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+%% 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 <<<