There wasno such confusion in the DOS versions of Mathcad. By the way, that corresponds tothe essence of the solved problem, for example, in finding the roots of asystem of equations in a Given/Findsolve block See Ch. 2. The roots of suchsystem returned by the function Find called numerically and not symbolically, makethe left part equal to the right one approximatelyrather than exactly.
Mathcad has a modewhich shows the result of an analytic conversion in the place of the sourceexpression. The problem of whether to show simultaneously both the formula andits result or to do it in turn is connected with, first, saving the spare placeon a screen and paper of a printer and, second the aim of calculation, itsdirection. If the aim of a calculation is mere practice, working out new dataand showing a result, the formula would be unnecessary. If the study of thecalculation is necessary for educational purposes, additional checking of theresult, or making some changes, the formulae would not be excessive. Mathcadhas the instruments to hide the formulas, which will be discussed further. Mathcadhas built in units for basic physical quantities as well as fundamentalmathematical constants, such as e the radix of natural logarithm or p ratio of circumference to itsdiameter, etc.
A detailed description of this Mathcad feature will be given insec. 1. 4. Mathcad also has aquite thorough reference book containing basic mathematical, physical, andchemical constants, which can be copied into a worksheet with correspondingunits. For example, Fig. 1.
However, using this operator is not highly recommended; it breaks cause effectrelation and confuses those who aim to study the document: a variable has adefined value but is not clear where it was defined. This users fail to take into consideration the difference in scope. Although,the global assignment operator can influence to top of the worksheet, forexample, damage plots created by QuickPlot. Thus we comment that some variablesare constants, while others are the first approximationto results see sec. 2.
5. 2. Such separating is useful to make a Mathcad worksheet well suited for Internetpublication see chapter 7. A Mathcaduser can change the style of a variable and insert a new variable with the samename into a worksheet. Fig.
1. Speaking about the texts andcomments Mathcad offers a set of style for dealing with text regions see sec. 1. 3 allowing the user toformat letters, words, paragraphs title of the first level, title of thesecond level practically as in MS Word. Otherwise, if the assigning variable has already been defined itsnumerical value is represented. So, one can check that the name is free: neither a user nor Mathcad hasdefined it yet.
Such technique allows us to omit a number of errors mentionedbefore. 1. 9. Second, one could forgetthat a user has already defined the variable and assign it something new. Thesituation when the same variable has one value in the first part of Mathcadworksheet and another in the second part is not right. Fig.
1. 10shows a very interesting and useful number 200 in the right bottomcorner of the counter Maximum number of undoable actionper document. This is new feature appeared in the 11 th version. Alas or fortunately, it is impossible in real life but is doable to someextent with computers. Just push the Undo button.
Earlier versions of Mathcad allow us to do that onlywithin the limits of a formula while the pointer is not taken away from it. Versions eleven and twelve of Mathcad allow the command Undo to affectthe whole worksheet. In Fig. 1. 10 the number indicated 200 is the numberof such steps.
Fig. 1. 11at point 2 shows us the tandem operators at work. Those two useful tandems points 1 and 2 inFig. 1.
1. 11 whichmakes analytic transformation before calculation. Theevaluation equal sign became a stumbling block for those who started Mathcadten or fifteen years ago, hearing of its unusual abilities for calculatingsophisticated formulae see Fig. 1. 1, making graphs see sec. 1.
6, animations see sec. 1. 7, solving equations andsystems of equations See Ch. 2. An error, unintelligible at first glance, had emerged: Mathcadinformed them that a variable is not defined.
Users tried to define that errorby operator of variable type being guided by experience of working withprogramming languages, but Mathcad has not such operator. Because of it the users continue to make the errors mentioned abovein connection with reassignment of variables. Through it Mathcad 11. 1Mathcad 11 was published and patch 11. 1 was released soon after itprovided the user with the mechanism for checking the reassignment of variablesand functions, built in and users, opened and hidden in closed regions seeFig.
1. 22 and in referenced documents see Fig. 1. 21. For this, thePreferences dialog box get the new Warnings tab see Fig.
1. 12. As theneed arose to have worksheets accessible by the Internet Mathcad acquired theso called Web Controls See Ch. 7 about Mathcad ApplicationServer shown in Fig. 1. 14.
Unlike the standard Mathcad Controls, they donot require the user to write or edit the programs in one of the supportedActive Scripting languages JScript and VBScript see Fig. 1. 13. It isnot difficult to write the program but the point is that such scripts areundesirable for Internet use, so that they were left out. Thenumerical evaluation operator has evolved too, or rather not the operator itselfbut its function.
Thus, current versions of Mathcad allow users to show numbersalso in non decimal systems, both as vulgar and decimal fraction, in scientificor in engineering formats, etc see Fig. 1. 6. 1. 10.
Such changes arecrucial from the standpoint of development of Mathcad Application Server See Ch. 7. 1. 15. If a userhas installed both Excel and Mathcad, entering of large volumes of informationcan be automated.
An Excel table is inserted into Mathcad worksheet by choosingInsert | Component. | Microsoft Excel and then in the Excel SetupWizard dialog box choose Create an empty Excel worksheetor Create from file. After that we can point the part of Excel tablewhere the information from Mathcad is transferred Inputs or converselythe information is transferred to Mathcad variable from Excel table Outputs. Fig. 1.
After entering data, the Exceltable can be removed to make Mathcad worksheet lighter, and in such case, andso that those who have not installed Excel can work with the worksheet. To assigndata to the variable M, choose Insert |Data | File Input. In addition, we can choose file format in opening list FileFormat and set the the comma asseparator instead of the tab character. The data itself Fig. 1.
20 showsit opening for review or editing in WordPad can be entered in Mathcad andwritten to the disk via command Insert | Data | File Output ormanually for example, in Excel, or generated by another program, or receivedby e mail et cetera. Besides the command for data changing shown in Fig. 1. 20Mathcad has a set of similar functions readprn, writeprn etc accessedby File Access from the InsertFunction dialog tab. The lengthof numeric literal writing to disk depends on the value of built in variable PRNPRECISION. The main disadvantage of savingdata in a file, rather than in Mathcad is that one can send Mathcad worksheetand forget about the file.
They can be written in another worksheet, andsaved on a user disk or on any computer in local network. To make them work forus, we must make a reference to thefile containing necessary data from the current worksheet. Fig. 1. 21 showssuch situation.
D disk contains Mathcad file sheet. The otherMathcad document which also shown in Fig. 1. The worksheet named sheet. mcd carries outthe calculations contained in the referenced worksheet which it is connectedto first by the command Insert | Reference. As a rule, the referenced documentscontains constants and functions covering a branch of applied sciences or having some generalengineering significance.
With such references, we can make, for example,dimensional a set of user functions in the form of DLL that were turned intobuilt in i. e. listed in the Insert Function dialog box see site . As a rule, Mathcad worksheet comprises three parts: area for source data, area for calculation, and area for results. We can delineatethese areas in Mathcad worksheet clicking Area from the Insertmenu. This command insert two horizontal lines near the mouse pointer, afterwhich the user moves the top line to the start of an area they wish to createand the bottom line to the end of their area see the top of Fig.
1. 22. What can we do with it?First, we safeguard the information against unintendededits by command Format | Area | Lock. After such command, theregion enclosed by the area can be viewed but not edited. The second command used separately or together withcommand Lock is Collapse and this hides the chosen area from auser. The result of these two commands on the area named Calculation shown at the bottom ofFig.
1. 22. The user types this name clicking Properties from thearea pop up menu. Otherwise, we can even hide a vestige of this area in Mathcadworksheet by this command. After such manipulations inserting, locking, andcollapsing an area a user can change the values of a and b and see the result variable c but cannot see and edit theformulae.
A Mathcad worksheet containing a collapsed area is like a piece ofpaper, the middle of which recedes into the background by several folds Japan origami. We canlock an area with a password, a row of symbols, and only knowing that we canunlock and expand an area. To protectthe worksheet, choose Protect Worksheet. from the Tools menu,shown in the bottom part of Fig. 1. 23.
Worksheet protection comes in threelevels presented by switches Protection Level in the Protect Worksheet dialog box. The first level, File,the lowest one, protect the worksheet from being saved in some formats, forexample as earlier Mathcad versions. Fig. 1. 24 shows two lists of fileformats for saving the current worksheet even without protection level File. In Mathcad 12 this protection level makes it impossible to save theworksheet as Mathcad 11, 2000i, or 2000.
Thatlimitation works at two other levels too. The second level protects existingoperators against change by the switch Contentbut new operators can be created. The third level, Editing, the highest one, does not allow us to change existingoperators or to create new. e. turn off protection from some operators as usual, from those definingsource information by the commands shown at the top part of Fig.
1. In all levels we can protect the worksheet with a password or withoutit options Password optionaland Reenter password. Some safeguarding mechanisms duplicate each other. We can protect a separate operator or several operators at once placing theminto an area Fig. 1.
22 or by technique shown in Fig. 1. 23. This duplication occurs because the protection mechanisms were not introduced simultaneously but from earlierversions to new. At the beginning, Mathcad 2000 it became possible toinsert areas into worksheets Fig.
1. 22 safeguarding it against editingonly Lock/Unlock. Then Mathcad 2001 we could collapse theseareas . Mathcad 2001i got the mechanismsto protect whole worksheet and some of its operators Fig. 1.
23 and1. 24. Mathematicshas a term correlation: for example,there are two enumerable sets, and each element of the first correlates with a single element in thesecond one. Ofcourse, Mathcad does not keep sets of angles,sets of correspondent sines etc but calculates this trigonometric function inaccordance with its built in algorithm. Another question is how accurate andquick these algorithms are.
r Some Mathcad operators have ambivalent contents. Is it an operator with two poweroperands, with the second operand equals to two, or squaring operator with oneoperand?The second example: 2K is it a multiplication operator or just a variable named 2K a name of this type is possible in Mathcad,see sec. 1. 2. 2 etc. Mathcadalso has mathematical operations executed both as operators and as functions.
For example, the exponent is calling as ex operator and as expxfunction. Basically, it would be convenient if all Mathcadoperators have such twins. First, it would give the users additional freedom inchoosing and second, enable them to introduce formulas in the text format thatwill be noted in Ch. 7. r Mathcadhas some particular functions and operators. We can understand how they workonly when we size up their mathematical meaning and the techniques of theircomputational execution.
These functions return the value depending on valuesof arguments, operandsand what is situated near them, and on additional specific adjustments. Thus,the function Findreturns different values for the same arguments depending on the firstapproximation in finding the roots of analytical system of equations that isthe function Find isappropriated; see the details in ch. 2. The second example, Fig. 1.
25 shows what is contained in the pop up menuappearing after right clicking the mouse for the definite integral operator. One may say that there isanother way of penetrating the inside, or interior of the function bypassingformal entrance, the list of the arguments. Advantage of the first form isthat where there are many arguments we can single out one or two and list themas formal arguments of the function. The rest of arguments we can consider byconvention as constant. The disadvantage is such non closed unlockedfunctions are difficult to transfer: we may forget about some externalconstants, and lose them.
A function of that kind is similar to Mathcadworksheet having some data in external files see Fig. 1. After pressing thiscombination, the color of pointer changes from blue to red that indicatesemergency state of Mathcad. 1. 1.
Fig. 1. Only onecharacter, which we cannot insert into the variable name or rather, we cantype but cannot see it then, is a period. Nevertheless, we can do thatcreating a variable with a text subscript, which has an invisible blank as aname before subscript and periods as subscript. For example, in Fig.
1. mcd. 1. 33. This blank is notfortuitous: some characters cannot stand in the beginning of variable names. First of all, that concerns digits 0 through 9.
Sometimes in certain Mathcad versions in combination with certainWindows versions some letters of Cyrillic alphabet cannot stand in the beginningof a name. Although, that restriction does not apply to a blank. Therefore, itis desirable to start a questionable name with a blank. A blatk or some of itmay use as a variable name making it invisible see sec. 1. 2.
3. There isone more reason to introduce the system index into Mathcad. As we noted before,the worksheet may contain different variables of the same name. To avoid errors these two objects must bedivided by styles. Although, that may be worse as the excess information makes aworksheet hard to read and study. First twooperators in Fig.
1. 37 insert the user function named + into a worksheet duplicating thebuilt in addition operator. Fig. 1. 37 also shows how to change the name of variable style from User 1 to invisible and the color of thevariables to white see New Style Name in dialog box EquationFormat.
This calculation technique was shown in Fig. 1. 11. 1. 44. Such displayingis useful especially if the operators forming variable values Re, Pr, and x speaking ofFig.
1. 44 are far from the operator forming the variable Nu that is of interest to us. Many users of Mathcad do not insert thecomments to the worksheets thinking that they are created for personal use andelucidative fragments could be inserted later. Still, if theworksheet is intended for personal use some comments in it will not besuperfluous. We may tangle even in own worksheet opening it after a time if ithave no any comment, for example a name. Theindicator that a mathematical operator is turned off the computation is a blackrectangle upward and to the right of it.
We can disable operators to transformit to a comment and, for example, to select the formulas for calculation fromthe list available or to speed up computation. For example, a three dimensionalplot can be disabled in checkout and be enabled again in ready worksheet. Ifone function is defined twice its first definition can be considered thecomment under the certain conditions. There are Mathcad worksheets containingwhole pages that are the comments as a matter of fact. The developer of suchworksheet suggests users to study the calculation, after that insert theirdata, and make computation.
1. 13 and in Web Controls see Fig. 1. 14 being used moreoften to design an interface. Therefore, we ought to type again duplicatecalculation operators after inserting the source data and hide this area asshown in Fig.
1. 22, for example. On theone hand, the comment consisting on variable names is a typical mistake of anovice user who does not know how to insert it right. Still, there is another extreme of this phenomenon the comment consisting fromvariable names. The most experienced users make all the comments or part ofthem as names of variables preparing their worksheets to publication in WebSee Ch. 7.
The point is that such comments are transferred in Web asgraphics without distortions while texts are displayed with wrong charactercoding. This reason is fortifiedwith that without units Mathcad worksheet is simpler to compile to environmentof the programming languages, for conversion of customized functions intobuilt in in the form of DLL. The third reason is major. It is connected withsome peculiarities and limitations of unit tool that cause a skilled user toseize units from a worksheet, almost finished, and turn them into comments:instead of If the work with dimensional values is notplanned, unit tool should be enabled via command Tools | WorksheetOptions. and switch None in Unit System tab.
Otherwise, unit tool may result in errors. For example, a user forgot to definea variable that has the similar name with a unit; nevertheless, the calculationis not interrupted. Another example. A user defines a function and wants to represent it asQuickPlot, i. e. 9.
5. 2. Mathcadsymbol mathematics work with units as with variables neglecting that one meteramounts hundred centimeters or sixty minutes are in a hour, etc. Symbolmathematics is foreign element in Mathcad; it was taken from Maple where unitappeared in the eighth version. Although, symbol mathematics is auxiliary toolin Mathcad integrated seldom into a calculation but operates auxiliaryfunctions. 5.
Mathcadlist of units includes not all physical and other quantities that we have todeal with. For example, there is no units of information bit, byte, units ofcost dollar, ruble, euro, etc can be necessary on calculations. What we have can do in Mathcad to make technological andeconomic calculations dealing with dollar, ruble, euro, etc?Fig. 1. 53shows one of techniques.
After that the cost will bedisplayed in candela see Fig. 1. 53 which we should replace with therequired unit pseudo unit of the cost. 6. Wesometimes meet formulas purely physical in essence and empiric in their formsee point 5.
Fig. 1. 4 gog equivalent fuel to produce 1kilowatt hour of energy. If the appropriate units are listed in thecomments, we can have this formula works correctly by the method described inpoint 4. Although there is another way such pseudo empiric formulaswere derived to free us from additional calculations connected with units. If we insert this formula, but not simplified, work of Mathcad unit toolwill not be disturbed see Fig.
1. 54 at the bottom. 7. Mathcaduses only absolute scale for units. Particularly, that means if the quantity equals to zero we need not to assign aunit.
Although, there are relative scales too. For example, wemeasure temperature in centigrade degrees relative scale but not in Kelvinabsolute scale. The dimensionless value, item, is often met incalculations. At that, the variable N remains dimensionless. Mole divided by Avogadro constant is item which could be used in calculations.
Fig. 1. Thus, in a calculationshown in Fig. 1. 58 cross section area of one tube and total area of alltubes have, strange as it may seem, different dimensions: mm2/it andmm2and they cannot be added.
The example shown in Fig. 1. 58 is strained, ofcourse. It isknown that Mathcad programming tool have no instruments to make comments inprograms and it did not enable us to have a local function there until versiontwelve. Nevertheless, Fig. 1.
Besides, the program contains subroutine Golden Ratio that has all features of a localfunction although the worksheet was created in Mathcad 11 in which thelocal functions are impossible. What is the matter?The matter is that allthese objects texts, pictures, operators and even working function Golden Ratio were placed on function Min GR that gives an illusion of a singlewell commented program. The thirddimension can be used to hide some information in Mathcad worksheet. There is awell known literature rule, and the authorof this book tries to follow it, to omit a word, a sentence, a paragraph, achapter if it is possible to do. Suchunnecessary information is hidden in close areas of the Mathcad worksheets or movedover the right border of a screen.
If it is impossible to do, we can overlapthis information by additional or neutral comment as shown in Fig. 1. 61. Therefore,considering screen and paper of printer to be flat, we should prefer simple 2 Ddimensional plots creating Mathcad worksheets. Real sophisticated problems maybase on set of functions with three, four, and five arguments which plots wemust represent correspondently in three , four , and five dimensional space.
Weshould remember that complicated plots often hide rather trivial calculationthat masks itself with perspective, lightning, fog and other options. Complicatedcalculation, having practical value, as a rule, is illustrated with simple,plane plots, which represent the basic law of calculation. Such calculationsinclude set of curves, for example, instead of surfaces looking agreeably buthard in work. The mostcommonly used plot on Mathcad worksheets is X Y Plot. It representslocation of a couple of elements components of two, three and more vectors inthe plane, tabulation of functional dependences of one argument.
Fig. 1. Variables x and y were placed in the appeared blank X Y Plot we can also insert there vectorsthemselves. After that simple manipulations vectors x and y also can be read from the disk or obtained in a calculation, not justinserted manually Mathcad represent a plot having a set of defaults. Inparticular, the points of the plot representing the location of the couple ofvector elements are connected with a broken line starting from the first pointthe first elements of vectors to the last one the last elements of vectors.
We can change the default settings via the dialog box Formatting CurrentlySelected X Y Plot. It is called, as the other dialog boxes for formattingMathcad objects, with mouse double click the graph. Cartesian plots canrepresent both couple of vectors and functional dependence themselves. 1. 63 but notthe names of vectors. After moving a pointer the plot will be produced over adomain from 10 to 10 if the correspondent function returns real numbers inthis range.
For example, square root of x will be represented within the rangefrom 0 to 10 because it returns imaginary numbers in the left part of x axis. At that, the problem comes to plotting of two vectors again: Mathcad dividesthe segment of plot to points, the number of which depends on plot size on thescreen on average it is 50 and on monitor definition, and computes the valuesof an argument and a function in these points, i. e. forms two vectors to plotthem. The example in thebottom of Fig. 1.
We should remember about it creating plots in Mathcad. Thesurfaces discussed below are created via the tabulation of a function of twovariables now. Then the correspondent grid of a matrix are risen at heightproportional to the values of function fx,y in the point Fig. 1. 65.
Besides, weshould remember of such a case. Here is a typical situation in mathematicalanalysis: two plots are created in one coordinate system, a function and itsderivative. That could be considered as an error, which Mathcad is to interruptwith such error message when meters are added with kilograms. In addition, we can remind a problem from the book for intrants: find thevalues of x of the given function in which this function is higher than itsderivative. They also doesnot agree to change the form of the problem but to keep the essence derivationand solving the unequality: to find values of x in which the derivative moreless than zero. From this it follows that we can draw two or more curves in asingle plot only if they are of the same dimension length, time, force, etc.
Otherwise, a reader will apply himself to the meaning of their cross pointswhile they have no meaning. 3. If we try to draw level lines of this function the graph would not be createdwithin default range of 5 to 5 for both xand y, but would be interrupted withan error message. We used this property showing the cost of a used car inFig. 1.
68. Unfortunately, graph of function of two variables has not sucha feature. It does not ignore complex values of the function in some sectionsof rectangular range but returns an error message. After that, many of usersgive up drawing a graph. Thelimitation shown in Fig.
1. 70 could be bypassed in the following way. Ifthe value of L lengthof the cylinder becomes negative, function S returns not the complex number but zero. Inaddition, we imposed a restriction to the problem: the ration of the tankdiameter to its length must be more than 0. 5. 5 had been inserted into function S its level lines became to were drawnwithin non rectangular trapezium shape variation range R and H non rectangular variation range of thefunction of two variables is discussed in Ch.
3 that considers the problem oftriangular diagram. Fig. 1. 73shows the stages in creating animation in Mathcad in unusual way: to generatepeculiar exe file. 4.
A user can set integer valuesfrom 1 to 100 for FRAME that vary the variable r from 1to 1. We select the portion of worksheet toanimate by dragging with the mouse. Fig. 1. After that, clicking Animate makes Mathcad change the valueof FRAME inspecified range and draw obtained shots pages of the notebook for their quickdisplaying turning the pages.
The animation clip see Play Animationfrom Fig. 1. 73 can be saved on the disk and played back without Mathcad. Why it is not an exe file!?Second, Submitbutton can be useful both in WebSheets and in worksheets, in traditionalMathcad documents. The point is that pushing of this button increase by onevalue of the variable connected with Submit after starting Mathcad in any mode its value equals to one. Thisvariable is a peculiar counter of pushing the Submit button, i.
e thenumber of calling to MAS. This feature of Submit button can be veryuseful in Internet and out of it: the formal variable can govern The Mathcadworksheet changing it, for example. Thus, Fig. 1. 75 show Mathcad worksheetthat illustrate the Newtonmethod of numerical finding zero of the function where pushing the Submitbutton allows us to see approximation step by step. 1.
1 and 1. 8 inDOS versions non graphic of Mathcad. 1. 33. When the cursor is red one canenter reserved characters noted before and some other into the variable andfunction name see sec.
1. 2. 2.