_{} _{}_{ }

The
traditional programming languages (Visual Basic, for example) paint the
data-sheets of programs in the following colors:

·
commentaries
– green color;

·
built-in
constructions of a programming language – blue color;

·
everything
else (user’s constructions) – black color.

If you really want it is possible to paint this
way Mathcad-programs (see fig. 59).

Color
in programs is a special story, which has been told about in the sketch «Color
in programs» (http://twt.mpei.ac.ru/ochkov/Color_in_Program/index.htm).

_{}_{ }_{}

When you
output zero dimensional value Mathcad shows needless formalism adding unit to
zero. This not quite corresponds with generally accepted standard of work with
dimensions: for example, we say that some equipment was on the grade (on the
height) 20 meters, but another one – on zero height, at that we do not specify
the unit (meters). If we speak about absolute scales of measure, then the
output of zero values we specify only dimension, but not units.

How
it is possible to remove unit of zero value in Mathcad-document is showed in
the fig. 60. For that it is enough to make
this unit … invisible one – to assign it the «invisible» style (we write in
white and white).

*Remark*. It is worth to hide unit only in that case if
it is known in advance that the insertion operator of numerical value will give
only zero value. At that it is necessary to remember that zero meters and zero
millimeters are not the same things if we speak about numerical (approximate)
mathematics of Mathcad.

One more example of hiding a
constant by change its color from black one to white color is showed in the fig. 60à.

In Mathcad there is undocumented
function time, which
returns the time (in seconds) of work of a computer[1]. The argument of such function may
be any number, zero for example and that is done in the first part of the fig. 60à, where the time of
addition of numbers of natural scale from 1 to million in Mathcad 2001 is
fixed. In the second part of the fig. 60à the function time is called already in the form of
the prefix operator, which operand is written in white and white and it is
invisible on the display’s screen (on the printer’s paper). As since this
argument (operand) does not have an influence on the value that has been
returned by the function (the operand). Hence there is no reason to show it.

*Remark*. Qualitative assessment of new machine
arithmetic that has been built in Mathcad 2001 is showed in the fig. 60à. This arithmetic allows quickening the
calculations.

See http://twt.mpei.ac.ru/ochkov/Invisible/Eng/Index.htm
too.

See http://twt.mpei.ac.ru/ochkov/Units/Part_1_eng.htm
too.

_{ }

As we can
see in the fig. 21 it is possible to control a dimension
of an input value out of a programs.

It is
showed in the fig. 61 how it is possible to carry on
such control inside a program when you call the user’s function. It is done by
already described in tip 21 technology: the
value that keeps controlled dimension is added and at the same time is
subtracted to the controlled variable. There are no changes with the value,
which keeps in the variable if it keeps necessary dimension (in the fig. 61 the variable diameter and length have to keep only the length and
nothing else, but the variable density – density). Otherwise the mistake appears (see three last «ablush»
operators in the fig. 61). The «program» variant
(see the fig. 61) only differs from the technology
that is showed in the fig. 21 that there is no
necessity to have the same name but different styles variables.

As well in the programs the function UnitsOf is to the point. This function
returns the dimension of its argument. The function UnitsOf has very interesting feature. It
has been inputted only in Mathcad 2000, however it already was in the program
but not in the documentation, Help and in Master function of Mathcad 8. It is
very remarkable fact: it turns out that new functions can appear in Mathcad
earlier than the documentation will tell about it. In early version of Mathcad
it is worth to «probe» new, unknown functions. At that it is not worth to
enable them in the calculations (*tip without number*). This tip fluently
spills over the tip 62, where it is recommended to
«be behind one step» in use different software. User of Mathcad 8 can input the
function UnitsOf in the
document if he has forgotten that there is not still such function. By the way
it is one more reason, which decides in favour of the method of input the
built-in functions with help of Master function, but not with help of manual type
of their names. Point is that the function UnitsOf (and similar) in Mathcad 8 is able
to be unfinished. The developer of Mathcad did not have time to shape it up: in
the program they left it (forgot?), but they did not describe in the
documentation, Help and in Master function.

See http://twt.mpei.ac.ru/ochkov/Units/Part_1_eng.htm
too.

_{}_{ }

Three
types of Mathcad-document are showed in the fig. 62
where the volume of a cone (as well this problem is considered in tip 30, 35, 36, 37 and 69) is calculated. The upper
part of the fig. 62 is the name of the document
(commentary[2]) and the horizontal line, which is
marked the *slammed area*, where all calculation is situated. If we *open*
the area, then user sees the middle part of the fig. 62:
the title and the opened area, inside of which there are three subtitles and
three new slammed areas (the sub areas – if we remember the word *structure*
in the title of the tip), which contents in its turn is showed (opened) in the
lower part of the fig. 62.

*Embedded
areas *(i.e. the
areas that are created inside the other range – the area of upper level) are
created by the moving the limits of the area of lower level (subtitle) in the
area of upper level (title). It is impossible to create new area inside already
created one. This feature of Mathcad it is possible to consider as a mistake or
limit. More over when a user try to open the embedded slammed area then
sometimes the system of Mathcad does not «understand» completely what the user
wants and it opens incorrect areas. The author removes this tip but gives new
one on base of it.

The
possibility of separation («enclosure») of the area (areas of the same level)
for the following its protection from the editing (Lock) in Mathcad-document
has appeared in the *seventh* version of the package. Apart in the *eighth*
version it is possible to slam (Collapse) these areas and we use it forming the
fig. 62. It is possible to say that «the age» of the
tools of creation and protection of an area is two «years», but the tool of
slammed is one year. Following such criterion it is possible to sort out all
tools of Mathcad and to draw enough sound conclusion: the «younger» tool the
higher probability to find a mistake or a defect[3] in it. It is not worth to use the
tools «with one passing year» for writing serious programs (*the main point
of the tip*).

It
is worth to formulate this tip to wide extent the following way: the newest
version of Mathcad it is possible to use in familiarization and educational
purposes. Some serious work it is better to do in the environment of rolled
versions.

This
principle of «the old horse, which does not spoil the furrow» touches not only
soft but also hard part of the computation systems. The computers, which
control the serious industrial projects, work under DOS, but not under Windows.

_{} _{}_{ } _{}_{ }

One of
the imperfections of the symbolic mathematics of Mathcad[4] is so that it is difficult enough
to automatize the analytical transformations in such a way that introduced
changes in the initial expression tell on all line of transformations at once.

One
of the solutions of this problem is showed in the fig. 63
– union of the operators «:=» (input of numerical value) and «®» (output of symbolic result).
Outputted expression (in our case it is any user’s function with the name y) as a by-product (see the icon of
the question after the title of the tip) forms new user’s function dy(x), with which all subsequent
transformations are possible (see fig. 63). Now if
we make changes in the initial function y(x) then the result will change.

The
process of the qualitative change of the tools of symbolic mathematics
«stiffened» on the seventh version of Mathcad. This explains the following way:
as we have said above the symbolic mathematics of Mathcad is not work of MathSoft,
Inc., but acquisition* of the elements* of the symbolic mathematics from
the famous package Maple made by the firm Waterloo Maple, Inc. In the tip 63 we underline (see once more the icon of the question
after the title of the tip), that the forming of the user’s function with help
of symbolic transformation (the line of the operators «:=» and «®») is by-product (undocumented
method). And symbolic mathematic is *by-product* (tool) of Mathcad.

*Tip
without number*.
Very often the «fruit» of the symbolic transformations «lie down» in
Mathcad-document badly. They are bad both in its form and its matter: reformed
by the operator «®» the
expression takes along the right pages across of the document; simplification
of the formulas is not done completely and etc. In this case the author
recommends to finish off all simplifications manually and to carry them in
Mathcad-document in the form of the figure following the technology in tip 56 relating to graphs of Mathcad.

_{} _{}_{ } _{}

In tip 16 the recommendations
about work with units of physical values wanting in the list of built-in in
Mathcad have been given. There is nothing difficult: *user's* units are
attached to *built-in ones *with help of *the factors* (MPa := 10^{6} Pa
end etc.). But this
rule cannot be applied for *temperature*. Point is that there are two
notions related to these physical values – *unit* of temperature (Kelvin,
degree Celsius, Fahrenheit degree and etc.) and *scale *of temperature
(Kelvin, Celsius, Fahrenheit and etc.).

Celsius
(or Fahrenheit if we speak about British system of measure) is the outboard
scale, which, nevertheless is widely used for *reflection* the value of
the temperature[5]. However the temperature of
absolute scale Kelvin (or Rankine in British system of measure) are used in
scientific and technical calculations. The formulas
of the conversion of the temperature from one scale to another one are simple
enough (see «Day’s tip» ¹74), but
above-mentioned technology (see tipå 16) of the input of user’s units does not work
in them.

One
of the solutions of this issue is showed on a simple problem in the fig. 64 – two *temperatures*
are given; we have to find a *difference* between them. It is clear, that it is not arithmetic problem, but
metrological one: all values have the
dimensionality of temperature. At that user can input and output the value of
temperature in any of four units and scales: Kelvin degrees (scale), Rankine,
Celsius and Fahrenheit. If there are no problems with degrees (scales)
Kelvin and Rankin (they are built-in in Mathcad: R = 1.8 K), then with respect to degrees
(scales) Celsius and Fahrenheit we have to use some ruses. These ruses are the
main point of the tip.

*Eight
objects* with the
names °C and °F are introduced into the calculation:

·
two
functions with the name °C (the first one °C(t) := (t+273.15) K style – Variables, and the second °C(t) := (T/K-273.15) – Units 1, at that the color a type of the name of the
second function is *white*. It is *invisible* on the screen of the
display[6])[7];

·
two
constants with the name °C (first one is °C := 1 style – Units 2, but the second one
is °C := K
– Units 3);

·
two
functions with the name °F (the first one is °F(t) := (t+459.67) R the style – Variables, and the
second one is °F(T) := (T-459.67) – Units 1, at that the color a type of the name of the
second function is *white*. It is invisible on the screen of the display);

·
two
constants with the name °F (the first one is °F := 1 the style – Units 2 and the second
one is °F := R
– Units 3).

The names of the objects are *the same*, but these are *different
objects*, as since they have different styles (Variably, User 1, User 2 è User 3).

When we work with temperature there are three situations. Above
mentioned function and constants help to react for these situations correctly:

Situation 1. It is necessary to ** input** the value of temperature on a
scale Celsius (or Fahrenheit). For it

Situation 2. It is necessary to ** output** the value of temperature on a
scale Celsius (or Fahrenheit). For this purpose in the operator «=» it is necessary to make the
variable that we are going to outtype as the operand of the

Situation 3. It is necessary to ** output** the value of a

The three above mentioned methods allow completely to realize work with
temperature: input of a value of temperature by any of four scales, output of a
value of the temperature, input and output of a value of a difference of the
temperatures.

Relative
scales meet in the engineering calculations not only during the work with the
temperature. If we want to know what pressure of water was at 100 m depth, then
we’ll obtain the reply: 10 atmospheres. But if we want to define any parameter
of water pressure dependent, water density for example, then we have to input into
the calculation not 10, but 11 atmospheres: to obtain *absolute* pressure
we have to add atmosphere pressure to *overpressure* (relative one).

Relative
scales are used, for example, in measure of length. The growth of an adult man
was measured in *vershoks* (vershok:=7/4 inch) in old Russia. But when one said
that some man has the growth of 12 vershoks, he had in mind two arshines and 12
vershoks (1 arshine=16 vershoks). Point is that the growth of an
adult man as rule kept within the range from 2 to 3 vershoks (from 142 to 213
cm). That’s why two arshines were usually left out, using relative (relatively
two arshines) scale of measure the length (this problem is described more
detailed in the sketch «Mathcad and some secrets of literature» – http://twt.mpei.ac.ru/ochkov/Gerasim/Gerasim.htm).

The
«tricks» that are used when you work with degrees Celsius and Fahrenheit (the prefix and postfix operators, the
invisible symbol of the operator) can be applied for work with decibels (see fig. 64a).

Bel
is the common logarithm of the ratio of two-dimensional values and the second
of them (denominator) is taken as some base. Decibel is correspondingly (see fig. 89) one tenth of the bel. Usually
decibels are connected with the acoustic intensity, but they can be applied to
any dimensional value. This way the value will be deprived its dimension.

See
http://twt.mpei.ac.ru/ochkov/Units/Part_1_eng.htm
too.

_{} _{}_{ }

The
question about points, on which the interval of construction of graph of the
function of one variable is divided, is toughed in tip 29. The number of such point is given either
by a user (see the first graph in the fig. 29) or by
the system Mathcad (the second graph). The description of the advantages of the
second technology is the main point of tip 29. In the
given tip we’ll speak about the defects of automatic choice of the points’
number and about ways of solution this problem.

If
a few lines, which visualize a few functions, are constructed at the same
graph, then, as rule, they are registered in different ways. Usually the color
is used for it in the screen of the display. But if it is necessary to print
the graph but a printer is black-and-white, then we have to mark in addition
the lines using circles, squares, criss-crosses[8]. In the case when the points are
near each other the marking signs can flow together in one line, which will
look like the openwork lace (see the second graph in the fig. 29).
On the other hand if we «disengage» the graph – to decrease the number of points,
which form it then the quality of the graph will become worse (see the first
graph in the fig. 65). Point is that the system
Mathcad draws the line[9]segments between the different
points and because of it the graphs are serrated. The way out is showed in the
same figure – in the fig. 65 where actually *two*
graphs of the same function are drawn. The first graph (the argument x1) is continuous line without marking
signs, but the second one (the argument x) is points without connecting line[10].

Two
methods of software-based forming of user’s function with the name η_{t} and three arguments P_{1}_{,} P_{2} and T_{1}[11], which returns … are showed in the fig. 66. But now we are interested in *the form of
the* function but not in what the function[12] returns, i.e. its content.

The algorithm of forming this function is very
simple: the values of three local variables I_{T}, I_{í} and q_{1 }are
calculated by line of the operators. The function returns the value of the
expression, which is formed of these variables – (I_{T} - I_{í}) / q_{1}. In the first function all operators are written abreast[13]. But in the second function all
operators are grouped so that the separate parts of the program are the
independent (see tip 30) notional blocks with
commentaries – calculations of values of the main variables which are part of
the «returnable» formula (I_{T} - I_{í}) / q_{1}. This increases appreciably the
«readability» of the program. Apart the separate parts can be withdrawn for
debugging and compilation. At the end of the fig. 66
this operation is executed about the variable x_{2}.

Two functions, which are showed in the fig. 66, differ not only by their *form *but also
by their *contents*. Point is that the call of the function h_{SS}[14]_{ }is duplicated in the different parts
at the second forming of the function._{ } It is made with a view of independent debugging of the program’s
parts.

_{ }

The
function minimum, which
is created with the help of programming, is presented in the fig. 67.
The function returns the coordinates of a minimum point of a function of many
variables (its name is the argument f) and it begins the search with the initial approximation points (the
vector X). The algorithm of search the
minimum is very simple. From the initial approximation point according to all
coordinates the step D is made (the third argument of the function minimum). In new places[15] the values of the minimizing
function are calculated. Where the function has minimum the crossing is made to
this point; the given point becomes new approximation point of minimum, but all
above-mentioned operations («probing» of the function close to the
approximation point) are repeated. If it turns that all new values of the
function increase close to the new point, then the step of search (D) will be decreased twice and all
operation will be repeated again. The search finishes when the value D becomes less then TOL
(the system variable of
Mathcad, which regulates the accuracy of calculations).

The
function minimum returns
not only *the vector* of values of the variable where the function is
minimum but also the *matrix* with the «history» of search. This matrix we
have to transpose (Ì^{Ò}), to define the number of the last
column (L) and to
calculate it (Ì^{<}^{L}^{>}). But such overload of the program
is not vain. The simplest but nevertheless very effective tool of debugging the
programs is observation the intermediate results (*the main point of the tip*):
the traces of search the minimum of two functions: õ^{8} + y^{6} and x^{8} + y^{6} + z^{6} are showed in the fig. 67.

_{ }_{}

It is possible to modify the function, which is
showed in the fig. 67 – to add one more argument:
maximum number of cycles of approximation to the minimum. The modified function
minimum is showed in the fig. 68.
*The main point of the tip*: when you debug a program, which has a high
probability of circularity, then it is worth to limit the number of execution
cycles. It is possible to take away the limitation of number of execution
cycles and to insert as the formal parameter N infinity (∞).

[1] What moment – it is the single question. Certainly there is not such information in the documentation of Mathcad, as since the function time is not documented. But we’ll be interested in the difference between two such values, i.e. the time of execution of one or another part of the program, but not the absolute value. By chance the author «met the function time, when he checked up «availability» of the same name variable for keeping the user’s value according to the method, described in tip 1.

[3] The author has found very serious mistake in the slammed tool. If in the area there is the operator that is inserted using the command Math Region from the menu Insert, then such operator turns off from the work after we slam the area. As we have said above the mistake was corrected.

[4] Symbolic mathematics (mathematics of analytical transformations) is enough «foreign body» in Mathcad. This way we can explain but not justify the fact that very little time is attended to symbolic mathematics in the given book.

[5] Celsius is not simply *widely*
used but it is *mainly *used in «human scale of measure «it is cold – it
is warm – it is hot».

[6] For we can see on the display it is
possible to change the color of *background* from white to blue, for
example.

[7] The symbol «°» (degree) is inputted with help of pressure Alt+0176.

[8] Lines on a graph can differ the thickness. But for identification the lines on the graph we do not need the quantitative differences (thicker – thinner) but we need the qualitative differences (if there are markers and what they are). One more way out is use dotted lines and the other lines with gaps. The firm-creator as it knew this defect in 2000-th version of Mathcad has provided for eventuality of marking the line on a graph by explaining text (or symbol – see the eagle in the figure of the bridge in the sketch «It is no trouble at all for us to built… a Bridge…» – http://twt.mpei.ac.ru/ochkov/Bridge/Bridge.htm). We have to refuse from the color on a screen of the display if potential user (viewer) is color-blind person: see the sketch «Color in programs» (http://twt.mpei.ac.ru/ochkov/Color_in_Program/index.htm).

[9] For example, the package Excel is able to smooth the points.

[10] This way the experimental points are marked on a graph.

[11] The function name and names of its formal parameters, as well names of local variables have textual index. Their defect is that all mix up textual index with numerical one.

[12] It returns cycle efficiency (coefficient of efficiency) of the simplest steam turbine cycle – the Rankine cycle, if it is interesting for somebody. By the way one of two units of temperature, which are built-in in Mathcad uses the Rankine name (see tip 64).

[13] For compactness a few operators are written on one line (see tip 33).

[14] h_{ss} is the enthalpy (h) of superheated steam. At the site http://twt.mpei.ru/orlov/waterprosteam
reader will find collection of the functions, which return heat-transfer
properties of water and water vapour. The main developer of the program is K.
Orlov (orlov@twt.mpei.ac.ru) – the
author’s student. He has done 55 % of work. This number is fixed during
registration of the program. The other 45% of work are distributed between À.À. Àlexandrov (alexandrovAA@mpei.ru), À.V. Ochkov (al@twt.mpei.ac.ru) and the author of the tips (ochkov@twt.mpei.ac.ru). The program WaterSteamPro™ is
transmitted by the technology shareware (see tip 55).

[15] There are four of them in two-dimensional problem; there are six of them in three-dimensional problem and etc. It is possible to optimize the search of a minimum if we’ll keep in mind the previous point. In this case when we optimize the function of two variables then, for example, we’ll have to «probe» not four but only three points.