Foreword the book by V.Ochkov “Units in Mathcad and Maple”
One good example of Error by using units in
calculation: Gimli
Glider
I tell you a funny story as an epigraph:
Dialogue during an exam:
Teacher:
What does a horsepower mean?
Student: It
is the power that a horse of one-meter growth and one kilogram weight has.
Teacher: But
where have you seen such horse?!
Student: But
you cannot look at it so simple. It keeps
in
Now days nobody is wonder at the fact that not
only candidates, engineers and students but also pupils solve their problems on
computer. But choice of program for these purposes astonishes. The question:
Why do a lot of people more often use Mathcad for calculation their problems.
Reply may be next: Mathcad has unique possibility[1]. Mathcad can operate as with simple
values as with physical values. Mathcad can be named physic-mathematical program.
Work in Mathcad is the
third step (and now days it is top one) of using computer engineering for
solving physic-mathematical, technical and educational problems. Two previous
steps are work with computer codes (for example with assembler language) and
with programming languages (BASIC, Pascal, C, FORTRAN etc). These two
technologies of scientific and technical calculations (computer codes and
programming languages) played spiteful joke with these calculations.
Dimensionalities of physical values and its units (meters, kilograms, seconds
and other ones) were excluded. Handmade solution of a
physical problem (school or student ones) as rule demanded and demands using of
dimension values[2].
Automation of such calculations (that is composition of programs for computer)
excludes «physics» of a problem. These programs keep only the numerical values
of variables, but corresponding them units programmer
has to «keep in mind». So when we interpret a calculation into computer
language it is necessary to follow the strict rule – all physical values must
1.
International
system SI will never be unique one though it is widely used. For instance,
2.
We
can image the process of creation of a program without its debugging. The main
tool of debugging is output intermediate results. So analyzed these results we’
3.
In
programs we have to put in some formulas that are derived basing both on
theoretical analysis of a problem and getting after static treatment of
experimental data. As rule coefficients of such formulas (for instance, see
picture. 1.28) are rigidly connected with fixed system of measure. These
coefficients demand conversion for using in the programs. It can bring in
additional error hence it will be cause of mistakes.
Some words about the design of the book.
When this book will be published at the same time
its Internet-version will appear (see http://twt.mpei.ac.ru/ochkov/unit/index.htm). Because of Internet version the design of
the book is special. For instance, hyper-links are displayed: see picture. 1.3, see below, see footnote 3
etc. Working with the book if a reader meets hyper-link he will have to go to
necessary page himself. In Internet-version the reader can go to new place and
come back easier (with help of help-link) – he has to click his mouse by
hyper-link. In Internet-version author will make alterations and addition. Author hope that readers will send them to address ochkov@twt.mpei.ac.ru.
Exclusion of units of physical values from the
calculations has told on the work in Mathcad. Many inexperienced users begin to
form the document in Mathcad-this way
They write dimension of some physical value as
a commentary but not as a factor of numerical constant. So they exclude the
dimension of this physical value from the subsequent calculations. For
instance, if we remember the language BASIC, above mentioned example will have
next form in it: P = 20: Rem Pressure in MPa. Here units are written in
commentary (in note – Rem). Using this algorithm a programmer can forget that pressure is in MPa, but not in bars. Finally he will get a mistake.
Mechanism of
the work in Mathcad with units of physical values allows:
·
to put in basic data in necessary system of
measure, in necessary units with necessary factors (kilo, milli
etc);
·
to put in a control of dimensions in
formulas, that we use in our calculation (for instance, we can
not sum kilograms and meters)[9];
·
to put out calculated values in necessary system and units,
to use convenient scale of the axes of plots etc.
One of the ways to solve this
problem is a failure of dimensional values. We can go to non-dimensional numbers that keep quantitative assessment of
physical phenomenon: Re[10], Nu, etc (see the table ??? in the third part of the book). When we say that
the pressure in a boil equals 50 atmospheres then we operate with
non-dimensional value. We mean that the pressure 50 times as much of
atmospheric one. And more radical way is from numerical characters we go to
qualitative assessments. We can say that pressure equals some number of
atmospheres, but we can also say that pressure is normal (high, low etc) and form
Mathcad completely support mathematics of work with dimension variables.
We’ll speak about it below and look through some simple examples.