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diff --git a/obsolete/macros/inrstex/inrsdoc/thmateq.tex b/obsolete/macros/inrstex/inrsdoc/thmateq.tex new file mode 100644 index 0000000000..e894a666e9 --- /dev/null +++ b/obsolete/macros/inrstex/inrsdoc/thmateq.tex @@ -0,0 +1,480 @@ +%-*-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 |