can use either \CS{Eq} or \CS{Eqs} to do this. For example, the sentence after \Eq{eigen;b} above was created by typing \TeXexample These may be referred to together as \Eqs{eigen} or may be referred to separately as \Eqs{eigen;a} and \Ep{eigen;b}. |endTeXexample \index{equations!multiple} \bigskip You will eventually encounter an equation that is so long that it has to be broken up into several lines, or you will want to print several equations together, lined up by their "$=$" signs. In plain \TeX\ you would use |\eqalign| to display these kinds of equations, or |\eqalignno| to display them with equation numbers. With \TeXsis\ you should use \CS{EQNalign} in place of \cs{eqalign} or \cs{eqalignno} to get automatically assigned equation numbers in aligned equations. As in plain \TeX, the alignment of the equations is controlled by |&| and |\cr|. An equation number is assigned to a particular line by putting |\EQN| followed by a label on that line, before the |\cr|. The label is everything {from} the |\EQN| up to the |\cr|. For example, \index{equations!alignment} \TeXexample |smalltt$$ \EQNalign{ {dp \over dt} &= - {\partial H \over \partial q} \EQN Hamilton-1 \cr {dq \over dt} &= {\partial H \over \partial p} \EQN Hamilton-2 \cr} $$ |endTeXexample produces the two aligned and numbered equations $$ \EQNalign{ {dp \over dt} &= - {\partial H \over \partial q} \EQN Hamilton-1 \cr {dq \over dt} &= {\partial H \over \partial p} \EQN Hamilton-2 \cr} $$ \cs{EQNalign} can be used both for typing multi-part equations or for splitting an equation which is too long to fit on a single line. However, when splitting a single long equation into several lines it may actually be better to use |$$\eqalign{|\meta{formula}|}| followed by |\EQN| \meta{label}|$$|. For example, in $$ \eqalign{ {\cal L}_{NC} &= {-g \over 2\cos\theta_W} \sum_i \bar\psi_i [ (t_{3,i}-2Q_i\sin^2\theta_W) \gamma_\mu - t_{3,i}\gamma_\mu\gamma_5 ]\psi_i Z^\mu \cr &\quad - e \sum_i Q_i \bar\psi_i \gamma_\mu \psi_i A^\mu \cr} \EQN{NC} $$ the equation number is centered between the two lines of the equation, which looks better. This equation was produced using \cs{eqalign} by typing: \TeXexample |smalltt$$ \eqalign{ {\cal L}_{NC} &= {-g \over 2\cos\theta_W} \sum_i \bar\psi_i [ (t_{3,i}-2Q_i\sin^2\theta_W) \gamma_\mu - t_{3,i}\gamma_\mu\gamma_5 ]\psi_i Z^\mu \cr &\quad - e \sum_i Q_i \bar\psi_i \gamma_\mu \psi_i A^\mu \cr} \EQN{NC} $$ |endTeXexample Closely related equations can be displayed together with the same equation number by following the label with a semicolon and a letter, as already described above. For example, the set of equations, $$ \EQNalign{ \nabla \cdot \vec D &= 4 \pi \rho \EQN EM;a \cr \nabla \times \vec E &= - {1 \over c} {\partial \vec B \over \partial t} \EQN EM;b \cr \nabla \cdot \vec B &= 0 \EQN EM;c \cr \nabla \times \vec H &= {4 \pi \over c} \vec J + {1 \over c} {\partial \vec D \over \partial t} \EQN EM;d \cr} $$ is produced by typing: \TeXexample |smalltt$$ \EQNalign{ \nabla \cdot \vec D &= 4 \pi \rho \EQN EM;a \cr \nabla \times \vec E &= - {1 \over c} {\partial \vec B \over \partial t} \EQN EM;b \cr \nabla \cdot \vec B &= 0 \EQN EM;c \cr \nabla \times \vec H &= {4 \pi \over c} \vec J + {1 \over c} {\partial \vec D \over \partial t} \EQN EM;d \cr} $$ |endTeXexample Note that the alignment of the equations is controlled by |&| and |\cr|, not by the arrangement used when the equation was typed into the manuscript file. Still, it is recommended that you arrange long sets of equations like these in some sort of clear pattern when you type them. This will both reduce errors when the equation is typed and make it easier for you to read the file when you are looking at it later. Remember, \TeX\ ignores all spaces and tabs when it reads material for equations or displayed equations. \index{equations!alignment} \CS{EQNdoublealign} is similar to \cs{EQNalign} but makes equations with two alignment tabs. For example, \TeXexample |smalltt$$ \EQNdoublealign{ \sigma_i^2 &= 1 & \EQN Pauli;a \cr \sigma_i\sigma_j + \sigma_j\sigma_i &= 0 \qquad & (i \ne j) \EQN Pauli;b \cr \sigma_i\sigma_j &= i\sigma_k \qquad & (i,j,k\ {\rm cyclic}) \EQN Pauli;c \cr} $$ |endTeXexample produces $$ \EQNdoublealign{ \sigma_i^2 &= 1 & \EQN Pauli;a \cr \sigma_i\sigma_j + \sigma_j\sigma_i &= 0 \qquad & (i \ne j) \EQN Pauli;b \cr \sigma_i\sigma_j &= i\sigma_k \qquad & (i,j,k\ {\rm cyclic}) \EQN Pauli;c \cr} $$ with both the equal signs and the parentheses aligned. The second |&| is required in the first line, because without it the |\EQN| would have aligned the equation number with the parentheses enclosing $i\ne j$. Plain \TeX\ also lets you display formulae on several lines without any alignment by using \cs{displaylines}. With \TeXsis\ you can use the similar control sequence \CS{EQNdisplaylines} to get an automatically assigned equation number for any line in such a display. The pattern is the same as in |\displaylines|, but you simply insert \cs{EQN}\meta{label} in front of the \cs{cr} on the line you want numbered. \CS{EQNdisplaylines} produces less space between the equations than would result from typing them separately, and generally looks better for closely related equations. All of the equation alignment macros are special cases of the plain \TeX\ macro \cs{halign}. Therefore, if necessary, the vertical spacing can be adjusted by inserting \example \cs{noalign}\cs{vskip}\meta{dimen} \endexample where \meta{dimen} can be either a positive or a negative dimension. \bigskip In a long document there are several different ways to number the equations. You could just start at ``1'' and count up for each new equation, but sometimes it is better to number the equations consecutively within each chapter or section of a document. In \TeXsis\ the default form for an equation number is $cc.ss.nn$ where $cc$ stands for the chapter number, $ss$ is the section number, and $nn$ is the equation number within that section. Whenever you move to a new section or chapter (using \cs{section} or \cs{chapter}, as described in Section~\use{sect.sects}) the number $ss$ or $cc$ is incremented by one and the number $nn$ is reset to one. If you are not using chapter divisions in your document then the chapter number $cc$ is not made a part of the equation number, and the number $nn$ is reset only at the beginning of a section. If you are not using section divisions in your document then the section number $ss$ is not made a part of the equation number, and the number $nn$ is reset to one only at the beginning of a chapter. If you use neither chapter nor section divisions in your document then neither the chapter number $cc$ nor the section number $ss$ will be made a part of the equation number, and the equations will simply be numbered consecutively, 1, 2, 3\dots. \index{equations!numbering} Although the system just described for forming equation numbers gives a result that is usually what you will want for your equation numbering, you can change the method by setting either of two switches. If you say |\showchaptIDfalse| then the chapter number $cc$ will not be made a part of the equation number even though you may be using chapter level divisions in your document. Similarly, saying |\showsectIDfalse| prevents the section number $ss$ from being used in the equation number. The opposites of these two switches are \cs{showchaptIDtrue} and \cs{showsectIDtrue}. The equation number $nn$ is reset to one only when a section or chapter number that is supposed to be visible is incremented, as you might expect. %>>> EOF TXSeqns.doc <<<