Tip 49. Formula-incognito

Fig. 49. Formula-incognito

When we watch the pictures of arrest of the offenders on TV we can see that as rule the faces of the official staff are closed by black boxes, which move within the screen synchronously with the motion of the policemen[1]. It is possible to lay the same boxes on the operators of Mathcad-document if the author does not want to open his secrets (see the fig. 49, where the mechanism of the forming of the matrix , which elements are the centres of the surface[2] net, is black-out). Similar boxes are laid by the change of color of background of the formulas[3] from white color to black one.

This tip was written for the seventh version of Mathcad, where it is impossible to slam the singled out areas of the document (see tip 50 and 62). But in eighth and later versions (in these versions it is possible to slam an area) the tip can be useful: it is possible not to slam the area with the operators-incognito but to protect from editing. This way we intrigue future user, for example, if we show that enough difficult calculation realizes by few number of operators. (As a matter of fact user can change the color of the variables and the constants of the whole document from black to white and he tears the mask of the formulas-incognito.)

Tip 50. Protection of the areas of Mathcad-document

Fig. 50. Protection of areas of Mathcad-document

It is worth to protect the areas of the Mathcad-document, which are not meant for editing by the author or the other users, by the password (see the fig. 50, where we slam the prepared upper part of the document with the formulas -incognito see the tip 49). The problem of use of any password is that we can forget it. The author knows how to unlock the password in the Mathcad-document however instead of telling this secrete he suggests one more tip.

As we beware of attracting attention not quite honest people the author asks people who have had the password but forgotten it to send the lock document to the address ochkov@twt.mpei.ac.ru, where it will be unlocked and forwarded. Payment for the service is new tip in the piggy bank of the author. The password (password) is inputted by the general technology (see the dialogue window Lock Area in the fig. 50): user inputs a password (here the password is two symbols, which do not print but change to the stars in order to avoid the risk that somebody could spies the password from the shoulder of the documents master[4]); then it is necessary to duplicate the password in the field Reenter password. It is possible to slam the area at the same time of input the password (see the tick in the field Collapse when locked). The area, which you have slammed, becomes the line (see the beginning of the fig. 62) where you can write the date of hiding the area (see the tick in the field Show lock timestamp). But it is possible to remove this line, inscriptions on it and the other tracks at all that they have not been seen on the screen of the display. Tip without number: if you look through a new unknown Mathcad-document it is worth to switch over to conditions of color extraction of the operators (the switch Regions in menu View see the fig. 1). In this case all invisible and slammed areas will be visible (they will be singled out by white color on the grey background).

Tip 51. Dimensionalities and symbolic mathematics

Fig. 51. Dimensionalities and symbolic mathematics

When reader familiarized with the tips 16 and 17 he made sure that the system of physical and the other values, which is built-in in Mathcad, allows to call this package not only mathematical one but also physical and mathematical package. Unfortunately units do not work in symbolic mathematics of Mathcad: as since it does not know , that a meter has hundred centimeters, an hour has 60 minutes and etc. But still it is possible to solve the problems analytically with dimensional values in Mathcad. At that the variables, which mean some built-in or users[5] units will behave as general (ordinary) variables in the expressions where you apply one or another operations of symbolic mathematics (simplification, root search, differentiation, integration and etc.).

For example, here is the solution of a little modified famous problem from the Chehov story Coach, which is solved by tools of symbolic mathematics of Mathcad. A merchant bought 138 arshines of black and blue cloth and he paid for it 250 thousand old rubles, 50 new (denominate[6]) rubles and 12 dollars. It is asked how many arshines did he buy of one and another cloths if the blue one cost 25 euro for one arshine but the black cloth cost 3 rubles?

In the fig. 51 the problem is solved by two symbolic methods using the function Find and using the operator solve. In this case units are simple general variables, which along with the other variables take part in symbolic (analytical) transformations.

Tip 52. Separate the variables of the program

Fig. 52. Separate the variables of the program

In the fig. 52 there is soft created users function, which returns density (r) of some matter () depending on temperature () and pressure (p). Two-dimensional spline interpolation is laid in the calculation: the kernel of the program is tabular values of the density (kg/m3) depending on discrete values of the temperature (50, 100, 150, 200 and 300 degrees by the Celsius scale) and on the pressure (1, 3, 4, 5 and 6 physical atmospheres)[7].

We try to obtain from the program not one but a few tips.

Tip 1

Two-dimensional spline interpolation, laid in Mathcad, has in mind that tabular values of the functions have to be kept in a separate square matrix, but their corresponding arguments values have to be kept in a rectangular matrix, which has two columns: the first column keeps discrete values of the first argument, but the second column keeps discrete values of the second one. But we choose more visual method: we note all figures in the matrix . At that the stub of the matrix (without upper element) keeps the value of the first argument, but the headline (without left element) keeps the value of the second argument. We write the commentary in the upper left corner of the matrix (in nine, if we use the language of footballers) for free space was not empty. The rest of cells of the matrix keep the values of the function (in our case it is the values of matters density).

The notation in the tables with the title and stub that we choose for initial data allows to edit them easily or simply to use for the inquiry: you look at the table-matrix and see at once that, for example, if t = 200  and p = 5 atm then the density of our matter is 365 kg/m3. Before the interpolation the matrix is laid out two independent matrixes with the help of built-in Mathcad-function submatrix (with two columns, which keep the vectors of the arguments) and V (square matrix, which keep the tabular values for the function).

If user wants to increase the speed of call of the function r, then it is worth to refuse the matrix and he have to input tabular values in the matrix at once (the name of this matrix reflects its structure: the first column is the discrete values of the temperature, the second one is the discrete values of the pressure) and V (density)[8].

Tip 2

The method, which is considered in this tip, is present at all tips of the book where we have to refer to the numbers of lines and columns of the matrix (see also tip 70 and others). So it is never worth to rely on the concrete value of the system variable ORIGIN (generally it is 0 or 1). It is better to make a link to this variable in the program. For example, in our function it is possible to write the following way:

<0> submatrix(, 1, 5, 0, 0)

But not:

<ORIGIN> submatrix(, ORIGIN+1, rows(M), ORIGIN, ORIGIN)

As a matter of fact this simplicity (see the first operator: <0>) can turn out worse then stealing. If our simplified function is found in Mathcad-document, where the default of the variable ORIGIN is broken (it may be equal not zero but some another whole number), then the bug will appear. The mistake will appear if we edit the matrix if we input new data in it.

Tip 3

Mathcad gives unique opportunity to change the type (Times, Arial, Courier and others), the size (8, 10, 12 points and etc.) and the style (bold, italic, underline) of the print of the variables, functions, and constants. It is impossible not to use this possibility. In our program the variables and the functions are divided by groups, which are united by the structure of data. Using the principle of continuous of the form and the contents we can understand the features of programming in Mathcad:

        predefined variables and built-in functions have bold type Arial Cyr size of 12 points (here it is possible to pick out the separate group of built-in and users units of physical values);

        local variables of a program as well type is Arial Cyr, but it is underline and less in size 10 points;

        users functions and variables, which are visible in whole Mathcad-document normal type (Variables);

        formal variables (arguments of the functions T and p) italic;

        as well numerical and textual constants can have different type (Constants and Small constant), which underlines the different groups of the constants in the matrix of table.

It is marked in the fig. 52 that the different groups of variables are attached to different styles of variables, which are given the different names[9]: Local variables, System variables and etc.

We can change , color of the type of names of the variables, the functions and the constants. This theme will be considered in the sketch Color in programs (http://twt.mpei.ac.ru/ochkov/Color_in_Program/index.htm).

Tip 4

The advantage of users function in the fig. 52 in comparison to its analogue, which is written in the environments of the traditional programming languages, is that our function has dimensional arguments (temperature and pressure) and returns dimensional value (density see tip 16, 17, 21, 45 and others). Unfortunately arrays of Mathcad can not keep the elements with different dimension. The way out may be the following: at the begging of the program we have to deprive the arguments (practically parameters) the dimension, reduced them to that non-dimensional value, which is used in the stub and title of the table, but at the end of the program we have to add the dimension of the tabular data to the outputting value of the function.

Tip 5

In the tips of the firm Mathsoft, Inc.(Tips of day see point 74) partial solution of the problem of use degrees Celsius in the calculations, which are not built-in in Mathcad by means definition of users function is shown. In our case you define the function with the name too, which is called as the postfix operator (150 ). This function converts non-dimensional degrees Celsius to the dimensional Kelvin (this method is more detailed described in tip 64). In the calculation we have two definitions of the function point is that they are different functions (the main point of the tip): in one function Latin letter is written, but in the other function Russian letter is written. It is done for user, that he has not puzzled over on what case to work during solution the problem and has not got stuck on unintelligible mistakes when instead of Russian letter is inputted Latin one and vice versa. This curious thing we very often meet with the letter , as since Latin and Russian variants are on the same key of the keyboard.

Tip 6

It is worth to fill empty space in a matrix up a textual constant, which explains its contents (see the left upper corner of the matrix M). It is possible to do without any mistakes as since numerical constants are non-dimensional in the matrix.

Tip 7

If the initial matrix of tabular data, on which sline-interpolation has to be conducted, is not squarte one but it is the rectangular one, then it is possible to recommend to divide this matrix by two cross squarte ones. So we can conduct interpolation using one of the matrixes depending on the concrete value of the long argument the argument that is fixed on the long side of the rectangle.

Tips may be more, but seven is beautiful number

Tip 53. Method of step-by step approximations

Fig. 53. Method of step-by-step approximations

Well-known problem about the evolution of an epidemic[10] is solved in the fig. 53: in a town, where 20 thousand people live (Healthy), unknown number of sick people appear (Sick), it causes the epidemic, which the simplest model is described by two formulas. It is asked, how many sick people were on the first day if there were 100 sick people on the thirteenth day.

The problem is solved by the method of step-by-step approximations: we set the number of sick people on the first day (this operator is the last in the document in the fig. 53[11]) and we keep a look out the variable Sick13, which value prompt to user how to change the variable Sick1 in the next approximation. We can fix the history of the approximations in the matrix, which is reflected at the end of the fig. 53. For it we have to write or copy manually the number triples, which are got at the each step of approximation to the solution, when the deflection (the last line of the matrix in the fig. 53) becomes equal zero (or rather nearly zero).

(Tip without number. The elements of the matrix can be reflected graphically, if we visualize step-by-step approximations). Continuation of the theme you could see in tip 91.

Tip 54. Touch up the names of the variables

Fig. 54. Touch up the names of the variables

As rule the process of creation of Mathcad-document breaks into two stages. At the beginning some draft of the document without the commentaries and with short names of users variables and functions is made. The main purpose of this stage is the input of the formulas and the debugging of the operators. At this moment user thinks not enough about that it is important not only the momentary right result but also the openness for optimization of the program, the study and the perfection by the author and/or the other people in the Mathcad-document.

At the second stage we bring in the commentaries, format numbers and graphs, change short (dumb) names of the variables and the functions to long (speaker) ones in the Mathcad-document. For this change it is useful to use the command Replace from the menu Edit. At that it is possible to bring in the symbols, which are not available at the direct print of the names in the names of the variables and of the functions at the same time. This replacement we can consider as the third way of writing of the non-standard names another two ways are described in tip 14 and 18. The creation of the variable with the inferior (bt) and upper (apostrophe, accent) indexes is reflected in the fig. 54. (The variables with the accent are widely used in the engineering calculations. For example, t' is the in temperature, but t" is the out temperature[12].) It is impossible to obtain the accent at the direct input of the variable as since the corresponding keystroke results in the rise of the inverted commas.

Here are some tips without number, ensuing from the foregoing:

1. It is worth to write manually (on letters ) the name of the users variable or function only one time. When you call this variable (the function) next times it is worth to copy it but not to print again. The method is especially useful when the name is long and/or includes special symbols, which are not available for manually print. More over it is possible to mix up Russian and Latin letters, which are similar by way of writing: c, a, and etc.

2. When you copy the functions it is useful to hold not only its name but also its arguments, then you can touch up them, change the formal variable, which are framed by the brackets and are shared by the commas, to the real variables or the constants (about the formal variables you could see in tip 83).

3. It is necessary to make an effort to input at once minimum commentaries in the Mathcad-document and not to put off this work later on[13]. Very often this later on is absent: the program has worked and given more or less acceptable result. But alas the program is closed for study and the subsequent processing as since it did not obtain good commentaries in due time.

4. It is not unreasonable to write the extensional Russian-language commentaries in the environment of some textual editor (in Word, for example) and only then to transfer them in the Mathcad-document, as since Mathcad has tolls for only English spelling check[14]. Apart in Word it is possible to check hand in hand with spelling syntax and punctuation, to sort out the synonyms for some words and much more that only real textual editor allows to do. It is possible to inculcate Word in Mathcad (or vice versa) for we do not carry the textual parts here and there (from Mathcad in Word and back).

5. The spelling mistakes are possible but they are desired in the names of users variables and functions. Thats why it will not be unnecessary to write the name in Word at the beginning, and then to carry it in Mathcad.

6. It is worth to input the built-in functions but the call of master function but not on letters print of their names. It will exclude some mistakes of Russian-language user (you input intersept instead of intercept, for example).

Tip 55. How to create shareware-programs in Mathcad

Fig. 55. How to create shareware programs in Mathcad

Shareware[15] is so-called free program that is speared under the motto: We have tried if you liked it and buy! The similar programs are available via Internet (the site www.download.ru, for example) or via D-ROM that are enclosed to different computer magazines (ComputerPress, Game.exe, Games Country and the other). As rule, the work with the similar program is connected with some limitations, which induce user to connect with the author for to register or to get full version of the program or the information of liberalization. Here you can see not only mercantile interests (see the motto above) but also the other reasons. For example the author, when he lays some shackles on the program[16], can lead the database of users, attract them for testing and for perfection of the program: Tried, liked, then register (praise, criticize, work together with me and so on and etc.)!

The example of shareware, close to reader is freeware Mathcad Explorer, which you can at the sites of Internet[17]. This program allows to work with Mathcad-documents but with some limitations: for example it is impossible to save on the drive the modified file[18] for it you have to buy the commercial version of Mathcad.

One of possible technologies of creation of shareware-Mathcad-document is reflected in the fig. 55. With the help of it our old problem about the cones volume is solved (see tip 35, tip 36 and tip 37). It clear that we can solve the other, more difficult program which solution is the intellectual property of the author that he want to protect via the mechanism shareware.

In the point 1 in the fig. 55 the variable Attention![19] has the information, which the author of the program here and further pushes under the nose of future user. This way the author reminds user that he works with the given program as if on credit . Then user is suggested to input initial data of the calculation geometrical sizes of the cone, which volume is calculated by the formulas in the frames (see also tip 62). To distribute ready program it is necessary to slam these frames with the input of the password (see tip 66). But before it we have to put down the calculated cones volume some sly method in the out parameter V: in 70 cases from 100 the variable V keeps the write result and in the other cases it keeps the contents of the variable Attention!

Here are the defects of the shareware-program that is showed in the point 1 in the fig. 55:

These defects are removed in the point 2 in the fig. 55. In the first place no it is possible to work with dimensional values owing to that the message of the author is not already textual constant as we have seen in the point 1, but it is users error message: in the similar case the recreated function Volume of the cone returns the right result in 70% of calls but the error message in 30% calls. In he second place our shareware becomes the general software after the author will tell user his registration number. In this case the variables Name and SerialNumber keep something different from that was written in the point 2 in the fig. 55: for example NN and 00000000. After that user will try the program and will make sure that the program returns right and necessary result (as before the calculation is hidden and protected by the password), he deals with the author and gets his registration number, which is generated by the function[20] in the point 3 fig. 55. When you input user name (for example, it was NN and becomes Orlov K.) and registration number (it was 0000000 and becomes 6796937) the program stops to remind that he work with the evaluation version. If the author wishes he is able to send by mail (or email), to let know by the telephone or to hand over at the private meeting the password as well, with the help of which the calculation was closed: to sell user not only the results of the calculation for different initial data but also the calculation. Registration number and password can be encoded in addition before sending.

The developer of the program can change the power (number 3) to another one in the function of the point 3. It is acceptably to change the algorithm of generation of the passwords to some especially secret one. The main point here is that the unique number is generated by the name (it may be fractional number: 135.834, for example).

Attention! The mechanism of creation of the trial function, which is showed in the fig. 55, is broken if at the beginning of the document we write Mathcad-operator rnd(x) := 0. In Mathcad, as well known (see tip 13) the built-in functions are defined and we can use it. It is said that it is possible to break any protection. But the protection, which described in this tip are not the protection per se, but the mechanism for reminding the user about his guaranties before the author. Any protection is meant for honest people, who come, for example, to closed door and pull it out they understand that the way is closed. A swindler begins to break open this door or even to beat it out and he does not guess that there is another open door and the way through it is not always longer one but in some cases is even shorter.

Tip 56. Speeding-up the work with three-dimensional graphics[21]

Fig. 56. Speeding-up the work with three-dimensional graphics

One of the defects of the work with three-dimensional graphics of Mathcad is too difficult to format the graph if we have large quantity of points: to turn, to change the colors and etc. While user wait for next repainting of the graph he asks himself not once: Why have I decided to do it?

This problem can be solved a few methods. One of these methods is not use three-dimensional graphics in Mathcad, but the application the other special-purpose programs for it such as Axum, for example. But the discussion of this method exceeds the bounds of the given book, thats why well not consider it. We try to solve the problem within the environment Mathcad. The main point of the method for decrease of the dormancy of three-dimensional graphics is the dynamic decrease of the number of displayed points. It may be done by two ways. In the first place it is acceptably to decrease the number of calculated points. However at that user can not always guess beforehand the necessary size of changing of the variables and their changing step: it may turn out that the built graph will not have all necessary information. As a result we have to calculate whole the matrix[22] anew. But if the calculation takes plenty of time, then it may be simpler not to touch the matrix but to format the graph at once[23]. It is possible to overcome this defect if we use the second method, which is suggested here as the tip.

The main point of the tip is that we construct three-dimensional graphs not of the initial matrix, but of the doubling one, in which we bring in only the part of points of the initial matrix. The illustration of the mechanism of construction is showed in the fig. 56. For dynamic changing of the power of the detail (of the quantity of points in the matrix, which is used for construction of the surface) is brought in the variable Detail. Then we bring in new index variables, we calculate the convert coefficient (the variable with the name k) and finally we fill the second matrix by the elements from the first one with the given detail. As a result three-dimensional graph of not the whole initial matrix but only of its part is constructed.

The advantage of the given method is that the above-mentioned graph reflects the whale size of changing of the variables, but with the large changing step. Using the choice of the value of the detail coefficient it is possible to reach that three-dimensional graph will quickly react to change its format. After it we can easily adapt its appearance and only then we assign the value 100% to the variable Detail (or place the reflection of the initial matrix). After it our three-dimension graph will look like we have thought.

Tip without number. Very often the difficult three-dimensional graphics applies the brake appreciably to the process of calculation by the operators, which are kept in the Mathcad-document. Quiet often it prevents the review, scrolling of the document. In this case it is possible to recommend to turn off the process of the recalculation and the repainting of the graphs for a certain time with the help of the switch Disabled Calculation. The more cardinal method of the shutdown of the graphics is the change it to the picture. For it the mold of the screen with the graphics is transferred to the environment of some graphical editor, then we finish off it there and new but already stark form is inserted in the Mathcad-document. (The principle of chopping of Gordian knot see the sketch History of one masterpiece (http://twt.mpei.ac.ru/ochkov/Lace/Lace.htm). There compromise method for review the unusual three-dimensional graphics of Mathcad is suggested: at first we are suggested to look at small picture, then we see enlarged picture on whole display, and only then viewer can decide if it is worth to download the corresponding Mathcad-document and view the masterpiece in all details to turn the volumetric structure, to touch its settings via the window of format and etc.

Tip 57. Work with the files on the drive[24]

Fig. 57. Work with the files on the drive

When you work with data files in Mathcad sometimes we have the necessity to check up if the necessary file already exists, if it is necessary to recopy it (it is done by the call of the built-in function WRITEPRN) or add it (APPENDPRN). The given problem is solved in the programming languages: if user would like to add non-existent file, it automatically creates as new one. The function APPENDPRN does not have the similar properties: the error message will appear if you want to add non-existent file.

The given problem can be solved using the different ways. But the simplest method is to address to the firm MathSoft with the request of making over the function APPENDPRN or of creation new one of similar nature but with the possibility of the creation of the file in the absence thereof. The second ay is to write the language the function, which either would change the function APPENDPRN or simply would check up the existence of the file on the drive. However the considered problem can be solved by means of built-in possibilities of Mathcad: via the call of the function READPRN with use of the operator on error and the additional array of two elements, which keep the file names.

The fragment of the Mathcad-document is showed in the fig. 57, which is illustrated the suggested tip. The main point of the method is that we execute the attempt of reading the data from the file before saving it. If we obtain the beneficial result (if the file with necessary format of data exists) then the variable, which keeps the file name for addition (the call of the function APPENDPRN) is assigned the initial file name, but the variable, which keeps the file name for creation of new file ( WRITEPRN) is assigned empty line. Otherwise (if the file does not exist on the drive) the above-mentioned operations are made in the reverse order. The control of right reading is made with the help of the operator on error, which executes the operations of the left operand in case if the execution of the operation of the right operand has been the cause of the error.

If the file does not exist on the drive then the function WRITEPRN will be called and the following calls will be of the function APPENDPRN[25].

The defect of this method is that every time the function of reading data from the file READPRN will be called. It may slow down very much the process of calculations if the file has large quantity of points.

Tip without number. It is possible to use the functions for saving the files on the drive for the transfer the value of the variable from the lower part of the Mathcad-document to upper one. It is necessary, for example, for realization of the step-by-step approximation method (see tip 53).

Tip 58. The function if and the operator if

Fig. 58. The function if and the operator if

Generally users of Mathcad for realization of the alternatives out of the program use the built-in function if from Master function, but inside the program the use the operator if from the programming panel. But it is possible to recommend to use the operator if out of the program too. (About the inclusion of the function if in the programs see tip 40.)

The users function, which returns the sum of income tax[26] depending on the sum of unsavoury income[27], is formed in the fig. 58. In the point 1 the function Tax is formed owing to multiple enclosure of the built-in function if. At that the function Tax becomes very long and difficult for reading and editing we have to write this function manually in the column[28] in the fig. 58. In the point 2 the function Tax is formed by the operators if and otherwise. The third variant of the program is perfected in the following directions:

        units of cost are brought in (see tip 18);

        work of the function is hastened owing to use of the operator return;

        the operator otherwise is removed as since all variants are sorted out and there is only single one.


[1] It is software-based method of the realization of incognito; there is the hardware-based method the sack with the cuts for eyes.

[2] We give the surface out to be colored Cartesian graph see the tip 23.

[3] By default the color of the formulas is black, but the color of background is white: we write in black and white.

[4] One of the ways to unlock others computers is the sending of page: the false dialogue window of the password inquiry. User who inputs his password thinks that he has mistaken and retypes the password. However the first inquiry does not disappear but falls into crackers (hackers) hands.

[5] Built-in or users variable variables are in respect to calculus mathematics of Mathcad. As we have said above symbolic mathematics of Mathcad does not have units.

[6] For use of visual methods the modified problem of .P. Chehov is attached to 1998, when there were both old and new (denominate) rubles.

[7] Among built-in units of Mathcad there are physical atmospheres (atm, or 760 torr) but there are not technical ones. But the technical atmosphere can be defined: at:= 1 kgf/cm2.

[8] Capital Greek letter P () coincides with way of writing of Latin letter . It may carry in some confusion in reading and in creation of the Mathcad-document from a sheet.

[9] It is possible to give the same names of two users styles in Mathcad. It may reduce to different kind of curious things.

[10] This classical problem wanders from one to another version of Mathcad and illustrates the features of work with the given package. The author of this problem is John Truxal and it is marked in the documentation of Mathcad. Nobody has been awarded such respect.

[11] Here the global assignment operator Sickº g is used see tip 10. At the expense of it the answer and new approximation are pulled together.

[12] In Mathcad 2000 it has been possible to mark the derivative of function by the accents (see the sketch It is no trouble at all for us to build a Bridge http://twt.mpei.ac.ru/ochkov/Bridge/Bridge.htm).

[13] Do no put off tomorrow that is possible to do the day after tomorrow.

[14] In Mathcad 2000 it has been possible to check spelling using three dictionaries: English and two American ones (optimise or optimize).

[15] Shareware is translated as part of the goods , part of program . But it is impossible to translate this neology word for word lets remember, for example, the terms software and hardware, which we can not translate word for word into Russian language.

[16] This way a horse is hobbled, for it can pasture but can not run far away.

[17] See for example www.mathsoft.com, www.softlin.ru and the other.

[18] Similar programs have the following prefix read-only. The way out: it is possible to save the created or rather modified programs at the site Collaboratory (see its description in the sketch Mathcad and Internet, or Network collective farm http://twt.mpei.ac.ru/ochkov/Collab/Collab.htm), if we mask them as new message. It is impossible to create new document in Mathcad Explorer but it is possible to take somebody elses one and to calculate your problem.

[19]Here ! is not the symbol of the factorial operator but it is simply the exclamation mark (see tip 23).

[20] The function sums the cubes of the ASCII-number of the symbols, which are come in the name of users. The author of shareware can develop and keep in secret his personal algorithm of generation of the registration number.

[21] The author of the tip 56 is K. Orlov

[22] Here we consider the old (till Mathcad 2000), matric method of construction of three-dimensional graph (see tip 34).

[23] We meet with this problem when we construct the surfaces by the functions of properties of water and water steam (http://twt.mpei.ac.ru/orlov/watersteampro/ru/index.htm). At that the calculation of the point takes whole Saturday, Sunday and Monday certainly we do not turn off the computer.

[24] The author of the tip 57 is K. Orlov

[25] As a matter of fact both functions are called, but in the first case the function APPENDPRN does not work as since its name is empty space, but in the second case the function WRITEPRN does not work.

[26] At the moment of publication of this book the function has become out of date united tax rate of 13% has been brought in.

[27] The main point of the term unsavoury money has changed a little in our time. In the old days money were considered unsavoury money if the tax had not been paid from them (gross, tare, net).

[28] In Mathcad we can write only the operator of addition in the column. For it the chord Ctrl+Enter is used.