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diff --git a/Master/texmf-dist/doc/texsis/base/TXSeqns.doc b/Master/texmf-dist/doc/texsis/base/TXSeqns.doc new file mode 100644 index 00000000000..d7c0e43aa90 --- /dev/null +++ b/Master/texmf-dist/doc/texsis/base/TXSeqns.doc @@ -0,0 +1,370 @@ +%% 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 <<< |