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-%-*-tex-*-
-\ifundefined{writestatus} \input status \relax \fi %
-\chcode{mateq}
-
-\def\cqu{\cquote{Having got our equations, we must proceed to
-carry out such operatins as we have neglected, taking care never to multiply
-when we can divide}{Rules for the Direction of the Mind, Rule XX,
-Ren\'e~Descartes (1596-1650)}}
-
-\chapterhead{mateq}{MATHEMATICS:\cr EQUATION\cr SETUP}
-\tex\ makes the typesetting of the most difficult expressions relatively
-easy. This chapter lists and shows examples of the special forms supplied by
-{\it plain}. For the finer points of mathematic equation setup, the reader is
-referred to the \texbook~[\cite{[Knut84]}]. For more advanced mathematics macro
-and accenting capabilities, the AMS\tex~[\cite{[Amst84]}] package is
-recommended.
-
-\intex\ supplies all that is in {\it plain} plus
-\beginlist
-\li $\bullet$ an autonumbering feature, using |\aneq|, |\aneqtag|, or |\autoeqnum|
-for equations,
-\li $\bullet$ a |\math| and |\displaymath|
-for those who don't like |$|,
-\li $\bullet$ a
-|\leftequationnumbering| and |\rightequationnumbering| that will magically
-shift all the equation numbers from the side specified by the form,
-\li $\bullet$ and a |\mathopen| command that will cause aligned equations to increase
-their spacing.
-\endlist
-
-
-The two basic commands |\math| and |\displaymath| can be used to invoke either
-scriptstyle or displaystyle mathematics in running text. For instance
-|\math{\sum_0^n}| gives \math{\sum_0^n} while |\displaymath{\sum_0^n}| gives
-\displaymath{\sum_0^n}.
-
-In addition there is a tagging mechanism that enables the
-symbolic referencing of equations, both forward and backward.
-
-
-
-\shead{eqcoms}{Command List}
-\begintwocolumn
-\hfuzz = 20pt %temporary
-\pla|\abovewithdelims|
-\pla|\atop|
-\pla|\atopwithdelims|
-\ext|\aneq \aneqtag|
-\ext|\autoeqnum|
-\pla|\bordermatrix|
-\pla|\cases|
-\pla|\cdot|
-\pla|\cdots|
-\pla|\choose|
-\pla|\ddots|
-\ext|\displaymath|
-\pla|\eqalign|
-\pla|\eqalignno|
-\pla|\eqno|
-\pla|\ldots|
-\ext|\leftequationnumbering|
-\pla|\leqno|
-\pla|\leqalignno|
-\ext|\math|
-\ext|\mathopen|
-\pla|\matrix|
-\pri|\noalign|
-\pla|\openup|
-\pla|\over|
-\pla|\overbrace|
-\pla|\overline|
-\pla|\overwithdelims|
-\pla|\phantom|
-\pla|\pmatrix|
-\ext|\rightequationnumbering|
-\pla|\smash|
-\ext|\sphantom|
-\pla|\sqrt|
-\pla|\underbrace|
-\pla|\underline|
-\pla|\vphantom|
-\endthreecolumn
-
-\shead{eqbasics}{Mathematic Equation Basics}
-To put some mathematics, either symbols or expressions, in running text you
-must surround them by a pair of single |$| like |$<expression>$|. An example
-is |$x^y$| gives $x^y$. Mathematics in running text will be broken at the end
-of line at spaces in the expression. To prevent a break but preserve a space
-insert the tie |~| into the expression. Thus |$x~=~y+1$|, giving $x~=~y+1$
-will prevent a break around the $=$ sign. To obtain display equations use
-form
-\begintt
-$$
-<display mathematics forms>
-$$
-\endtt
-There should be {\bf no} blank lines or |\par| within
-|<display mathematics>|. Mysterious errors will result. To have a
-display expression end a sentence, the last character before the equation
-number should be a period.
-
-Superscripting and subscripting can occur only in mathematics mode. The |\@|
-is the superscript character and |_| is the subscript character. The braces
-|{<super,subscript characters>}| are necessary to tell \tex\ what material
-goes where. The example below shows the result of different use of braces.
-
-$$
-2^xyz_uvw : 2^{xyz}_{uvw} : 2^{xyz_uvw} : 2^{xyz_{uvw}}.
-$$
-is given by
-\begintt
-$$
-2^xyz_uvw : 2^{xyz}_{uvw} : 2^{xyz_uvw} : 2^{xyz_{uvw}}.
-$$
-\endtt
-
-Finally, in commands that produce alignments, the |&| is used to separate
-items and the |\cr| to end a line. Examples of their use will be found in
-such commands as |\matrix| and |\eqalign|. Equation numbers are put in using
-either the automatic numbering commands |\aneq|, |\aneqtag|, |\autoeqnum|, or
-the normal manual numbering forms |\eqno| for right hand numbers or |\leqno|
-for lefthand numbers.
-
-
-
-
-\shead{mateqcomforms}{Command Forms}
-This is a list of commands that are given in {\it plain}, along with some
-examples and the form of the parameters for their use.
-\beginblockmode
-
-\mbr
-\pla\@|\above.. \atop.. \over.. functions
- {<parm1> \atop <parm2>}
- {<parm1> \choose <parm2>}
- {<parm1> \over <parm2>}
- {<parm1> \atopwithdelims<left delim><right delim> <parm2>}
- {<parm1> \overwithdelims<left delim><right delim> <parm2>}
- {<parm1> \abovewithdelims<left delim><right delim>
- <bar thickness> <parm2>}|
-\nbr
-These forms are all used for putting |<parm1>| on top of |<parm2>| as might
-happen in a fraction. The difference between them is that |\atop| puts no
-line or bar between the |<parms>| while |\over| does, and |\choose| puts in
-no bar but does enclose the |<parms>| in large parens as is normally done in
-a binomial coefficient. The forms |\...delims| allow for the specifcation of
-delimiters around the |<parms>| that may be different than parens. Finally
-the |<bar thickness>| in the |\abovewithdelims| allows for the thickness of
-the line or bar between the |<parms>| to be a specified |<dimen>|. As an
-example
-$$
-{37 \over 45} : {n \atop \alpha} : {10 \choose r}
-: {f(x) \overwithdelims\{\} g(y)}
-: {\alpha \atopwithdelims][ \beta}
-: {209 \abovewithdelims\langle\rangle 2pt h(y)}
-$$
-is given by
-\begintt
-$$
-{37 \over 45} : {n \atop \alpha} : {10 \choose r}
-: {f(x) \overwithdelims\{\} g(y)}
-: {\alpha \atopwithdelims][ \beta}
-: {209 \abovewithdelims\langle\rangle 2pt h(y)}
-$$
-\endtt
-\mbr
-\ext|\aneq \aneqtag{<tag>}|
-\nbr
-\ext|\autoeqnum{<tag>}|
-\nbr
-When |autonumbering| is on these commands will automatically number
-equations. |\aneq| and |\aneqtag{<tag>}| inserts a bracketed equation number
-in the correct place in the equation depending on whether it is an
-|\eqalign|, |\eqalignno|, or |\leqalignno|. The |\autonumeq{<tag>}| inserts a
-number without brackets. The |<tag>| allows for references to the equation.
-
-If |autonumbering| is off, the |\aneq| results in nothing and the two forms
-with |<tag>| insert the |<tag>| --- without the |< >| of course.
-
-For example, |autonumbering| is on,
-$$
-x \equiv y \aneq
-$$
-is given by
-\begintt
-$$
-x \equiv y \aneq
-$$
-\endtt
-and
-$$
-\eqalignno{ f(x)&\approx K x^{-3/2} & (\autoeqnum{eqfx}.a) \cr
- &\approx 0 \qquad x \gg 0 & (\ref{eqfx}.b) \cr}
-$$
-is given by
-\begintt
-$$
-\eqalignno{ f(x)&\approx K x^{-3/2} & (\autoeqnum{eqfx}.a) \cr
- &\approx 0 \qquad x \gg 0 & (\ref{eqfx}.b) \cr}
-$$
-\endtt
-The |eqfx| is the equation tag. However, the
-tag for |eqfx| does not make it to the margin! A better way to do this, and
-one that guarantees that the tag for |eqfx| makes it to the margin is as
-follows:
-\begintt
-{\silenttrue \autoeqnum{eqfx}}
-$$
-\eqalignno{ f(x)&\approx K x^{-3/2} & (\ref{eqfx}.a) \cr
- &\approx 0 \qquad x \gg 0 & (\ref{eqfx}.b) \cr}
-$$
-\endtt
-The \@|\silenttrue| turns on a switch in |\autoeqnum{<...>}| that prevents it
-from writing out its value.
-In this case, the |\autoeqnum{<...>}| is in vertical mode
-so that the tag for |eqfx| will now make it to the margin.
-The command |\ref{eqfx}| will produce the
-(equation) number corresponding to |eqfx|. This construction allows
-both lines of the equation to have the same main equation number.
-
-\mbr
-\pla\@|\bordermatrix{<top 0>&<top 1>&<top 2>&...&<top n>\cr
- <side 1>&<first row>\cr
- ...
- <side m>&<last row>\cr}|
-\nbr
-The |\bordermatrix| places parens around the |<rows>| in the same way as
-|\pmatrix| and then places the |<top i>| items above and the |<side i>| items
-to the left hand side outside the parens. Thus
-$$
-\bordermatrix{ R\backslash C& Col 1 & Col 2 & Col 3 \cr
- Row 1 & a & b & c \cr
- Row 2 &\sum_{\ell=1}^n & & \alpha \cr
- Row 3 & v=0 &\gamma & 1000.0\cr}
-$$
-is given by
-\begintt
-$$
-\bordermatrix{ R\backslash C& Col 1 & Col 2 & Col 3 \cr
- Row 1 & a & b & c \cr
- Row 2 &\sum_{\ell=1}^n & & \alpha \cr
- Row 3 & v=0 &\gamma & 1000.0\cr}
-$$
-\endtt
-
-\mbr
-\pla\@|\cases{<first math>&<first condition>\cr
- ...
- <last math>&<last condition>\cr}|
-\nbr
-This allows for a very simple way of stacking a set of possibilities in an
-equation and putting a large brace on the left hand side. The |<...math>| is
-the mathematics, and is in display math mode, and the |<...condition>|
-is the condition and is in internal horizontal mode \dots\ not
-math mode. Thus
-$$
-G(x) = \cases{\vert x\vert & for $x<0$ \cr
- x^2 & for $x\ge 0$ \cr}
-$$
-is given by
-\begintt
-$$
-G(x) = \cases{\vert x\vert & for $x<0$ \cr
- x^2 & for $x\ge 0$ \cr}
-$$
-\endtt
-\mbr
-\pla\@|\eqalign ...
- \eqalign{ <left first row> &<right first row>\cr
- ...
- <left last row> &<right last row>\cr}
- \eqalignno{<left first row> &<right first row>&<first eq.no>\cr
- ...
- <left last row> &<right last row>&<last eq.no>\cr}
- \leqalignno{<left first row>&<right first row>&<first eq.no>\cr
- ...
- <left last row> &<right last row>&<last eq.no>\cr}|
-\nbr
-All three of these commands allow for a list of equations, or expressions to
-be aligned on a {\bf single} point within the expression. This is usually a
-relation such as $=$ or $\gg$ although any point is possible. The
-|\eqalignno| and |\leqalignno| have an extra field at the end of each row for
-an equation number, which will be printed on the right for the former and on
-the left for the latter. Note that in both cases the actual |<eq.no>| field
-appears on the right. Any of the |<left ...>| or |<right ...>| or
-|<... eq.no.>| may be omitted.
-A major difference between |\eqalign| and |\eqalignno| is that the latter
-always is as wide as the page while the former is its natural width.
-|\openup<dimen>|, immediately after the opening |$$|,
-is used to increase the spacing between the lines of an
-|\eqalign| or its relatives by |<dimen>|. A font relative
-|<unit of measure>| such as |ex| is recommended with |1ex| to start.
-To obtain a lefthand equation number with
-|\eqalign|, or ordinary display mathematics, use |\leqno|.
-
-\intex\ supplies automatic equation numbering and ways of having a document
-change from left to right hand equation numbering or vice versa. See
-|\lefhandequationnumbering| and |\righthandequationnumbering|.
-For automatic equation numbering use |\autoeqnum{tag}|, |\aneq| and
-|\aneqtag|.
-
-An example of an |\eqalign| is
-$$
-\eqalign{F(x) &=\int_{-\infty}^x H(y)dy \cr
- &=x^{3/2} \cr
- G(z) &\ll 1 \cr}
- \eqno (12)
-$$
-which is given by
-\begintt
-$$
-\eqalign{F(x) &=\int_{-\infty}^x H(y)dy \cr
- &=x^{3/2} \cr
- G(z) &\ll 1 \cr}
- \eqno (12)
-$$
-\endtt
-Notice the equation number is centered on the right hand side. An example
-for |\eqalignno| is
-$$
-\openup 1ex
-\eqalignno{F(x) &=\int_{-\infty}^x H(y)dy \cr
- &=x^{3/2} &(2.i)\cr
-\noalign{and}
- G(z) &\ll 1 &(2.ii)\cr}
-$$
-is given by
-\begintt
-$$
-\openup 1ex
-\eqalignno{F(x) &=\int_{-\infty}^x H(y)dy \cr
- &=x^{3/2} &(2.i)\cr
-\noalign{and}
- G(z) &\ll 1 &(2.ii)\cr}
-$$
-\endtt
-The use of |\noalign {<text>}| places the |and| at the margin and maintains
-the alignment of the $=$ and $\ll$. This technique will not work with
-|\eqalign| because it does not necessarily take up the full width of the page.
-Note again the placement of the equation numbers.
-
-\mbr
-\pri\@|\eqno <reference> \leqno <reference>|
-\nbr
-|\eqno| and |\leqno| are used to put an equation |<reference>|, usually a
-number, at the right or left hand side, respectively, of an equation. It can
-be used with any display mathematics except |\[l]eqalignno|.
-
-\mbr
-\ext\@|\leftequationnumbering \rightequationnumbering|
-\nbr
-These commands when used with the auto equation numbering commands |\aneq|,
-|\aneqtag{<tag>}| or |\autoeqnum{<tag>}| will result in completely left or
-right hand equation numbers whether |\eqalignno| or
-|\leqalignno| is used. However |\eqno| and |\leqno| still result in right and
-left hand equation numbers respectively.\footnote{\dagger}{These defaults are
-subject to negotiation and change. It is quite easy to make the equation
-sides independent of |\eqno| and |\leqno|.}
-
-
-
-\mbr
-\pri\@|\noalign {<text>}|
-\nbr
-This is used in mathematics to insert text that you want at the left hand
-margin between lines in an |\eqalignno| or |\leqalignno|. See the
-|\eqalignno| example.
-
-\mbr
-\pla\@|\openup <dimen>|
-\nbr
-This is used in display mathematics |$$\openup ... $$| to increase spacing
-between lines in an |\eqalign|, |\eqalignno|, or |\leqalignno|. The dimension
-should be |<decimal number>ex| for font size independence. It does not work
-with |\cases| or |\matrix|.
-
-\mbr
-\pri\@|\over ... \under ...
- \overline{<math>} \underline{<math>}
- \overbrace{<math>} \underbrace{<math>}|
-\nbr
-These commands put lines and braces on top or under the |<math>| expressions.
-Thus
-$$
-\overline{\alpha\beta\gamma} : \underline{2^x_y} :
-\underbrace{\overbrace{H(x)=u+g(x,y)}}
-$$
-is given by
-\begintt
-$$
-\overline{\alpha\beta\gamma} : \underline{2^x_y} :
-\underbrace{\overbrace{H(x)=u+g(x,y)}}
-$$
-\endtt
-
-
-\mbr
-\pla\@|\dots ... \cdot \cdots \ldots \ddots|
-\nbr
-These produce dots of various flavours. Thus
-$$
-[\cdot] : [\cdots] : [\ldots] : [\ddots]
-$$
-are given by
-\begintt
-$$
-[\cdot] : [\cdots] : [\ldots] : [\ddots]
-$$
-\endtt
-\mbr
-\ext\@|\mathopen{<dimen or glue>}|
-\nbr
-This will cause all of the horizontal alignments, such as |\eqalign|,
-|\eqalignno|, |\halign|, |\cases|, |\matrix|, |\bordermatrix|, |\pmatrix|,
-and the table macros within the {\bf group} to be spread apart by
-|<dimen or glue>|. Its effect on tables with rules or lines is probably
-undesirable. {\bf It should be used only within a group such as that implied
-by |$$ ... $$|.}
-
-\mbr
-
-
-\mbr
-\pla\@|\matrix{<entry 11>&<entry 12>&...&<entry 1n>\cr
- <second row>\cr
- ...
- <last row> \cr}
- \pmatrix{<several rows>\cr}|
-\nbr
-The |\matrix| entry allows for the production of matrices or arrays with an
-arbitrary number of rows and columns. Thus
-$$
-\left\{
- \matrix{ a & b & c \cr
- \sum_{\ell=1}^n & & \alpha \cr
- v=0 &\gamma & 1000.0\cr}
-\right\}
-$$
-is given by
-\begintt
-$$
-\left\{
- \matrix{ a & b & c \cr
- \sum_{\ell=1}^n & & \alpha \cr
- v=0 &\gamma & 1000.0\cr}
-\right\}
-$$
-\endtt
-Note that the braces |{..}| were in addition to the rows generated by
-|\matrix|. |\pmatrix| is identical to |\left( \matrix{<rows>\cr} \right)|.
-This just saves the placement of the braces.
-
-
-\mbr
-\pla\@|\phantom{<math>} \vphantom{<math>} \sphantom{<hlist>}|
-\nbr
-|\phantom| builds a box the same size as |<math>| but prints a blank.
-|\vphantom| is a box of the same height and depth as |<math>| but of zero
-width \dots\ a custom made strut. |\sphantom| is used when
-strange expansions such as messages create errors using a |\vphantom|.
-All three are used for controlling spacing.
-
-\mbr
-\pla\@|\sqrt{<math>}|
-\nbr
-This puts a square root sign around a |<math>| expression. Thus |$\sqrt{2}$|
-gives $\sqrt{2}$.
-
-\mbr
-\pla\@|\smash{<math>}|
-\nbr
-This leaves the |<math>| expression in a box of height and depth zero. It is
-useful for certain kinds of superposition.
-
-\endblockmode
-\ejectpage
-
-
-
-\done