In the life of each of us, events happen, the memory of which lives for a long time. One of such events for me was participation in the conference “Metaphorical maps in the work of a psychologist”, which took place last October in Moscow.

Two days of wonderful, intense, professional interaction with colleagues, sharing experience and knowledge, getting to know new products, two days of interesting meetings and just human communication... Such events energize so much that you feel elated for a long time to come.

The most valuable resource of the conference is, of course, people. Organizers, masters, participants - so different, but infinitely interesting in this diversity. One of these “jewels” in my “box of memories” is Oksana Stepanova. An amazing person ... You know, sometimes it seems to me that fairy tale therapists eventually become a bit of magicians themselves.)))

I carefully keep the author’s set of cards received from Oksana as a gift “Magic Helpers from good fairy tales and from time to time I get important tips from them.
Our acquaintance with Oksana continued after we went home - Oksana went to Krasnodar, and I returned to my native Minsk.

And, although now we can communicate only with the help of Internet technologies, our communication is still filled with warmth and mutual respect for each other's professional success. I am very pleased to see how much energy and love Oksana puts into her developments, what interesting author's products she creates, and how much important work is going on in the Idyll center.

And I, for my part, really appreciate Oksana's opinion about my professional findings and new ideas. Oksana enjoys using one of my author's developments - matrices for working with metaphorical maps "POPPY Fields", and I am pleased that my product is in such good hands and benefits people.

And to you, friends, I proudly recommend my author's products - sets of matrices for working with metaphorical maps "POPPY fields" and "POPPY clearings". This is a good help for those professionals who work with MAC, and experience shows that the use of matrices in the work of a psychologist is very effective, since it allows you to solve several problems at once.

I can note the main advantages of the products: convenient, visual schemes, wide coverage of the subject of client requests, soft “input” of the client to work with a metaphor, and a decrease in the level of resistance in the client. I am proud that these products are not only well thought out and structured, but also executed at a high technical level, so working with them is both convenient and pleasant. I hope that they will be interesting and useful for you, friends!

Check out more detailed description products can be found by following the links below.
If you have any additional questions contact me by email This email address is being protected from spambots. You must have JavaScript enabled to view. and I will definitely provide you with all the necessary information.

And in the near future I will share with you some of the "highlights" that I use when working with "POPPY fields" and "POPPY glades".

... I believe that there will be new meetings. We will again find ourselves at the same time in the same place, and remember last year's conference, master classes, the Golden Metaphor awards that Oksana and I received at the closing of the event (thanks to colleagues for acknowledging our products!), share the accumulated news and new plans.
After all, life does not stand still, but a pleasant “aftertaste” after such meetings remains ...
Ekaterina Radchenko, psychologist, MAC practitioner, author of PUZZLE-maxi, Poppy Fields, Poppy Fields products, author and host of intensive training programs.

• “Loved the game! The cards are large and dense, I think they will last us a long time. We play with the whole family: at first it was difficult, but then you swing and the speed game begins. My husband and I, as adults, did not have any advantages, it seemed that the daughter even quickly finds the right combinations. We also have a neuropsychological game "Try it again", we decided to combine them, because. puzzle cards, which are designed to make the game more difficult, are very similar to the cards from "Try it again." Now we play like this: we select in advance simple cards from "Try to repeat" with poses that can really be repeated. Then we shuffle and open one card from the deck of brain teasers, memorize it, and then put it face down. Whoever finds the right combination must repeat the pose from the closed card and shout "To the country". If the posture is correct, then you can take the combination and open a new card from the puzzle deck, if not correct, then the participant can try again after one of the opponents tries to pick up the combination.”

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“The book (1st part) really liked my children. We listened with pleasure and asked many questions. After each chapter there are exercises that get more difficult from chapter to chapter. Therefore, it is better not to read the book before going to bed, but to stock up on time for an interesting dialogue with children. The exercises are designed in such a way that they give a field for creativity to parents and children, depending on specific situation. My children especially enjoyed painting their portraits, inhabiting the houses of kindness, generosity, etc. After the section on perception, we had fun and began to come up with our own exercises for the 5 senses. The children also liked to act out a fairy tale with the participation of 4 types of temperament. Perhaps the most favorite game to know yourself and your character. We improved it a little, added several qualities that were not proposed by the author. For example, honesty, cunning, self-esteem. Each of us filled out 4 sheets - 1 about ourselves and 3 other family members. While filling, talking, clarifying, explaining, clarifying, portraying and even laughing. My children love such tasks where you can learn more about yourself, show another person his portrait and see yourself through the eyes of another. They remember such moments and from time to time ask to repeat them. By the way, when you decide to do such a thing with your children, do not forget to write a name and date on each sheet. Everything changes. Save these sheets. After a while, you can return to them, do it again and see what will change and what will remain the same. I am very glad that the author decided to make a continuation of the 1st part of Psychology for Toddlers. Children are looking forward to the new adventures of Yulia and her dad. There is little children's literature on the market aimed at knowing oneself, one's own inner peace. Even fewer quality publications. The tale of the most spiritual science of Igor Vachkov is based on best achievements psychological science behind last years, written plain language and essentially invites children and adults to an amusing trip. A journey that works for the development of a child and an adult. I am pleased to recommend for active reading to parents, teachers and everyone who is interested in the development of the child's personality.”

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“I looked at the topics of the authors’ Ph.D. dissertations, they are very far from practice preschool education. It seems that all work is based on conclusions, and not on results. scientific research. All information has long been known to scientists dealing with this problem. The authors-philologists are completely unaware of psychological and pedagogical research in this area, and there are quite a lot of them. The content of the work is reminiscent of the WRC bachelor's or master's degree in Teacher education, philological education manifests itself in places. That's all. Thanks to the authors for the abstract work.”

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“A wonderful program to develop the emotional intelligence of children. I am a teacher-psychologist, I have worked in kindergartens for 14 years. Worked with children on various good programs. For the last 2 years I have been studying with the senior and preparatory groups in the Life Skills Program. It differs from other programs in that the theoretical base is very well written, all practical tasks are tied to theory, and there are many explanations of what, why and how to do it. There are some easy ones and some very difficult ones. The kids don't seem to be able to handle them. No, they manage. And the kids love it.”

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“Great metaphorical cards! The structure is unusual: the deck consists of 31 sets of photographs (each set contains 3 cards). You can work both with sets (instructions will come to the rescue), and with individual cards (according to the standard principle). There are a lot of possibilities for using the deck! The quality of the cards themselves is also very good. Thanks to the publisher for continuing to look for something new in the world of metaphorical cards!”

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“Sets are so-so. The old model, in some places with drawings of the 2007 calendar, and the poster with emotions is generally useful and there are valuable quotes. For example, the bill of human rights. But it’s easier to find them yourself on the Internet, order a print from a printing house than overpay for delivery.”

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“I am a child psychologist and have worked in kindergarten 12 years old. During this time I led group lessons various programs, including this one. I think it's a GREAT program. And it is interesting for children, and it is interesting for a psychologist to work and see what happens, how children change. Highly recommend even though there are many others now good programs. The only thing is that there should be a maximum of 6-7 people in the subgroup for everything to work.”

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“I express my gratitude to the author for the depth of consideration of the issue. After getting acquainted with the book, the superstition about what is given to some children and not to others disappears. There is an understanding of the process of formation of literacy. In fact, the book gives: 1. Understanding how literacy is formed in different children. 2. A simple step-by-step literacy tool. Regards, Mikhail."

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“A book for thinking teachers and responsible parents. Helps to better understand the origins of problems. Written good language, the author presents specific material in an accessible and exciting way. I `m teaching foreign language, but even for me the book turned out to be useful in terms of methodology and psychological aspects.”

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“Hello! I want to say thank you for the program "Year before school: from A to Z". I work as a teacher-psychologist and last academic year led a group on psychological preparation children to school. This year I have a similar task, but unfortunately in online stores, including yours, there are no workbooks for this program in stock. Is the publication of this product planned in the near future?”

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“The second deck - and even more delight :) I was waiting for the release for almost a year, after acquiring the “about you” deck. And not in vain!!! This is another masterpiece by Irina Logacheva and a team of psychologists. Of my 25 decks, these two are the most :) Very interesting images, plots ... and the artist's work is simply magnificent. Yesterday I tried it at work - a real pleasure, and the same positive customer reviews about the deck. Beauty and professionalism!”

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“I recently purchased a preschool kit. The emphasis in this game is on the development of fine motor skills and the cognitive sphere of the child. The manual is very detailed with illustrations. Parents and children can easily play this game at home. I especially want to praise the card: it depicts a lot of characters, and therefore it will definitely not be left without the attention of children.”

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“Thank you for these cards. This kit is one of the most used in my work with clients in many areas, from initial counseling to corrective developmental activities. Moreover, it is interesting and effective to use these cards in prevention.”

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“Gorgeous book. Many thanks to Inna Sergeevna for the work with which she illuminated not easy life children in orphanages. The book changed my view not only on disadvantaged children, but also helped me find an approach to my own. ”

I have been teaching psychologists how to work with the MAC for many years and have noticed how differently the acquaintance, understanding and mastering of this amazing tool goes. Someone tries to direct the whole process in a logical direction, someone only in an intuitive way, someone immediately begins consultations with metaphorical cards during training, and someone cannot start even after a year. Everyone has their own pace, their own motivation, their own tasks. But when working with MAC there is one very important thing - questions. You must master the art of asking questions. Without this, a full-fledged consultation using the MAC is not to be expected. And it’s just the most difficult thing for some novice psychologists to work with questions.
But now there is a way out. Our colleague Ekaterina Radchenko has created special matrixes of questions, which, in my opinion, will be very useful, especially for those who are just learning how to work with metaphorical cards. Thanks to this, you can perfectly work out a variety of problems: partnerships, and your career, and self-awareness, and low self-esteem etc.

I am pleased to share a fragment of the consultation and the Sphere of Life technique with the permission of the client.

Considered something like a wheel of life balance, covering all areas of the client's life. We used a deck of Subpersonalities and OH.
Instruction.
1. From the Sub-Personalities to the closed one, get the cards and decompose into all the questions of the finished matrix.
2. Open, discuss.

3. Then, from the OH to the closed one, get a pair of cards (picture-word) and place them near the area where the client is not satisfied and wants to change it.

4. Open, discuss, summarize.
I will not give all the comments of the client, try, looking at the photo, to assume for yourself what he could answer. But, as an example, I will tell you about those where changes are needed.
So:
"In finance, I'm like a first grader. I'm constantly learning, but it's not mine, definitely.
How can I improve my financial condition? Well, I thought about it. I need to find a decent, mature man. I will give him my youth, attractiveness, respect - and he will give me financial support. For me, this is honest and understandable.
My career looks really like on the map. I'm afraid to go out into the world and declare to myself. I sit behind a chair like a little girl. And how can I deal with this fear? Develop a habit, as on this card, loudly declare yourself, your desires and opportunities! I intuitively felt that this was the way it should be!
My leisure time also leaves much to be desired.
I always amuse everyone, like a jester. Judging by the following cards: as long as I communicate with others, like a caring mother, thinking about their well-being, interests, forgetting about my desires, I will remain a cleaver.
Well, my living conditions. I'm like a kid in a tank who doesn't know how to drive it properly. They really don't suit me. Too many things to watch, control, etc. How can this be changed? You know, I think we need to get rid of the excess. Eliminate everything that I do not need, that interferes and requires my time and energy ... "

I repeat: such ready-made matrices can serve as a good help in consultations both for specialists and for their clients.
Let me remind you that we have started another enrollment to the IAC Training School. Details at the link: http://ohcards.ru/news/651/

Tags: Victoria GOLOBORODOVA, training in working with MAC, training in metaphorical cards, MAC training school, distance learning metaphorical associative cards

Another notation for an inline formula provided by LA TE X is to write \begin(math) at the beginning of the formula and \end(math) at the end (in other words, the inline formula can be formatted as an environment named math).

The switch formula LA TEX allows you to surround on both sides not only with pairs of dollar signs, as provided by the standard, but with \[ (at the beginning) and \] (at the end). Alternatively, you can style the off formula as an environment named displaymath. In the same file, you can use both standard and LA TE X notation for formulas.

These alternative notations are fully equivalent to standard TE X's (with dollar signs), with one important exception: if off formulas are denoted with LA TE X's rather than TE X's notation, then it's possible to make the off formulas not centered , but pressed to the left (see p. 159).

3. A set of matrices

First, we will explain how to type matrices with the amsmath package connected (which is better and more convenient in all respects), and at the end of this section, we will tell, for the sake of completeness, about those matrix typing tools that are available in "pure" LA TE X ( without connecting additional style packs).

So let's assume that the amsmath package is included. Then for a set of matrices enclosed in parentheses, it is worth using the pmatrix environment. Here's how it works:

Matrix rows are separated using the \\ command (you do not need to end the last line with the \\ command), and elements within the same row that belong to different columns are separated from each other using the & symbol. The text that corresponds to one line of the matrix on print does not have to fit into one line of the TE X file; in one line of a TE X file, you can put text corresponding to several lines of the matrix on print. In short, TE X's "end of line equals space" principle also applies in the matrix environment.

Rectangular tables of formulas are not only enclosed in parentheses; respectively, the environments bmatrix, vmatrix and Vmatrix are defined, which differ from pmatrix only in that instead of parentheses the table is enclosed in square brackets, respectively, vertical dashes | | and double vertical dashes k k. There is also the matrix environment, which prints only a rectangular table, without any parentheses. By combining the matrix environment with a couple of delimiters, you can get a more exotic looking bracketed matrix.

If you need matrices with more than ten columns, you need to change maximum amount columns by writing something like this in the preamble:

(After that, the maximum number of columns in the matrix will be twenty; in TE X'nic language, this action is called "assigning a new value to the MaxMatrixCols counter"; see chapter VII). You can also give this command not in the preamble, but at the beginning of the off formula, which includes your matrix; then the permission to increase the number of columns will be valid only for the matrices included in this off formula.

Here's how to type Pascal's triangle using the matrix environment:

The source text for it looks like this:

\setcounter(MaxMatrixCols)(20)

&&& 1 && 2 && 1\\ && 1 && 3 && 3 && 1\\

& 1 && 4 && 6 && 4 && 1\\ 1 && 5 && 10 && 10 && 5 && 1 \end(matrix)

(note, by the way, that in this example, the empty table elements at the end of the line are omitted, so the number of characters & in different lines of the table

miscellaneous). If we didn't increase MaxMatrixCols, then the last line would cause an error message.

To get a horizontal row of dots in a matrix that extends over several columns, use the \hdotsfor command; its required argument is the number of columns occupied by dots. In the example below, pay attention to the placement of & signs in the lines containing \hdotsfor:

You can also adjust the density of dots obtained using the \hdotsfor command: in the optional argument (it comes before the required one), you can specify decimal- dilution factor. If you say \hdotsfor(5) instead of \hdotsfor(5), then the dots will go one and a half times less often.

Along with horizontal rows of dots, vertical and diagonal dots have to be used in matrices. To set them, the \vdots and \ddots commands are used:

a 11a 12

a 21a 22

. . .. . .

a n1a n2

The \vdots and \ddots commands can be used not only in matrices, but also anywhere in mathematical formulas.

Along with the matrices used in offline formulas, sometimes you have to put a small matrix into the inline formula. Naturally, both the sizes of symbols and the intervals between them in such a matrix should be more modest. The smallmatrix environment is intended for such purposes (it also becomes available when the amsmath package is included). Here is an example of its use:

\end(smallmatrix)\bigr)$

As you can see, you have to put parentheses around such a small matrix yourself. The smallmatrix environment does not have any options with ready-made brackets.

Now, as we promised, we will tell you what options for a set of matrices remain if you do not connect additional packages. In this case it is necessary to use LA TE X's array environment. Here's how to get the example from p. 72:

Compared to what pmatrix gives, the differences are as follows:

1) Parentheses around a matrix typed using the array environment must always be set independently.

2) The \begin(array) that opens the environment must be followed (in curly braces, since it's an array environment argument) with a so-called matrix preamble describing how many and what columns the matrix should have. In our case, the preamble is three letters ccc. This means that the matrix has 3 columns (one letter per column), and that the contents of each of these columns must be centered in the column (c stands for "centered"). (In addition to c, the preamble can be either an l, meaning that the corresponding column will be left-aligned (left), or an r, meaning that the column will be right-aligned (right).)

IN the rest of the syntax is the same as for the pmatrix environment and its counterparts. Commands \ldots, \vdots and \ddots you can still use it, but \hdotsfor - alas, no. There is also no analogue of MaxMatrixCols for the array environment (since the preamble already determines the exact number of columns). Surroundings

The use of smallmatrix in "pure" LA TE X (without connecting additional packages) is also not provided.

4. One over the other

In this section, we will talk about those cases when it is necessary to place one symbol above the other in the formula. In sec. 1.2 has already dealt with a particular case of this problem: the setting of "limits" at the sign of the sum, integral, or something else of that kind. We will now consider the general case.

4.1. The simplest cases

To begin with, consider the following possibilities for placing one part of the formula on top of the other:

1) The top of the formula is a little above the line, the bottom is a little below (similar to the fraction created by the \frac command, but possibly without the slash).

2) The lower part of the formula is flush with the rest of the text, the upper part is above it.

3) A horizontal curly bracket is drawn above or below the formula fragment, and another formula fragment is located above or below this bracket.

Let's go through these options one by one.

Let's start with one addition about the \frac command described in the first chapter, which specifies fractions. If a fraction specified using the \frac command occurs in an inline formula, then its numerator and denominator are printed in a rather small font, which is not always acceptable. To avoid this, you can use the \dfrac command by including the amsmath package: then the font will be larger. If a fraction in an inline formula is included in the exponent or index, then sometimes it makes sense to specify it using the \tfrac command (again, so that the font is not too small; this command is also available when connecting amsmath). Here are some examples:

Now about how to arrange the parts of the formula "in the same way as in a fraction", but without a fractional line. There are two (unfortunately, mutually exclusive) ways to do this: with and without the amsmath package.

If you have the amsmath package included, you can achieve the desired effect using delimiters and the smallmatrix environment:

Of course, if you have a lot of such formulas in the text, it is unthinkable to use such long notations: you need to develop an abbreviated notation based on smallmatrix (read in Chapter VII how to define “macros with parameters”).

For the most common case of "binomial coefficients", when the delimiters are ordinary parentheses, the amsmath package provides a special \binom command that works similarly to \frac:

The \binom command also has counterparts \dbinom and \tbinom related to

To it in the same way that \dfrac and \tfrac are related to \frac.

IN amsmath also provides a "generalized fraction" construct for creating commands similar to \frac and \binom. By definition, a generalized fraction is a formula fragment arranged as follows: the left delimiter, then the fraction (the thickness of the fractional bar can be arbitrary, including zero), then the right delimiter. Recall that delimiters are brackets and similar symbols that can automatically change size (p. 67); in a generalized fraction, delimiters may not be present (so that an ordinary fraction is really a special case of a generalized one). To set a generalized fraction, the \genfrac command is provided with six arguments. To understand how it works, let's look at an example:

The first and second arguments to the \genfrac command are the left and right delimiters, respectively; the third argument is the thickness of the slash (if the thickness is zero, then the slash is not printed); the fourth argument specifies the font size for the numerator and denominator: if you leave it blank by writing just () instead of (0), then TEX will choose the size itself; the number 0 means that the size of the characters will be the same as when using the \dfrac command (in Section 5.2 you will learn that in TE X'nic terminology this is called displaystyle), the number 1 is the size as when using the \tfrac command (it same textstyle), the numbers 2 and 3 set even smaller sizes; finally, the fifth and sixth arguments are the numerator and denominator proper.

If you leave the third argument empty, writing just () instead of the curly braces that contain the thickness, then the default thickness of the solidi (it is 0.4 points) will be selected. If the first and second arguments are left empty, then there will be no delimiters (however, if a left delimiter is specified, then a right delimiter must also be specified). For example, \dfrac(x)(y) is the same as

\genfrac()()()(0)(x)(y)

In particular, our example with the Christoffel symbol can be written as

$\genfrac(\()(\))(0pt)()(ij)(k)$

Of course, the \genfrac command is not good on its own, but as a raw material for defining macros tailored to your specific needs.

Now about what to do if you do not include the amsmath package.

In this case, it is convenient to use the TE X command \atop:

IN this case we also used the \left and \right commands to set curly braces of the required size.

For binomial coefficients, there is the TE X \choose command:

(n\choose k)=\frac(n{k!(n-k)!}!}

Pay attention to the curly braces in which we enclosed the expression n\choose k: the \choose command places the part of the formula from the opening curly brace to \choose on top, and the part of the formula from \choose to the closing curly bracket below. If there were no curly braces,

the whole fraction n would go down! along with an equals sign.

The \atop command determines what goes up and what goes down, according to the same rules as \choose. In the example above with \atop, we did without curly braces, since in the mathematical formula their function is also performed by the \left and \right commands.

When the amsmath package is included, the \atop and \choose commands cannot be used.

An interesting use case for fractions is the so-called "continued fractions":

The result doesn't look good. In sec. 5 explains why it went so badly and how to fix it "manually", but in practice it's best to include the amsmath package and do this:

If you want some of the numerators in the continued fraction to be not centered, but turned off to the left or right, you need to say \cfrac[l] or \cfrac[r] instead of \cfrac, respectively.

Another case when it is necessary to print two formulas of the same size, one under the other, occurs when the expression for summation indices spans several lines. In this case, after including the amsmath package, use the \substack command:

The single argument to the \substack command contains formulas that must be under the sign of the sum (or product, or any other "limit operation"); strings are delimited by \\ (as in environments designed for a set of matrices).

Consider the case where the bottom of the formula must remain at the row level. To achieve this effect, LA TE X's \stackrel command is used. This command has two arguments: the first is what will be above the line, the second is what will remain in the line:

If the text to be written above the arrow is long, the \stackrel technique will give unsatisfactory results. In this case, by including the amsmath package, you need to use the \xleftarrow and \xrightarrow commands, which are specially designed for putting inscriptions above and below the arrows. In the mandatory argument of these commands, the inscription is placed above the arrow, in the optional argument - under the arrow (the optional argument, if any, is placed before the mandatory one). If the caption is long, the size of the arrow will automatically increase:

Finally, to draw a horizontal curly brace under the expression (and perhaps also make a signature under this brace), you need to use the \underbrace command. The argument of this command is the fragment of the formula under which you need to draw a bracket; the caption under the parenthesis, if needed, is formatted as a subscript. For example, such a formula

1 + 3 + 5 + 7 + . . . + (2n − 1) = n2

| (z ) n terms

is obtained as follows:

\underbrace(1+3+5+7+

\ldots+(2n-1))_(\mbox($n$ terms))=n^2

If you have the amsmath package included, it is wise to use the \text command instead of \mbox.

The horizontal curly brace above the formula fragment is generated by the \overbrace command, the inscription above it is formatted as a superscript. In one formula, there can be horizontal curly braces both above and below the formula fragment:

In our example, the bottom horizontal bracket was placed entirely inside the top horizontal bracket. You can also make sure that the upper and lower horizontal brackets do not contain one another, but overlap, but this requires additional tricks (p. 93).

4.2. Multi-line off formulas

TEX never hyphenates offline formulas, so if your formula doesn't fit on a line, you'll need to break it into separate lines yourself. The first thing that comes to mind for beginners is to arrange each of these lines as a separate off formula using $$...$$ and write these off formulas in a row. In this case, the vertical distance between the two lines turns out to be too large, so that by eye they do not

considered as part of the same formula. In this section, we describe how to correctly organize such a partition.

As in the case of matrices, the most convenient (and recommended by us) tools are opened by including the amsmath package; we will begin with their description, and at the end we will describe the modest tools for typing multiline formulas that are available without connecting additional packages.

So, let's say you have connected amsmath. Then the easiest way to set multi-line off formulas is the multline environment:

The first of the lines is printed turned off to the left, the last - turned off to the right, the rest of the lines are centered. Like the equation environment, the multline environment must not be enclosed in $$ signs. As you may have noticed, the formula, designed as a multline environment, is automatically numbered. To avoid this numbering, you need to use the "asterisk option" - the multline * environment.

In fact, the first and last lines are printed not close to the margins, but with an indent equal to \multlinegap. The value of this parameter can be changed in the usual way by writing in the preamble something like

\multlinegap=.5in

To make some of the middle rows not centered, but turned off to the left, you need to use the \shoveleft command, writing, say,

\shoveleft(+46+47+48+\ldots)\\

instead of +46+47+48+\ldots\\. For justification to the right, the \shoveright command is used in a similar way.

When there are several off formulas in a row, you can not format each of them with $$ or the equation environment, but use the gather environment:

When using gather, formulas must also not be enclosed in $$ symbols. Each of the formulas collected in gather is automatically numbered. In order for a formula so numbered to be referenced (otherwise, why number it?), it must be labeled by prefixing \\ with the \label command (see examples of labels and references in section 2.1; details in section IV.9 below) .

If some of them do not need to be numbered, put the command \notag immediately before \\. If you don't want to number any of the formulas, you can use the "asterisk option" - the gather* environment.

When breaking a formula into parts, it is often desirable to arrange the lines one below the other so that they are aligned in a certain way. To achieve this effect, it is convenient to use the split environment:

\begin(equation)

1999 = 1000 + 900 +

(5) 1999&=1000+900+{}\\

Breaking the formula into lines is still specified using \\ , and the & sign precedes the characters on which alignment is performed. For technical reasons, a formula split into lines using split cannot be specified using $$ signs (which is why we used the equation environment in the example). On the other hand, because of the equation, our formula got a number. If you do not need numbering, you can either write \notag before \end(equation), or use the equation* environment, which does not number formulas.

Split formulas can also be used inside gather or align environments (the latter will be discussed below), with or without asterisks.

It is often necessary to print one or more aligned columns of formulas. The align environment is intended for these purposes:

equality. In our example, the second & in the line separates the first column of formulas from the second, the third & is aligned in the second column, the fourth &, if there was one, would separate the second column from the third, etc. Still not needed $$ signs, each line of equations is automatically numbered, which can be suppressed by writing \notag before \\, and there is still an asterisk version of align* that does not number formulas.

With proper use of the align environment, there should be an odd number of & characters in the line. Namely, if we have n columns with equations, then there are n - 1 signs & separating the columns from each other, plus n more signs - one for each column, and in total (n - 1) + n = 2n - 1.

A useful use of align occurs when successive off formulas contain text comments. It is desirable that these comments be aligned. Here's how you can achieve this with align:

Notice the two ampersands separating the comment from the formulas (see the text in small print above). It is also worth noting that, as with the multline and gather environments, formulas specified with align cannot be formatted with dollar signs.

It is not always convenient to include comments on calculations directly in formulas. Sometimes you want some of the comments to go on a separate line. The \intertext command allows you to do this so that the alignment is not broken:

Along with the align environment, which gives an entire off formula at once, there is an aligned environment that can be used as part of a larger formula. Here's how you can use this environment to set up a system of equations:

To create a system-wide curly brace, we used the \left and \right commands, and the \right command has an “empty delimiter” - a dot (see Section 2.5).

Finally, another type of multi-line off formula occurs when the expression on the right side of an equality should look different in different cases. For this case, the amsmath package provides the cases environment. Let's demonstrate how it works with an example:

|x|=\begin(cases) x,&\text(if $x>0$;)\\ 0,&\text(if $x=0$;)\\ -x,&\text(if $ x<0$.} \end{cases}

Now that you have familiarized yourself with the possibilities of typing multiline formulas using the amsmath package, let's talk about what can be done in this direction without connecting additional style packs.

Systems of equations can be typed using the array environment like this:

x^2+y^2&=&7\\ x+y & = &3.\\

We assigned one column to the left side of each equation, to the equals sign, and to the right side. In doing so, we asked that the left-hand sides of the equations be right-aligned (hence the r in the preamble), the right-hand sides

aligned to the left (l in the preamble), and the equal sign was centered in its column (hence the second letter in the preamble is the letter c).

You can notice that the spaces (spaces) before and after the equal sign are larger than allowed by typographic rules (and than is obtained when using the aligned environment from the amsmath package). Unfortunately, this is difficult to deal with; it's easier to get a kit that includes the amsmath package.

If you want individual equations in the system to be numbered, you can use the eqnarray environment. It works in the same way as the array environment with the rcl preamble in the above example, but it automatically prints its number for each equation (similar to how the number is automatically printed for a turn-off formula created using the equation environment - see Section 2.1). If you mark any equation with the \label command, then you can refer to it later using the \ref or \pageref command. Example:

Note that the eqnarray environment does not create a curly brace that encloses a system of equations. In this example, the ~ character between "s."

And \pageref is set so that the word "with." and the page number did not fall on different lines (see p. 103); for similar purposes we used this symbol

and secondarily.

When using the eqnarray environment, you do not need to write $$ signs (just as you do not need to write them when using the equation environment).

If you do not want to number all equations, you need to mark the equations that you will not number with the \nonumber command (immediately before the \\):

Z ∞ e−x 2 dx =√ π

−∞ √

\begin(eqnarray) \int_(-\infty)^\infty e^(-x^2)dx & = & \sqrt(\pi)\nonumber\\

(10) \sqrt(576) & = & 24 \end(eqnarray)

Finally, if you don't want to number the equations at all, you can use the "star version" of the eqnarray* environment.

The array environment can be used not only in offline formulas, but also in inline formulas, although the result usually looks ugly. The eqnarray and eqnarray* environments create only off formulas.

You can also use the eqnarray or eqnarray* environment to break the off formula into multiple aligned parts:

Note that we preceded the first + in the second line of the formula with a pair of opening and closing curly braces; this is done so that the + sign does not come too close to the first character of the second line in print, which, combined with increased spacing around the equal sign, would be too much (you can experiment on your own). The nature of the described effect is explained below in Sec. 5; it is partially taken into account in the amsmath package (unfortunately, different versions of this package may give different results).

4.3. Set of commutative diagrams

To type "commutative diagrams" in LA TE X, you need to include the amscd style pack. Let it be done. Then the commutative diagram takes the form of a CD environment. For a reader familiar with AM S-TE X, what follows can be explained in one sentence: between \begin(CD) and \end(CD) you must place exactly the same text that is written in AM S-TE X in a similar case between \CD and \endCD (see ). For everyone else, it is more convenient to explain the rules for a set of commutative diagrams by an example. Consider the following diagram:

With the amscd package connected, it is typed as follows:

0 @>>> E’ @>f>> E @>g>> E’’ @>>> 0\\

@. @VVpV @VVqV @VVrV @.\\

0 @>>> F' @>f>> F @>g>> F'' @>>> 0 \end(CD)

The first row in this entry corresponds to the top row of the chart. An arrow pointing from left to right is specified by the @>>> construct (and an arrow from right to left is specified by the @ construct).<<<); если над стрелкой надо поставить какую-то надпись (например, просто букву), то нужно ее разместить между первым и вторым знаками неравенства; чтобы надпись

turned out under the arrow, it is necessary to place it between the second and third signs of inequality.

The second line defines the vertical arrows. The @VVV construct specifies a downward arrow; if an inscription is needed to the right of the arrow, then it must be placed between the second and third letters V (in order for the inscription to be to the left of the arrow, it must, of course, be between the first and second letters V). The vertical arrow pointing up is specified by the @AAA construct (the letter A is the maximum approximation to the arrow pointing up); to the right and left of it, you can also make an inscription (in a similar way).

The @ construct. sets an "empty" arrow (in our case - between two zeros); it is necessary so that LA TEX does not lose count when figuring out which columns to put vertical arrows in.

Let's describe the work of the CD environment more accurately. Each commutative diagram is treated by the CD environment as a table consisting of alternating "horizontal" and "vertical" rows. Each "horizontal" line consists of formulas interspersed with horizontal arrows. All horizontal rows must have the same number of formulas. If some of the places intended for formulas should be left blank, then leave a space in this place or, if you prefer, write (). There must be an arrow between each pair of formulas. If any of these arrows are not needed, @ should be put in their place. ("empty" arrow).

Each "vertical" line consists of vertical arrows. There should be as many of them as there are formulas in any of the horizontal lines. If some of the vertical arrows are not needed, put @ in their place. (empty arrow).

If the inscription with the arrow pointing down (and, therefore, given by the @VVV construct) itself contains the letter V, then you need to put it (the inscription) in curly brackets - otherwise TEX will not be able to understand which of the letters V refers to the inscription, and which - to the designation of the arrow. Similar measures must be taken if the inscription with the arrow pointing up contains the letter A (and also, of course, if the inscription with a horizontal arrow contains the sign > or<, хотя ввиду математического смысла таких надписей последнее менее вероятно).

Along with arrows, horizontal and vertical "stretched equal signs" occur in commutative diagrams:

As you can see from this example, such signs are given by @= (horizontal) and @| (vertical). Notice also how we've bracketed the V in the caption for the left vertical arrow.

The \pretend construct. . . The \haswidth of the AM STE X system (see the book) is not supported in LA TE X.

Mathematicians know that in commutative diagrams there can be not only horizontal and vertical arrows: there are also slanted, curved, and dotted ones. . . The capabilities of the amscd package to print such arrows are not enough; if you need such more complex diagrams, you should use the XY -pic style package (see appendix E).

In "pure" (without connecting style packs) LA TE X, a set of diagrams is not provided. In the most extreme case, if there is neither amscd nor XY -pic, you can do this:

\begin(array)(ccccccccc) 0&\longrightarrow & E' & \stackrel(f)(\longrightarrow)& E & \stackrel(g)(\longrightarrow) & E'' & \longrightarrow & 0\\ &&\downarrow \lefteqn(p)&&\downarrow

\lefteqn(q)&&\downarrow\lefteqn(r)\\ 0&\longrightarrow & F' & \stackrel(f)(\longrightarrow)& F & \stackrel(g)(\longrightarrow) & F'' & \longrightarrow &0

The result is almost the same diagram as in our first example (although the vertical arrow letters will be larger than the horizontal ones, because the \stackrel command makes the letters smaller). The only thing that needs clarification here is the \lefteqn commands. They are needed so that the vertical arrows with inscriptions are correctly centered. If these \lefteqns are omitted (and write p instead of \lefteqn(p), etc.), then the vertical arrows with captions will not be centered, but shifted to the left.