Tip 1 Do not use assignment operator  The Mathcad software package is frequently called a super-calculator because it displays a numeric answer for almost any expression containing constants, variables, operators and functions. To obtain the result you need only type �=� after the expression:

125 + a2 � sin (b) = 2.753

Of course, the variables included in this expression (a and b) should be defined as numeric values beforehand. An assignment (input) statement in the Mathcad environment is shown as �:=�, and is often typed with the �:� (colon) key. Mathcad conveniently inserts the second symbol �=� itself, giving:

a: =

However, it is better to use the �=� key, rather than the:� key, when assigning a value to a variable in a Mathcad worksheet. If the variable has not yet been defined, then Mathcad will automatically convert the �=� operator (output) to the �:=� operator (input). If the variable has been defined, then its numeric value is displayed. By using the �=� key, it is possible to determine whether or not the given variable is free ; i.e. neither defined by the user nor used internally by Mathcad. This provides security against a number of potential mistakes. First, a system variable can be overwritten, for example by typing �: = 5, m: = 1, A: = 2 ( � is the base of the natural logarithm, m and A are units of length and current, respectively). Secondly, one may forget that the variable has been already assigned, and assign a new value to it. It is usually not desirable for a variable to have one value in one part of a Mathcad-worksheet and another in a later second part.

You can turn off the automatic replacement mode, when the operator �=� is replaced by �:=� (this hybrid of two operators is called a Smart Operator), by choosing �Options� from the �Math� menu, and selecting the �display� tab� (see appendix 1).

Note. The figure shows a Mathcad-worksheet in a mode when separate operators are displayed on a white background, while free space in the document is gray. From this point on this mode will be used to display Mathcad-worksheets if it can help clarify the tip.

Tip 2 Symbolical output instead of numerical  Sometimes it is better to use the operator (symbolic value output) instead of the �=� operator (numeric value output). Figure 2 illustrates three reasons for choosing symbolic output:

Example 1: the �=� operator displays up to 15 decimal places in the mantissa, whereas the operator displays up to 250 decimal places. (Numeric value output instruction in the Symbolic menu allows up to 4000 points).

Example 2: the operator allows exact answers by outputting all significant digits (Example 2.2), or by representing a number as a simple fraction (Example 2.4).

Example 3: the operator, in conjunction with the expression k:=k (for symbolic mathematics, this frees a variable from any numeric value), displays a numeric answer that also includes symbolic constants (this result can also be obtained by simply inserting a variable after the answer, in the same manner as physical units; see Example 3.3).

The operator for symbolic calculation, ��, instead of numeric calculation, �=�, can solve a task where some of the variables have no numeric value � see Fig. 2b., which shows the solution of the �chemistry-economics� task (If gold is selling for \$320 per ounce, how many atoms of gold can you buy for one dollar?).

After inputting the initial data Mathcad highlights the \$ symbol (the \$ symbol is entered using Shift-Ctrl-k) in red because it is an undefined variable, with no numeric value. This is not a problem, however, if the answer is displayed on the screen using the �� operator, instead of the �=� operator.

Tip 3 Combination of symbolical and analytical output   Sometimes it is necessary to combine symbolic and numeric output operators in one expression. The figure shows three example applications:

Example 1: in an arithmetic expression it is possible to see the answer both in the form of a simple and a decimal fraction.

Example 2: when solving an equation one can see both exact and approximate, answers.

Example 3: An expression can be simplified before calculation. This operation can be performed automatically by the �Optimize� mode, but Mathcad only shows the numeric answer. In our example we once again observe both analytical and numeric answers.

Tip 4. Constants in the numerical answer  Many users do not know that the basic operator in Mathcad (see Tip 1), �=�, the operator for numeric output, has two operands rather than one:

= []

The Mathcad user fills in the first placeholder with an algebraic expression, and the numeric value appears in the second placeholder. The third placeholder may contain (square brackets indicate that this second operand is not necessary) any constant or variable that has been defined earlier.

Usually system variables or built-in Mathcad physical units are inserted in this placeholder:

L = 200 ◦ cm or L = 2 ◦ m

The figure demonstrates another two examples:

Example 1: In the Russian cartoon �Thirty-eight parrots� the length of a Boa is measured in Parrots, Marmosets and Elephants. If to assign (to select) animal�s size, than Boa will �fit� in 38 parrots and one more parrots wing, in 5 Marmosets or in 2 Elephants. �You see, in parrots I am much longer!�.

Example 2: The solution of a trigonometric equation quite often contains the number p. It is possible to place this number in the second operand of the �=� operator. This can be used to obtain the exact answer, and also to find all the answers.

PS

The above figure represents a typical fragment of a Mathcad-worksheet: a user types some text describing a variable (Time), inserts a variable (t), assigning it with a numeric value and possibly a unit (360 sec), and then displays it with the same or some other (min) units. One can pack the whole Mathcad-worksheet with such sets of three regions and solve a problem this way, as shown. This methodology is inconvenient, however, when moving fragments to a new location, because it is possible to lose regions included in the fragments. Furthermore, if after movement the output operators are arranged a little bit higher than the input operator, then the relationship of cause and effect will be broken and the error message �This variable or function is not defined above� will be displayed. (Unnumbered part of Tip: when creating a new Mathcad-worksheet it is worth placing operators in columns, with only one on a line. In the final document you may convert this so that there are a number of operators on one line of the worksheet, thus making it more compact).

Another method of calculation is shown at the bottom of the figure. A comment was written with six spaces on the end; then, at the center and at the end of this series of spaces mathematical areas (select �Math Region� from the �Insert� menu) were inserted. These mathematical areas were filled with input and output operators. In this manner, the single �conglomerate� region obtained can be dragged from place to place without any fear of losing one part. It is possible to transform the three regions at the bottom of the figure, constituting a certain fragment of the calculations, into a single region: text including six mathematical areas. Although one could try to force the whole calculation into one text region, however, such a region becomes difficult to edit so it is important to know where to stop.

The above figure represents a typical fragment of a Mathcad-worksheet: a user types some text describing a variable (Time), inserts a variable (t), assigning it with a numeric value and possibly a unit (360 sec), and then displays it with the same or some other (min) units. One can pack the whole Mathcad-worksheet with such sets of three regions and solve a problem this way, as shown. This methodology is inconvenient, however, when moving fragments to a new location, because it is possible to lose regions included in the fragments. Furthermore, if after movement the output operators are arranged a little bit higher than the input operator, then the relationship of cause and effect will be broken and the error message �This variable or function is not defined above� will be displayed. (Unnumbered part of Tip: when creating a new Mathcad-worksheet it is worth placing operators in columns, with only one on a line. In the final document you may convert this so that there are a number of operators on one line of the worksheet, thus making it more compact).

Another method of calculation is shown at the bottom of the figure. A comment was written with six spaces on the end; then, at the center and at the end of this series of spaces mathematical areas (select �Math Region� from the �Insert� menu) were inserted. These mathematical areas were filled with input and output operators. In this manner, the single �conglomerate� region obtained can be dragged from place to place without any fear of losing one part. It is possible to transform the three regions at the bottom of the figure, constituting a certain fragment of the calculations, into a single region: text including six mathematical areas. Although one could try to force the whole calculation into one text region, however, such a region becomes difficult to edit so it is important to know where to stop.

Tip 5. Typical set of three operators  The above figure represents a typical fragment of a Mathcad-worksheet: a user types a some text commenting describing further actions a variable (Time), puts into calculationinserts a variable (t), assigning it with a numerical numeric value and with or without possibly a dimensionality unit (360 sec), and then �prints displays� it out and accompanies it with the same or some other (min) dimensionality units. One can pack the whole Mathcad-worksheet with such sets of three operators regions and solve a problem put by this way, as shown. But this technology methodology is inconvenient, however, for when carrying moving fragments to a new place location, because it is possible to lose operators regions included in this the fragments �on the way�. Furthermore, if after movement the output operators are arranged a little bit higher than the input operator, then a the relationship of cause and effect will be broken and the error message �This variable or function is not defined above� will be displayed: �This variable or function is not defined above�. (Not Unnumbered piece part of Tip: when creating a new Mathcad-worksheet it is worth to placeing operators in columns, with only one on a line. In the ready final document you may proceed convert this so that there are �from a line to a surface�, placing some a number of operators on one line of the worksheet, thus making it thereby more compact.).

Another technology method of calculation is displayed shown at the bottom of the figure: . a A comment was written with six spaces on the end; then, in at the center and in at the end of this chain series of spaces mathematical areas (select �Math Region� from the �Insert� menu) there were inserted mathematical areas (Math Region instruction in the menu Insert). And these mathematical areas were filled in with input and output operators. Thus, In this manner, anthe single obtained �conglomerate� region obtained can be dragged from place to place without any caution fear of losing one part. So iIt is possible to transform the three bottom operators regions at the bottom of the figure, constituting a certain semantic fragment of the calculations, into a single operator region: text including six mathematical areas. Although one may could try to �push�force the whole calculation into one commentary text region, however, but such a operator region becomes difficult to edit so: it is necessary important to know where to stop.

Tip 6. Test of data input  Very frequently variables in computation have a limited range of reasonable values. In this case it is expedient to test correctness of entered value. The figure �asks� user to assign variable t with a value of room temperature. Let us assume that it can not be less than minus 20 degrees Celsius (at this point addition statement is functioning as logic addition: AND) and more than 40 degrees Celsius (units of measurement � are accepted by default in this example). And if the user makes an intentional or an accidental mistake, then variable t takes a string value and blocks further numerical computations. (Expression containing function if can be enclosed in collapsing frameworks in order to protect user from embarrassment.)

Tip 7. Input of a bulky matrix with standard elements  To put into calculation a vector or a matrix the Mathcad users usually resort to the instruction Matrix� in the menu Insert (or press button with the matrix image). This technique displays a number of disadvantages. First, in this case the size of a matrix is bounded by the value of 100 elements, while any matrix can keep up to 8 millions elements in Mathcad. Many users solve this problem by creating several matrices and �gluing� them together into a large one using stack and/or augment operators. The second disadvantage is in that all elements of a matrix defined through the Matrix instruction are unfilled. In this case matrix elements have to be inserted manually. And maybe this is a reason for a limitation of 100 elements, which can be inserted manually yet.

Solution of this problem is shown in the figure: a matrix containing any standard element (we have set it as 1.0) is automatically created and displayed. This matrix can be of any size. Then it is copied to the right operand of the assignment statement and subsequently edited. (We have assumed here, that the matrix elements � are numbers ranged from 1 to 2. We should not type units and points manually � it is sufficient to edit zeroes. Through the use of function Matrix it becomes possible to record various numbers in such semi-finished matrix). (Not numbered piece of Tip. It is better to insert such matrices into a Mathcad-worksheet by introducing a table from Excel.)

Tip 8. Output of a bulky matrix  There may appear some difficulties not only at input of a bulky matrix (see a piece of Tip 7), but also at displaying its elements. By default there is displayed only upper left corner of a bulky matrix in the form of a table. Its sidehead keeps line numbers and headline keeps column numbers. A matrix which is not entirely displayed has vertical and horizontal scrolling regions with a runner to the right and at the bottom of a table. To look through the whole matrix one should move the runner. (Scrolling technology). Another possibility � output of a bulky matrix not in the form of a table but in the form of a matrix. In this case you can look through it by scrolling the Mathcad-worksheet window itself.

And if a bulky matrix has to be printed out (unfortunately, a sheet of paper has not scrolling regions), then we can recommend to print this matrix-table several times, having preliminarily displayed its separate regions with the help of the runner (see the figure).

Tip 9. Do not pass multiplication sign  Beginning with the 7-th version the Mathcad environment does not strictly require a multiplication sign to multiply a constant by a variable. This method seemingly brings Mathcad-worksheets more closer to documents written in handwriting: 2 �, not × �, 5 kg, not × kg and so on.

But here are three reasons against this technique that runs counter to the programmer tradition to put a multiplication sign (asterisk) between two factors: �, 5 kg and so on.

Firstly; speaking about numerical rather than symbolical mathematics of Mathcad, design 2 × is more or less clear: two is multiplied by variable a. And expression 2 a carries (at least, superficially) some ambiguity: this is both the product of 2 by a and a postfix operator named a, which operand is a constant (two).

Secondly; the behavior of expression 2 a is not known in the case when this construction is applied in symbolical mathematics.

The figure illustrates the third �contra� argument. If two is reduced by unit, then it should remain unit, but by no means two and furthermore with minus? The answer has appeared to be evident, but this simplicity is, indeed, �worth than larceny�. The point is that not unit is subtracted from two, but two is multiplied by minus unit in the given example. This service of Mathcad (programmer does not have to enter a multiplication sign) has turned out to be an ill turn (involuntary pun). There is another ambiguity: what is k a � is that variable k, multiplied by variable a, or is that a variable, with a name of three symbols, where the second one is a space.

Tip 10. Global assignment at the system solution   The Mathcad environment offers the built-in function Find for solution of the system of algebraic equations and inequalities. A system is commonly solved this way: the first approximation to the system is being set with the help of assignment statements �:=�. Then the key word Given is being written, a system of equations and/or inequalities is being set (it is not allowed to use only Boolean operator �¹�) and, at last, function Find is being called. This function returns values of its arguments, converting a system of equations into a system of identities. The answer is presented as a column-vector. Besides, in a number of cases in order to solve the system the standard values of built-in variables TOL and CTOL should be changed. These variables influence on the accuracy of calculations. Very often this technique requires successive approximations to the right answer (and there may take place not only one approximation). In this situation it is advisable to change the above-mentioned sequence of operators (but not sequence of their satisfaction) and to use the successive approximation technique, described in a piece of Tip ?. This technique implies that the first approximation is �omitted� by means of �º� operator, and other initial settings are placed more closer to the answer. If the system is bulky and display can not find room for it then the shown sequence of operators will obviate the necessity for scrolling a Mathcad-worksheet to adjust input data and to browse the answer. (Note. The described successive approximation technique is impossible to be automated through programming since the key word Given is not inserted into the program body).

Tip 11. Superfluous brackets  Many Mathcad functions have a vector or a matrix (array) as their argument. Vectors and matrices represent a collection of scalar values contoured with brackets. On the other hand, brackets also are an attribute of the function call operation. This feature leads to appearance of double brackets at the conventional call of a function with a row-vector argument (see the first operator in the figure). Double brackets can embarrass a user who may try to remove a �superfluous� pare of brackets not understanding a reason why it is impossible. The way out � lies in the call of a �matrix� function in the form of a prefix operator, which operand does not need a �brackets�� design (the second operator in the figure).

Tip 12. Set of three Boolean functions  The Mathcad environment does not present built-in Boolean (logic) functions And and Or, which are an integral part of all programming languages and which support implementation of all sorts of logic constructions, cycles and alternatives. And what is to be done here!? Firstly, it is necessary to recollect, that �logic AND� and �logic OR� have also such names as �logic multiplication� and �logic addition�: the given functions are realized through customary multiplication and addition operators (see item 1 in the figure).

Secondly, logic And and logic OR functions can be defined (see item 2 in the figure). In so doing there can be defined any quantity of Boolean arguments � that is elements of associated vectors of multiplication and addition operators. (Not numbered piece of Tip. It is possible to define two pairs of functions with various names �-And���-Or to satisfy both Slavophiles and Westerners.)

Thirdly, the Mathcad environment already has built-in functions that are well suited for logic tasks, � they are min and max (see item 3 in the figure). Moreover, they are also convenient for the following reasons:

It is not necessary to �litter� a Mathcad-worksheet with supplementary built-in functions AND � OR;

Functions min and max accept both matrix, column-vector and row-vector arguments;

Functions min and max accept not only Boolean expressions (yes � no, 1 � 0) as their arguments (array elements), but also real expressions, varying smoothly from zero to unit. Thus it is possible to operate not only with precise but also with fuzzy logic.

Tip 13. The built-in function redefinition  Mathcad offers user an opportunity of renaming and even redefinition of built-in functions. Renaming seems more or less intelligible (the old substance in the new frame): a new user function is being defined which is completely or partially coincident with one of the built-in functions. What is the reason for that? Items 1 and 2 of the figure illustrate two examples of the built-in function renaming. First, it is possible to replace a short English name of the function with a more integrated national one (item 1). We obtain a certain hybrid of a function name and a comment. Renamed function in item 1 is being called as a prefix operator (fx), that makes possible to get rid of brackets and to make the name-comment absolutely natural. This effect is also aided by three spaces in a new function name (see a piece of Tip 14).

The opposite object is being achieved in item 2: the built-in function name (augment - the �horizontal� confluence of two matrices) is abbreviated to one symbol. It allows drastic reduction of the program size because this program use the renamed function repeatedly.

Item 3 demonstrates redefinition (the old substance in the new frame) of the built-in function stack to visualize its essence. Function augment, as noted above, accomplishes the �horizontal� confluence of two matrices � its arguments are positioned also �horizontally�. Function stack is dedicated to the �vertical� confluence of two matrices but its arguments are positioned still horizontally. Redefinition of function stack � replacement of two arguments by one column-vector with two elements � restores �geometrical� logic of two functions for matrix confluence described above.

Tip 14. Special symbols in the name of variables and functions   The most significant distinction between the Mathcad variable names and their analogues in conventional programming languages is that variable names in Mathcad allow Greek letters and subscripts.

Programming environments always place certain restrictions on the variable (function) name. As compared to conventional languages Mathcad has less restrictions: the variable name in Mathcad can include Greek letters (Cyrillic letters deserve particular investigation � see a piece of Tip ?) and subscripts. All these peculiarities of Mathcad enable names assigned to different variables long ago in mathematics (p, for instance), physics, chemistry etc. On the other hand, Mathcad-variables have, in a manner of speaking, a bad �heredity� preventing the variable name from insertion of some symbols: �\$�, �&�, �?� etc. The point is that these and other symbols (keyboard keys) are loaded with a purpose of input in a Mathcad-worksheet of sum operators (�\$�), integration operators (�&�), differentiation (�?�), product (�#�) etc. But, if you are eager for that, it is possible to enter special symbols and spaces into a name of user�s function. And, moreover, Mathcad permits such �wild� name of a variable as a space or a chain of spaces.

This technique of the restriction removal from the variable (function) names has its own history. Tip ? recommends to use string variables � series of symbols framed by inverted commas � for comments of the Mathcad-program operators. Such series may include spaces and other symbols (�\$�, �?�, �&� and so on). Sometimes when editing such Mathcad-program there may happen �a glitch�, that will result in a loss of inverted commas by string constants. Constants will �turn red� indicating an error situation. But every cloud has a silver lining � it is possible to take advantage of this situation (transformation of a string constant into the variable name), and that is described in the figure. The �ablush� series of symbols should be copied and subsequently utilized as a name of a variable or a function. The story is to be continued in the next piece of Tip.

Tip 15. Enumerated variable names   Previous piece of Tip shows how to lift some restrictions from the Mathcad variable names � for example, how to insert special symbols and spaces into a name. But some restrictions on the variable names still remain. In particular, it is impossible to start a name with a number. But� It is meant an Arabic number (by default).And there also exist Roman numbers apart from Arabic ones. The figure illustrates solution of a conventional linear programming task: a furniture factory can produce two models of chairs at the cost of 70 US dollars and 12 Euro. On this order human and material resources are allocated. Besides, quantity of boards, cloth and time for producing of each chair is known:

 Chair Board consumption, meters Cloth consumption, square meters Time consumption, man-hours The first model 2 0.5 2 The second model 4 0.25 2.5 Resources 440 65 320

The question is to project manufacture of chairs so that their cost is highest possible.

There are two unknown variables in this task which are enumerated and have special symbols within their names (minus sign and space): I-st chair and II-nd chair. Special symbols are also applied for user units of measurement: physical (length and area), financial (US dollars and European currency), calculating (pieces) and mixed (man-hours). American currency unit is put into calculation on the principle �Time is money�: the given calculations does not use a physical quantity �time� and it can be temporarily �loaded� with another duty. Certainly, it is easier to write down: \$US := 1, and pieces := 1 (our pieces are �attached� to gram-molecules, which are also not equipped at the given calculations), but it is fraught with such a mistake:

\$US + pieces = 2

There should appear not a numerical answer (two), but an error message �Incompatible dimensions�.

Physical and other quantities are used in the described calculation not only for making it more �readable� but also for the dimension conformity control in the inserted formulas.

Tip 18. Once more about special symbols in the variable names   Authors have published their technique of insertion spaces and special symbols into the variable names, described in the previous piece of Tip, on the Collaboratory site (http://webserve.mathsoft.com/mathcad/collab) of Mathsoft (Mathcad designer firm). That was quite intriguing:

I have spaces in var-name := 10

Can you do it? := 2

I have spaces in var-name + Can you do it? = 12

There were two kinds of responses to this quiz:

a) How do you contrive to do that?

b) I have been inserted special symbols into the variable names for a long time, but I did not know, that it is possible to enter spaces too.

The second response leans upon the fact, that one may use special chord (Shift+Ctrl+k) for names of functions and variables to block the Mathcad system perception of special symbols (space, \$ and so on) as instructions (the name envelopment with cursor) or operators (sum, differential and so on). The figure displays definition of the variable with two spaces and �\$� symbol within its name (Cost in \$US). And the second variable with a space inside (Russian ruble) is obtained by editing the first one. It is impossible to insert spaces and special symbols at editing. To do that the above mentioned chord is required.

Tip 19. The invisible function and function-werewolf   The previous piece of Tip recommends to write down not Arabic (it is impossible) but Roman numbers starting the variable name to enumerate variables.

But many people can count Roman only up to twelve (lets recollect clock-dial). The figure shows the function of Arabic to Roman numbers conversion. (We face Roman numbers very frequently, keeping financial payments: amount of money � is an Arabic number, and its cash fragmentation � a Roman one. Our program will prove useful to the automatic teller operation, for example).

But it is not the very purpose to cite this program. The object is to give the reader another piece of Tip.

A lot of Mathcad built-in functions and operators have their inverse analogues: sin ¾ arcsin, x ¾ ln(x), differentiation � integration etc. Producing a pare of user functions (direct � inverse), one may act in a such way: produce the first function, and define the second (inverse) function employing the first one. If the function has a real argument then the task is reduced to the equation root search, and if argument is integer, the task may be solved through selection.

Apotheosis of the technique of insertion special symbols and spaces into the variable names � is a user variable or function, with a name of single space. The question is, what can this variable or function give a user except a headache.

The figure illustrates operation of such anonymous function. This function returns a Roman number, if its argument is an Arabic one, and, vice versa, it returns an Arabic number, if its argument is a Roman one. Single function incorporates both direct and inverse algorithms. The given function determines the direction of conversion identifying argument�s type. If it is of string type (if IsString � see the figure), then Arabic number is searched, otherwise (otherwise) � a Roman one.

The anonymous function allows not only recalculation without overloading a Mathcad-worksheet with unnecessary words and symbols ("XXV" =25 is clear without any comments), but also simulation of calculations (subtraction, root extraction, sine computation) with Roman numbers (see the end of the figure). The focus is that our function is called not conventionally, but in the form of prefix and postfix operators: there are no brackets framing an argument, and the symbol of operator itself is invisible. Thus it leads to the absolute illusion of Mathcad�s ability to operate with Roman numbers without any supplementary tools � operators and functions.

There follow some pieces of Tip from the figure:

1.�� Sometimes it is expedient to give an invisible name to an object of a Mathcad-worksheet. There can be a space or a chain of spaces.

2.�� In specific cases it is expedient to combine direct and inverse calculation algorithms into a single function. The reason is that these algorithms may have the same base (our pairs of Roman and Arabic numbers, for example).

3.�� The function displayed in the figure can appear useful for a partial encryption of messages. For instance, you may send such a message to your trade agent: �Minimum price is MM, and maximum � MMCD�. If you apprehend that the competitor, you protect the message by encryption, is acquainted with Roman numbers, then another pair of numbers may fit: Arabic numbers are, for example, Fibonacci numbers, (1, 2, 3, 5, 8, 13 and so on indefinitely), and their Roman analogues �are their serial numbers (�1�, �2�, 3�, �4�, �5�, �6� and so on indefinitely).  The reader has likely already noticed that some pieces of Tip in the book interfere each other. The Tip 11 has recommendations to exclude superfluous brackets from Mathcad-expressions. And now we will discuss insertion of auxiliary brackets.

The 1-st item in the figure once more time notes that the Mathcad environment, as well as all other programming environments and mathematics on the whole, value involution operator �superior� to multiplication operator, which in its turn, is valued �superior� to addition operator. These brackets once again accentuate the given hierarchy of operators, which can be familiar not to everyone (see example 2 in the figure, a piece of Tip ?).

Item 2 by means of function (treelike operator) augment simulates a reshuffled pack of cards � a row-vector with 52 elements (cards). In the pack simulating �tree� brackets look far from superfluous (compare items 2.1 and 2.2).

In the 3-rd item a Boolean function Verdict is formed, which returns results of voting (the participant votes (1 � pro, 0 � contra) are stored in a vector-argument X): verdict is approved when more than two thirds of voters vote �pro� (function mean returns an arithmetic mean of the array elements). Brackets here also stress an algorithm of function operation and the satisfaction sequence of operators �>� and �:=�.

Tip 21. The variable dimension control   Sometimes user of the finished Mathcad-document has to input not a simple value of the input variable, but a value with a fixed dimension of a physical quantity. Input of a dimensionless quantity or a quantity with another dimension should result in an emergency halt of the Mathcad-worksheet execution. Such scenario is realized by operators of the figure: user should set a length into the variable d, i.e. he should set a constant multiplied by either a built-in or user unit of length. An input dimension is checked as follows: variable d (it deserves an individual examination) is firstly raised by and immediately reduced by a unit meter (m). The value of the variable d does not suffer any variations but, in the case that the length is added by anything else, there occurs an error. The focus of the displayed Mathcad-worksheet is in that it contains two different variables of the same name d (it is feasible in the Mathcad environment � see a piece of Tip ?.?.). To avoid possible muddle one may give another name to the second variable � D, for example.

Tip 22. Graphical solution of the linear programming task   Mathcad-worksheet in the figure of a piece of Tip 15 solves the linear programming task. Many mathematical program packages (Maple V, for instance) offer graphic facilities for visualization of a such task: a plane polygon is constructed edging the feasible solution area of the task.

The Mathcad environment does not offer special visualization facilities for solution of the linear programming task. But with a certain skill this problem is solvable by means of a conventional Cartesian diagram � see the figure. The essence of the problem is described in a piece of Tip 15.

To solve the linear programming task graphically there are defined two functions Constr (Constraints) and �� (criterion function, Russian �������� ��������) with one argument Chair1 (quantity of chairs of the first model). Application of symbolical mathematics at the function output is dictated rather not by needs of inequality solution (solve) � that is easy manually, it serves for the graphical solution essence illustration: straight line of the of criterion function slides upwards (animation of Mathcad is very pertinent here) � solution will be achieved at the moment when the straight line �takes leave� of a polygon. The graphical solution in question both suits for an integer task (a polygonal line is substituted by a chain of points) and for a nonlinear task (line segments are replaced with portions of a curve: convex or concave).

Tip 23. Color Cartesian diagram  The given piece of Tip extends the subject of the Mathcad non-typical graphics, started in the previous piece of Tip.

Mathcad (as well as, however, almost all mathematical packages) offers the three-dimensional diagram coloring facilities: for example, surface vertexes can be colored more �warmly� while a concave remains more �cold�. Unfortunately, this principle cannot be applied to a two-dimensional diagram. But if something impossible is very desirable, then it becomes possible. The figure exhibits a single argument function diagram, so that � see comments to the right of the diagram. Unfortunately, this book is not colored, but the reader can trust the author, that this line on the diagram is displayed with all rainbow hues. The trick of this figure lies in that it presents not a Cartesian diagram but a colored surface, turned to a spectator in a special way (with one �side�). It is worth to hide operators and the forming matrix , which elements � are values of the function under analysis, by locking up an appropriate area (see the top of the figure). This operation produces an absolute illusion of a colored Cartesian diagram.

(It is expedient to �pull� operators, the forming user function y(x) and diversity limits of the diagram (x ���-x ���) up to the diagram. Operator �º� (global assignment) proves useful here, it spreads its influence not only downwards (that is executed by operator �:=� (�semi-global� assignment) but also upwards).

Tip 24. Graphical solution of the set of two algebraic equations  To solve the problem of this piece of Tip heading one should graph two Cartesian diagrams and verify a point of their intersection. But this task may appear arduous for systems are not consistently represented in a suitable for graphing form: y1(x):=�, y2(x):= �. Equation sets are generally assigned in a different way � f1(x, y)=0, f2(x, y)=0 � variables in this notation are very seldom dividable to reduce this task to a straightforward one:

 x + y = 5 y1(x):= 5 � x � y = x2 y2(x):= x2

The figure depicts this problem solution: matrix M is filled up by means of two loops with a parameter (x value and y value). The matrix elements display status of equations on the x-y plane: if selection of x and y values shows any expression to reverse its sign in the given point, then corresponding element of matrix M takes the value unit, and zero � otherwise. We are scanning the diagram area as though as �popping up� (oscillographingon the level line diagram) our curves (at the center of Bernoulli lemniscate (�entwined with ribbons�). How long will this �scanning� take � this is another question. Constant 300 allows compromise settlement between speed of the diagram construction and its quality.

??? There is used an English (Chair1) rather than Russian (I-�� ����) variable name for the Mathcad 8 pro symbolical mathematics is not capable of operating with such variables.