— a software package that can be used for various purposes. You can think of it as a stuck-in calculator, which can not only calculate arithmetic operations, but also integrate, multiply matrices and graph functions. At the same time, this system includes daily programming, which is procedural (parallel), object-oriented and applied in one bottle. In addition, you can integrate with MatLab, and also allows you to call external program compilation procedures onto Cі Fortran. Zagalom vin allows you to create a powerful program for obvious reasons. This middle ground allows you to create prototypes for various technical and scientific developments by then writing code with other languages.
Such diversity of features of this middle ground can complicate its conversion into a distant victorious process. As a rule, the cob is cultivated using the method of ticking with the vikoristanny of the administrated Help. It’s a pity that I didn’t understand the riddles of my Russian friends. Here are three instructions in English that you can read in order:
All three can be found with private access. In this section I will share my knowledge of working in Maple and putting the robot into practice and fine-tuning.
Start naturally with the installation. The system is not cheap, but for information you can find it on various torrents, for example https://rutracker.org/. Yak for Linux, so for Windows.
Having installed, we launch and most importantly the column with Pallettes|Workbook on the right is a window with a large number of icons divided into two parts:
You can now get acquainted with a variety of ready-made applications of various topics in mathematics, programming, and natural sciences - by clicking on the icons on the right side of the window and launching related documents Maple. These documents can be edited and saved. We begin the independent robot by clicking on the right side of the window New Document or else New Worksheet. The difference between these two types is small, as shown in the table. We will continue to vikorist Worksheet. The rows to be added are marked here with a [> . Maple commands are shown in bold font. After the [> icon, you can enter commands that end with either a dotted line or a doubletted one. For example, for an arithmetic operation:
sin(3.)+1;
After the command it is embossed Enter or a bear sign! beast on the menu. Respect the difference between such a team and two similar ones: sin(3)+1;і sin(3)+1: The result will not appear on the screen - because... The command is completed with the sign:.
This sign is used if you do not need the result, and it is too cumbersome. For example, after entering a value: a:=Pi:
We will not require confirmation that a is older than 3.1415…. We can continue to vikorize like this:
b:=2*a;
the result will appear as $b:=2\pi$. Respect that we invite you for help := . The primary sign of zeal of vikoryst is expressed in a slightly different way. Maple knows what Pi means the number $\pi$. To extract this numerical value, you need to use a special function like this:
b:=evalf(2*a);
As a result, we remove the value of the number pi with the accuracy that is currently being calculated in Maple. And respect here! Maple You can use a variety of significant numbers. For promotional items 10 characters. This number can be easily changed. I write on the document itself like this:
restart: Digits:=16:
Thus, the document will have 16-digit numbers. This number can be changed, the maximum value for your OS can be obtained using the command
kernelopts(maxdigits);
I have a maximum value of Digits = 38654705646. Please note that I am also a victor on the cob team restart: This command is very handy, since you edit the document within an hour of one session, for which you do not need to restart Maple closing and opening yogo. You just press the button after correcting !!! at the top panel Maple And everything will be overhauled with new updates of all changes.
U Maple There are many ways to present the function.
Method 1. Assignment of functions to additional operator ( := ): I guess it’s called for, for example:
> f:=sin(x)+cos(x);
How to specify the specific value of the change X, then you will see the value of the function f for whom X. For example, if you continue to live the front butt and calculate the values f when , record the following:
> x:=Pi/4;
After the demise of these teams X May have a given value.
To avoid specifying a specific value at all, it is better to use the substitution command subs((x1=a1, x2=a2,…, ),f), where the figured arms have the symbols xi and their new meanings ai(i=1,2,…), which trace should be substituted for the function f . For example:
> f:=x*exp(-t);
> subs((x=2,t=1),f);
All calculations in Maple behind the minds they vibrate symbolically, so that the result is misplaced by clearly irrational constants, such as these. To find the closest values to the numbers with the floating coma, follow the command evalf(expr,t), de expr- Viraz, t- Accuracy, expressed in numbers after the coma. For example, by stretching the front butt, the numerical value of the function is brought closer:
> evalf(%);
Here is the Wikorista symbol ( % ) to call out the front command.
Method 2. Significant functions behind an additional functional operator, which is assigned to a set of variables (x1, x2, ...) one or a few virasivs (f1, f2, ...). For example, the assigned functions of two substitutions for an additional function operator look like this:
> f:=(x,y)->sin(x+y);
Reversing this function is done in the most mathematically analogous way, if instead of the arguments of the function, the specific values of the variables are assigned to the handles. By pulling the front butt the function values are calculated:
Method 3. For additional commands unapply (expr, x1, x2, ...), de expr- Viraz, x1, x2, ...- A set of changes, such as those that are stored, can be converted into expr y functional operator. For example:
> f:=unapply(x^2+y^2,x,y);
U Mapleє possibility of assigning non-elementary functions of the form
call for help
> piecewise(cond_1, f1, cond_2, f2, …).
For example, function
register in this way.
MATHEMATICAL RESEARCH ZAVDAN
U MAPLE
PARTINA I
FEDERAL AGENCY FOR INVENTORY
State lighting installation
“Nizhny Novgorod State University named after. »
MATHEMATICAL ZAVDAN IN MAPLE
faculty for students who are learning
direct preparation 010100 – “Mathematics”.
Nizhny Novgorod
UDC 621.396.218
Unraveling tasks for MAPLE. Part I. Commanders: , : Basic-methodical technology. - N. Novgorod: Publishing House of the Nizhny Novgorod State University, 2007. - 35 p.
Reviewers:
Associate Professor of the Department of Chefa, Faculty of Computational Mathematics and Mathematics,
Ph.D. n. ,
Associate Professor of the Department of Physical Education, Faculty of Physics,
This is a practical guide to understanding the capabilities of the analytical calculation package Maple. Consistent learning of topics and a new task will allow you to master, step by step, the basic techniques of working in a mathematical system Maple.
Basic methodological development is intended for students of the 2nd and 3rd courses of the Faculty of Mechanics and Mathematics.
UDC 621.396.218
© Nizhny Novgorod State
University im. , 2007
Computer algebra systems – the price of new technologies from scientific research and awareness. In recent years, there has been a widespread expansion of such systems of legal significance, such as Maple, Mathematica.
The Maple system is included in the integrated Scientific WorkPlace system and is used by many leading universities around the world for both scientific research and the initial process. The Maple core is included in other advanced packages, such as MathCad, MathLab.
The opportunity is given to allow the cob to advance to the technology of the Maple vicor system, to remove the first beginnings, after which it can independently grow in more fine feeds of the Maple vicor system. I would like to point out that this technology does not necessarily describe the Maple system. Vaughn is assigned to the first staff to teach mathematics students the most basic mathematical tasks under the supervision of Maple.
1. COB VIEWS. TYPES OF DATA
The dialogue with the system proceeds in a “nutrition-supply” style. The command begins with the symbol > and will end at the coma point ( ; ), or double ( : ). To execute a command you need to press the key Enter. If there is a dot at the end of the command, the result of the command or a notification about the strike will be displayed on the screen. The double box at the end of the command means that the command will be entered, otherwise the result will not be displayed on the screen. Symbol # Use wiki to enter text comments. Also, to enter text, use the T key on the toolbar. To go back to entering commands, press the > key. To click on the result of the previous commands, the symbols %, %% or %%% are used. Team restart scales the result of all leading teams.
Changes in Maple are characterized by their type. The names of variables in Maple can be composed of letters, numbers and many special characters, but they can also begin with letters. There is no deduction for the dowzhine of the name. In addition, Maple differentiates between small and great letters. To assign a specific value, the operator is required to := . Changes can be used in mathematical expressions and functions without prior designation.
Let's take a look at the features of recording data in Maple for numeric, row and multiply types.
Viraz goes up to the whole type ( integer), which consists of a sequence of numbers that are not separated by the same signs. Virazis of the form a/b, de a, b – the whole number is placed before the shot type ( fraction). Up to the number with a floating point ( float) appear in the form of a. b, a. V. b. The same type of numbers float can be written in the display form a*10^b. Complex numbers ( complex) in Maple are written in algebraic form: a + I * b, where a, b are speech numbers.
Row viraz tipu string- whatever the end sequence of symbols, on the other side is placed at the upper side of the foot. The sequence of symbols, taken from the collar foot, is represented by the symbol ( symbol).
Bezlich ( set) in Maple it is specified to rearrange the curly arms of the elements of the multiplicity. For example,
> A:=(x^n$n=1..6);
> B:=(a, a, b, b, b, c);
https://pandia.ru/text/78/155/images/image003_72.gif" width="197" height="26">
To create an array, use the command array(i1..j1, i2..j2,..., M), which rotates the array with elements from the list M.
Collapse to elements of a multiplicity, list, array is obtained from the assigned indexes of the element in the square arms.
> V:=array(1..2,1..2,1..2,[[,],[,]]);
https://pandia.ru/text/78/155/images/image006_53.gif" width="16 height=19" height="19">
An array can be specified with a command like V:=array(1..2,1..2,1..2,); , having re-signified the value of V to the additional assignment operator.
At Maple you can write letters of the Greek alphabet in a printed form. For this reason, the command line is called the Greek letter.
> beta+Gamma+delta;
Zavdannya 1.1
1. Specify the multiplier A, which is the sum of the whole numbers from 3 to 20, and the multiply B, which is the sum of the squares of these numbers. Find the ratio, the difference, the difference of the multipliers A and B. Find the strength of all the subtracted multipliers.
2. Select more list and more than one array.
2. ARITHMETIC OPERATIONS, FUNCTIONS. REVISION OF ARITHMETIC
VIRAZIV AND RISHENNYA RIVNYAN
2.1. Calculation in Maple
To write mathematical expressions, Maple uses the add (+), add (-), multiply (*), sub (/), reduce (^), and assign (:=) operators. The order of mathematical operations is standard.
butt.
> (a*b^4-(a*b)^4)/7;
The main constants in Maple are defined as follows: Pi- Number π, I- one i is visible, exp(1)- the basis of natural logarithms e, infinity- Infinity, true- truth, false- nonsense. The following signs of equalization are used: <, >, >=,<=, <>, = .
The Maple system, however, successfully copes with both symbolic and numeric calculations. For the promochans, rozrahunki are carried out symbolically.
butt.
>1/2+123/100+ sqrt(3);
The part of the expression that has the sharpest number, written with a floating coma (float), will be calculated approximately.
butt.
>2+ sqrt(2.0)- Pi;
All standard calculations are carried out in ten significant figures. A number of numbers can be changed by issuing the command > Digits: = n.
In order to remove the value of the expression from the numerical view, use the function
https://pandia.ru/text/78/155/images/image012_43.gif" width="414" height="19">
2.2. Setting the function
Maple has a great number of functions. The assignments for basic elementary functions are being overridden.
Let's look at a number of ways to implement new functions:
1) giving a change to the active virus
Name of change: = viraz;
Change name (list of parameters): = viraz;
butt.
> f(t):= cos(t)^2+1;
> f(0);
With this method of assigning a function, in order to calculate the values of the function at any given point, it is necessary to use the operator to assign the values of the variable (parameters) or to use the substitution operator subs.
https://pandia.ru/text/78/155/images/image018_28.gif" width="106" height="33">
> value(%);
https://pandia.ru/text/78/155/images/image021_25.gif" width="100" height="33">
>x:= Pi: y:=10: f;
Team value(viraz) Vikorist is used to calculate the value of the virus.
Please note that after assigning variable x a specific value x:=a, variable x will no longer be unvalued. You can reverse the status of an undesignated change using the command > x:= evaln(x); or take it away from the command > x:=’ x’ ; or redeem everything assigned by the command restart.
2) Assignment of functions to an additional functional operator
Function names: = list of parameters-> viraz;
Returning to a function specified in this way is represented by the standard form: the name of the function (a, b, …), where a, b, … are the specific values of the variables.
butt.
> f1:=(x, y, z) -> x^(y^ z);
> f1(2,2,2); f1(x,2,2);
https://pandia.ru/text/78/155/images/image024_25.gif" width="25" height="26 src=">
3) The function can be installed after an additional command
unapply (viraz, parametr) How does the expression transform into a functional operator?
butt.
> f2:=unapply(sin(x)^2+2*exp(y^2),x, y);
> f2(Pi/4,1);
https://pandia.ru/text/78/155/images/image027_22.gif" width="189" height="107">vicoris team
https://pandia.ru/text/78/155/images/image028_21.gif" width="248" height="77">
> f1:=convert(f, piecewise);
> f2: = unapply(f1, x);
> f2(-1/2); f2(-1);
https://pandia.ru/text/78/155/images/image032_20.gif" width="196" height="27">.gif" width="73" height="49">. Forgive the expressions.
3. Find out the meaning of the virus 
. To create a complex number, a function is used evalc.
4. Write the function 
without module sign.
5. Install 
and find f(-10)+3f(-1)+f(3).
6. Install a function 
Using the functional operator, you can find the values for x=-1, y=.
2.3. Reinvention of mathematical expressions
Maple has a wide range of capabilities for analytical transformations of mathematical formulas. These include operations such as bringing similar ones, decomposing them into multipliers, opening the arms, bringing the fraction to a normal appearance, and many others.
At Maple, you can transform both sides and sides.
To see the left (right) part of the mathematical expression A = B, use commands
lhs(Viraz);
rhs(Viraz);
To see the figure (sign), the following commands are used:
number(Viraz);
denom(Viraz);
butt.
>F:=(a^3+b)/(a-b)=3*a^2+b^2/(a+b);
>numer(rhs(F));
>denom(rhs(F));
To see the song part virazu, list and multiply, use the command
op(i, Viraz), where i is the number that indicates the position of the virus.
butt.
> x+ y+ z; >op(2,%);
Gif" width="12" height="12 src="> isolate(Rivnyannya, viraz);
butt.
> P:= 2* ln(x)* exp(x) -3* exp(y)+7=10* ln(x) - exp(y):
> isolate(P, y);
> R:=5*(x^2)*sin(x)+1=5*sin(x):
> isolate(R, sin(x));
1) Bringing similar members to the displayed change is carried out by the command
https://pandia.ru/text/78/155/images/image047_14.gif" width="439" height="28">
2) You can visualize multipliers using additional commands
https://pandia.ru/text/78/155/images/image050_16.gif" width="186" height="56">
>factor(x^4-3*x^3+2*x^2+3*x-9);
>factor(x^3+x-3*sqrt(2));
>factor(x^3+16, (2^(1/3),sqrt(3)));
>alias(w=RootOf(x^3+2*x+1)); factor(x^3+2*x+1,w);
https://pandia.ru/text/78/155/images/image055_15.gif" width="504" height="26 src=">
> convert(%,radical);
https://pandia.ru/text/78/155/images/image057_17.gif" width="303" height="57">
> factor(x^2+x+1,complex);
Gif" width="12" height="12 src="> expand(viraz, options), where in the options you can specify a viraz, the bow should not be opened. This command is also used for working with exponents and reducing trigonometric expressions to trigonometric functions of simple arguments.
butt.
>expand((x+1)*(x+2)*(x+3)*(x+4));
>expand((x+y)*(x+3), x+3);
>expand(exp(a-n*b+ln(c)));
>expand(tan(3* x));
4) You can bring your dribble back to normal appearance with the help of an additional command
https://pandia.ru/text/78/155/images/image063_16.gif" width="60" height="54">
>normal(sin(2*x+3+4/(x-1)+5/(x-2)), expanded);
5) To recreate viruses that will take revenge on radicals, a team will be created
rationalized" in order to avoid irrationality among the bannermen, " expanded" to open all the arms.
butt.
> (7+5* sqrt(2))^(1/3);
> radnormal((7+5* sqrt(2))^(1/3));
> a := sqrt(3)/(3^(1/2)+6^(1/2));
rationalized");
6) Simplification of expressions is carried out by the command
DIV_ADBLOCK515">
butt.
>(sqrt(2)+sqrt(3))*(sqrt(2)-sqrt(3));
>simplify((sqrt(2)+sqrt(3))*(sqrt(2)-sqrt(3)));
> f:=(1-cos(x)^2+sin(x)*cos(x))/(sin(x)*cos(x)+cos(x)^2); simplify(f, trig);
Options also have assumptions about the value of the argument. For formal symbolic transformations of richly meaningful functions, you can specify symbolic.
butt.
> g:=sqrt(x^2);
> simplify(g, assume=real);
> simplify(g, assume=positive);
>simplify(g, symbolic);
The simplify command allows you to change the expression for the tasks of the minds (the minds are indicated in the curly arms).
butt.
> f: = -3 * x * y + x + y: simplify (f, (x = a-b, y = a + b));
In such situations, it may be difficult to change the situation behind the scenes using additional commands
https://pandia.ru/text/78/155/images/image076_12.gif" width="276" height="54">
>simplify(B, trig);
>convert(%,tan):
>simplify(%);
7) You can unite the parts of the viraza following the singing rules with the help of a command
https://pandia.ru/text/78/155/images/image079_12.gif" width="94" height="25 src=">, when specifying the option ln Potency is generated. So, just like for simplify in the options, you can enter symbolic.
butt.
> combine(exp(sin(a)*cos(b))*exp(-cos(a)*sin(b)),);
>combine((a^3)^2+a^3*a^3);
Gif" width="12" height="12 src="> solve (rіvnyannya, change).
It’s important to overreact in the curly arms through a coma. If you do not specify a set of changes in the command parameters, a solution will be found for all changes that take part in the competition. If it is necessary to balance the alignment system, then the alignment of the system is indicated in the curly arms through a coma. The result of issuing the solve command will be a list of solutions for this task, since the search cannot make a decision and the solve command cannot be found, and the output will not appear every day. The list of solutions can be used in the same way as the original list.
butt.
> eq:=(x-1)^3*(x-2)^2;
> s:=solve(eq);
> solve(x^4-11*x^3+37*x^2-73*x+70);
https://pandia.ru/text/78/155/images/image086_12.gif" width="349" height="22 src=">
>e:=solve(AX);
>rhs(e); subs(e, z);
DIV_ADBLOCK517">
butt.
>e1:=(x^2-y^2=1,x^2+x*y=3);
> s:=solve(e1,(x, y));
> _EnvExplicit:=true:
> solve(e1,(x, y));
The maximum number of solutions that can be found using an additional solve command is given by the values of the global variable _MaxSols, which has a value that is equal to 100. How to add a global variable _EnvAllSolutions significance true, then when an infinite number of solutions has been solved, the solve command for certain reasons can generate a response in the form of a virus, which will replace the changeable type. For example, for trigonometric equations, the answer is integer parameters of the form _Z~.
butt.
> _EnvAllSolutions:= true:
>solve(sin(2*x)=cos(x), x);
https://pandia.ru/text/78/155/images/image094_11.gif" width="274" height="51 src=">.gif" width="12" height="12 src="> fsolve (equalities, changes, options).
In the options you can specify the interval in which the search for roots is carried out, you can also specify complex - to find all complex roots, or the option maxsols=n– to find the n least roots of a polynomial. If the value is set by a polynomial, then the command fsolve If all the nearby speech roots are found, in the literal expression the fsolve command will find only one numerical root, equal, or another root can be searched for, changing the search interval so that the root can be found without entering again.
butt.
> fsolve(x-cos(x));
https://pandia.ru/text/78/155/images/image097_10.gif" width="641" height="23">
For maximum recurrence, the command is stagnated
https://pandia.ru/text/78/155/images/image098_10.gif" width="255" height="22 src=">
> rsolve(e1,f);
> rsolve((e1,f(0)=1,f(1)=2),f);
With the help of the solve command, you can also determine the inequities and the systems of equalities and inequalities.
butt.
> solve(ln(x)<1, x);
https://pandia.ru/text/78/155/images/image102_8.gif" width="119" height="23 src=">
> solve((x-y>=1,x-2*y<=1,x-3*y=0,x+y>=1),(x, y));
https://pandia.ru/text/78/155/images/image104_7.gif" width="180" height="56">.
2. Forgive the expression.
3. Forgive the virus.
4. Find similar expressions and calculate its value at a=-3, x=1.
5. Forgive viraz a) 
; b) 
.
6. Avoid irrationality with the famous person.
7. Virat, in radicals.
8. Bring that A, B, C are the kuti of the tricut.
9. Express through width="164" height="41">;
b) https://pandia.ru/text/78/155/images/image120_7.gif" width="88".
11. Unfold the penis 
into multipliers over the field of operational numbers and over the field of rational numbers. Find out the layout of the radicals.
12. Factor the term into multipliers over the field of action numbers and over the field of complex numbers. Find out the layout of the radicals.
13. Unleash jealousy 
.
14. Untie the system of ranks 
.
15. Untie the system of ranks 
and forgive me the confession.
16. Numerically know all the solutions to the equation 
.
17. Find three numerical solutions.
18. Unleash the system of nervousness.
19. Release anxiety.
3 . WAKE-UP SCHEDULES
This part is devoted to the capabilities of the Maple V system for visualizing highly complex calculations.
3.1. Two-world graphics
For wake-up schedule functions f(x) in one change (in the interval https://pandia.ru/text/78/155/images/image132_6.gif" width="69" along the axis Oh) vikory team
plot(f(x), x=a..b, y=c..d, options),
de options- Option or set of options to set the style of the daily schedule. If they are not specified, there will be vicoristan installations for the calculations. You can also customize images from the toolbar. To do this, press the left mouse button onto the image. After this, the active buttons are in the bottom row of the panel. You can also find out the coordinates of a point on the graph. To do this, you need to move the cursor to the required point on the graph and click the left mouse button. Left-handedly the bottom row of buttons on the panel will show the coordinates. You can also customize images using the context menu. You can click with the right mouse button.
Basic command parameters plot:
title=”text”, de text- The title is small (the text can be deleted without legs, so as not to fill the clearings).
coords=polar – installation of polar coordinates (by default, Cartesian coordinates are installed).
axes- Setting the type of coordinate axes: axes=NORMAL- Primary axes; axes=BOXED- Graph with frame and scale; axes=FRAME– axis centered on the lower left cut of the baby; axes=NONE- Without axles.
scaling- Set the scale of the baby: scaling=CONSTRAINED– however, the scale along the axes; scaling=UNCONSTRAINED- The graph is scaled to fit the size of the window.
style= LINE- Displayed by lines, style= POINT seen in speckles.
numpoints=n- Number of calculated points on the graph (based on calculations) n=50).
сolor– setting the color of the line: English name for the color, for example, yellow- zhovty etc.
xtickmarks=nxі ytickmarks=ny- Number of marks along the axis Oh and axes Oh obviously.
thickness=n, de n = 1,2,3...- Tovshchina line (for washing n=1).
linestyle=n- Line type: continuous, dotted, etc. (depending on n=1- Without interruption).
symbol=s - type of symbol used to represent dots: BOX, CROSS, CIRCLE, POINT, DIAMOND.
font=– setting the font type to display the text: f sets the name of the fonts: TIMES, COURIER, HELVETICA, SYMBOL; style sets the font style: BOLD, ITALIC, UNDERLINE; size- Font size in pt.
labels=- Writing along coordinate axes: tx– along the axis Ohі ty– along the axis Oh.
discount=true- Indication for encouraging continuous developments, in which asymptotes are not visible on the chart.
butt. Check your schedule https://pandia.ru/text/78/155/images/image134_1.jpg" width="292 height=292" height="292">
Weekly graphics of a function specified parametrically
Command for help plot It is also possible to create graphs of functions and tasks parametrically y=y(t), x=x(t):
plot(, parameters).
butt. Create a graph of a parametric curve https://pandia.ru/text/78/155/images/image138_2.jpg
Everyday graphics of a function specified implicitly
For a weekly schedule of an implicit function F(x, y)=0 vikorist team https://pandia.ru/text/78/155/images/image139_5.gif" width="80" height="27">.
>with(plots):implicitplot(x^2+y^2=1, x=-1..1, y=-1..1);
Gif" width="12 height=12" height="12"> textplot(, options), de x0, y0– coordinates of the point where the text begins 'text'.
Displaying many graphic objects per drawing
If one little one needs to learn a bunch of graphical functions, you can quickly use the command
plot((f1(x), f2(x), …), options);
If it is necessary to draw a number of graphic objects that are drawn using various commands, then the result of the commands must be changed in the following ways:
> p:= plot(…): t:= textplot(…):
When you put it on the screen, what you see is not displayed. To display graphic images, you need to enter the command from the package plots:
display(, options).
butt. Create function graphs https://pandia.ru/text/78/155/images/image142_6.gif" width="73" width="59" "> for one little one.
> with(plots):
> p1:=plot(sin(x), x=-5..5, y=-2..2, thickness=3, color=blue):
> p2:=plot(cos(x), x=-5..5, y=-2..2, thickness=3, color=green):
> p3:=plot(tan(x), x=-5..5, y=-2..2, thickness=3, color=black):
> p4:=plot(ln(x), x=-5..5, y=-2..2, thickness=3, color=red):
> display(p1, p2, p3, p4);
https://pandia.ru/text/78/155/images/image146_5.gif" width="297 height=24" height="24">, then for this you can use the command inequal 3 packages plots:
inequals((f1(x, y)>c1,…,fn(x, y)>cn), x=x1…x2, y=y1..y2, options)
The figured arms have a system of irregularities that indicate the area, then the dimensions of the coordinate axes and parameters. Using additional parameters, you can adjust the thickness of the line cordons, the color of open and closed between, the color of outer and inner areas:
.gif" width="12" height="12 src=">optionsexcluded=(color=yellow)- Setting the color of the external area;
.gif" width="12" height="12 src=">optionsclosed(color=green, thickness=3)– setting the color and line of the closed cordon.
Zavdannya 3.1
1. Create a schedule https://pandia.ru/text/78/155/images/image148_6.gif" width="75" height="43">.
3..gif" width="72" height="20">, at the frame.
4..gif" width="83" height="23 src=">
> plot(1-sin(x^2), x=0..2*Pi, coords=polar, color=black, thickness=4);
5. Create a hyperpain chart: .
6..gif" width="75" height="20 src=">) inscribed in an ellipse. Sign the lines in bold italic font.
> with(plots):
> eq:= x^2/16+ y^2/4=1:
> el:=implicitplot(eq, x=-4..4, y=-2..2, scaling=CONSTRAINED, color=green, thickness=3):
> as:=plot(, color=blue, scaling=CONSTRAINED, thickness=2):
> eq1:=convert(eq, string):
> t1:=textplot(, font=, align=RIGHT):
> t2:=textplot(, font=, align=RIGHT):
> t3:=textplot(, font=, align=LEFT):
> display();
7. Create an area surrounded by lines: , , .
> with(plots):
> inequal((x+y>0, x-y<=1, y=2}, x=-3..3, y=-3..3,
optionsfeasible=(color=red),
optionsopen=(color=blue, thickness=2),
optionsclosed=(color=green, thickness=3),
optionsexcluded=(color=yellow));
3 .2. Three-dimensional graphics. Animation
Graph of a surface defined by an explicit function
The function graph can be generated using the vikoryst command
plot3d(f(x, y), x=x1…x2, y=y1…y2, options).
The parameters of the command are often shared with the parameters of the plot command. Up to frequently used command parameters plot3d also said
light=- Defined highlighting of the surface created from a point with spherical coordinates ( angl1, angl2). The color is indicated by parts of red ( c1), green ( c2) and blue ( c3) colors that are found in the intervals.
style=opt sets the style of the baby: POINT-points, LINE- Lines, HIDDEN- Mesh with visible invisible lines, PATCH- Zapovnyuvach (installations for zamovchuvannyam), WIREFRAME- Mesh with invisible lines shown, CONTOUR- Lines of the river, PATCHCONTOUR- Replenishment and lines of the level.
shading=opt defines the function of the intensity of the refill, whose value is similar xyz- for getting ready, NONE- Without rose painting.
Three-dimensional images can be manually adjusted using the additional command option plot3d, and the vikoryst context is not the program menu. To do this, right-click on the image. To open the context menu for adjusting the image. The commands of this menu allow you to change the image color, lighting modes, set the type of axes, type of lines. Similarly, as for two-dimensional graphics, you can activate the bottom row of buttons on the toolbar by clicking the left mouse button on the image. You can rotate the image using additional buttons on the panel or by pressing the left mouse button.
butt. Apply the surface at the same time to the lines of the level
https://pandia.ru/text/78/155/images/image160_0.jpg" width="321" height="198">
Surface graph defined parametrically
If a surface is required, it is specified parametrically: x=x(u,v), y=y(u,v), z=z(u,v), then these functions are overreacted in the square arms of the team:
plot3d(, u=u1..u2, v=v1..v2).
butt. Pobuduvati tor.
> plot3d(, s=0..2*Pi, t=0..11*Pi/6, grid=, style=patch, axes=frame, scaling=constrained);
https://pandia.ru/text/78/155/images/image162_4.gif" width="99" height="24">, I will be looking for additional commands for the package plots:
implicitplot3d(F(x, y,z)=c, x=x1..x2, y=y1..y2, z=z1..z2), This indicates the level of the surface and the size of the baby behind the coordinate axes.
Graph of space curves
In the package plotsє team spacecurve for a random spatial curve, specified parametrically: .
spacecurve([ x(t), y(t), z(t)], t= t1.. t2) , de zminna t changes to t1 to t2.
Pobudova several trivial figures on one chart
Team plot3 d Allows there to be a handful of figures at once, shuffling around in the open space. To do this, it is enough to replace the description of one surface with a list of descriptions for a number of surfaces. Why the function plot3 d It has unique capabilities - it automatically calculates the points of the webbing of shapes and shows only the visible parts of the surface. This creates images that look completely natural.
butt. Vikonati pobudovu two surfaces 
within width="39".
> plot3 d({ x* sin(2* y)+ y* cos(3* x), sqrt(x^2+ y^2)-7}, x=- Pi.. Pi, y=- Pi.. Pi, grid=, axes= FRAMED, color= x+ y);
Animation
Maple allows you to display images on the screen using additional commands animate(two-world) and animate3d(trivimirni) from the package plots. The essence of animation with these functions lies in a series of frames, each frame of knitting with the values of variable t. Among command parameters animate3dє
frames– number of animation frames (depending on frames=8).
For images that are collapsing, it’s best to use the context menu.
Zavdannya 3 .2
1. Create a surface graph.
2. Stay cool 
:
3..gif" width="65" height="21 src=">.gif" width="173 height=53" height="53">.gif" width="95" height="48 src=" >.gif" width="71" height="23 src=">.
Enter the name of the baby, write the names of the axes and set the new scale along the axes.
6. Paint an object that is collapsing - a Moebius strip.
4 . MATHEMATICAL ANALYSIS
Let's take a look at the main functions of the advanced task of mathematical analysis, embedded in the Maple package.
4 .1. Between functions and differentiation
Payments between the teams are due
.gif" width="12" height="12 src="> Limit(Viraz, x = a, parameters).
butt.
>Limit(ln(cos(a*x))/(ln(cos(b*x))), x=0)=limit(ln(cos(a*x))//ln(cos(b*x ) ))), x=0); width="215" height="58 src=">
The differentiation in Maple is based on the other team
DIV_ADBLOCK519">
https://pandia.ru/text/78/155/images/image182_4.gif" width="262" height="54">
10. PROGRAMMING AT THE MIDDLEMAPLE
The Maple mathematical package allows computer scientists to compile computer programs, procedures and libraries. For this purpose, the package needs to have a wide range of commands and a design similar to high-level algorithmic programming language.
10.1. Brain operator
Maple's mental operator begins with a reserved word if And it’s bound to end, in a word fi It has the following structure:
if Umova then viraz 1 else viraz 2 fi ;
This design makes it possible to place, depending on the value of the logical mind, either Viraz 1 (like the mind is true) or Viraz 2 (like the Mind's mind). Both expressions 1 and 2 can follow any sequence of commands from the Maple package. The mental operator can make notes in a short view:
if Umova then viraz 1 fi ;
[> restart;
[> x: = 4;
x:=4
[>if x>4 then print ('x>4'); else x:=x^2;
print(2*x); fi;
32
p align="justify"> For the implementation of flexible minds, it is necessary to develop a new version of the mental operator, which may lead to the structure.
if Umova 1 then viraz 1 elif Umova2 then viraz2... elif Umova n then viraz n else viraz n +1 fi ;
As follows from the structure of this operator, the contribution of minds can be practically unbound and implemented through the use of a service word. elif . How you can express it differently is the sequence of Maple commands.
[> restart;
[>x:=8:
[>if x
x:=c
10. 2 . Loop statements
The Maple mathematical package for implementing a cyclic computational process includes several types of operators in the cycle. The core of all loop operators is the sequence of commands placed between service words do і od . The loop operator of the over-reinforced type, which is found in almost all algorithmic languages, has the following structure:
for Change cycle from cob value of the harvest cycle by time period for increasing the value of the change cycle to end value of the change cycle
[>for i from 0 by 4 to 8 do i od;
0
4
8
The operator for a loop like “yet” in Maple looks like this:
while Umova do viraz od ;
Once the body cycle (viraz) ends until the hour when the meaning of the logical mind is truly accepted, as the mind is hypnotic.
[> restart;
[>n:=0:
[>while n
1
2
9
The preceding operator of the cycle is a symbiosis of the two preceding ones and has the following structure:
for Change cycle from cob value of the harvest cycle by value of increased amount while Umova do virazi od ;
For this operator, the calculation cycle is completed until the hour when the logical expression is true, and the change in the cycle changes to the cob value from the specified time.
[> restart;
[> for y from 0 by 2 while y
0
2
4
6
The fourth statement in the assignment cycle for working with analytical viruses and the attack structure:
for Change cycle in viraz 1 do viraz 2 od ;
Here the body of the cycle of expression 2 is converted, since the symbolic change given to its names sequentially takes on the values of the skin operands of algebra 1. It is significant that the work of this construction lies in the internal supply virus 1. So if virus 1 is a sum, then it During the alternating cycle, the daily value of the skin supplement is determined, as is the solid content of the skin substance.
[> restart;
[> a:=5*x^2+x+6/x;
[> b:=simplify(%);
[> for m in a do m; od;
[> for m in b do m; od;
10.3. Function procedures
Maple function procedures can be specified in two ways. To create procedure-functions, the first method is to use the vikory symbol ( ) and is determined by the offensive structure:
Function names: = (list of formal parameters) viraz;
The name of the function is indicated by a set of characters in the Latin alphabet, the list of formal parameters is entered through whom. Viraz is a Maple command that implements the body of a function procedure.
[> f1:=(x1,x2)->simplify(x1^2+x2^2);
[> f 1 (cos(x), sin(x));
1
Another way to define procedure-functions is the vikory command unapply It has the following structure:
These are the functions:= unapply (Viraz or operation, list of changes);
This method of assigning procedure-functions to a new function is via an input or if the virus passes the vikory function as a function.
butt .
[> f3:=unapply(diff(z(r)^2,r)-2,z);
[ > f3(sin);
[ > combine(%);
10.4. Procedures
Any procedure in Maple begins with a title, which is the name of the procedure, followed by an assignment sign and a service word proc , Further in the round arches through whom formal parameters are indicated. The procedure will inevitably end with a service word end . All definitions and commands linked between service words proc і end add up the body of the procedure.
Name of procedure: = proc (List of formal parameters); command (or virazi); end ;
If the procedure is important, this click will affect them. The value that is rotated is the value of the remaining specified operator (command) from the body of the procedure, in which the type of result of the procedure is the same as the type of value that is rotated.
[> f:=proc(x,y);x^2+y^2;simplify(%);end:
[ > f(sin(x),cos(x));
1
When writing procedures in Maple, you can use low commands and service words, in addition to a designated minimum set of obligations that allow you to describe changes, determine the exit from the procedure, and notify about errors.
When describing the formal parameters of a procedure, you can specify their type using a double checkbox. With such a description, Maple automatically checks the type of the actual parameter and provides notifications about changes in different areas with the type of the formal parameter.
After the title of the procedure, a part of the procedure can be described, which is supported by a new gap. When describing local changes that are used in the middle of this procedure, you can use the description, which is specified by the service word local , after which it is necessary to indicate the names of local changes through the pass. The choice of global variables for a procedure can be specified using a service word global , which may be located in the description part of the procedure.
To exit the procedure in any place of the body and the result of the work from the required command, you can use the command RETURN ( val ), de val – the value that rotates, as a different type can occur when leaving different places of the procedure.
For an emergency exit from the procedure, you can use the command ERROR (‘ string ’) , here string – a notification that is displayed on the monitor screen in an emergency situation. Thus, the basic structure of the procedure can be represented as follows:
Name of procedure: = proc (List of procedure parameters) local list of local changes, induced through coma; global overflow global changes, induced through coma; RETURN ( val ); ERROR (‘ error in body of procedure ’);… end ;
[>
[ > examp(-1);
[> examp(0);
[ >examp(2);
11. METHODS OF ENTERING AND RECORDING INFORMATION
AT SREDOVYSHCHIMAPLE
To save names (identifiers) and their values in external memory in the file with them name . txt you need to enter the command:
save list of names of changed ones, recovered through coma, “file names with extensions” txt ”;
What is the extension of the indicated symbol? m , The file will be recorded in the internal Maple format, with all other extensions in text format. To display the information saved in the file on the screen, use the command
read “ File name ”;
[> restart;
[> examp:=proc(x) local y,w; Global z; if x
[ > examp(-1);
[> examp(0);
Error, (in examp) Variablex = 0
[ >examp(2);
[ > read "nnn.txt";
To record an entire file instead of the screen, you can use the following two commands.
Persha team
writeto ("file name")
As a result of this command, all information that fits on the screen will be saved in a file with designated names. Moreover, if the instructions file is stored in external memory, then the information that is saved will be replaced with a new one.
Another team
appendto ("file name")
Allows you to add information that is displayed on the screen after this command to the end of the file.
[ > f:=12;
[> f1:=factor (y^2-3*y); save f,f1, "n1.txt";
[> appendto("n1.txt");
[> solve(x^2-3*x+2=0,x);
As a result of the victory of the team save f , f 1, " n 1. txt "; a text file will be created n 1. txt , We respect this information:
f:= 12;
f1:= y*(y-3);
and as a result the victory of the team appendto (" n 1. txt "); Instead of the file I will look:
f:= 12;
f1:= y*(y-3);
[ > solve ( x ^2-3* x +2=0, x );
2, 1
The Maple package has a few commands for displaying information on the screen. The simplest of them are commands
print (perelik Maple
lprint (perelik Maple -Virazhen, what to play it safe through a coma);
Moreover, since nothing is assigned to the change, then their names are displayed, otherwise their values are displayed.
[> x:=y^2: print (x, "primer 1", y, factor(x-5*y));
[> x:=y^2: lprint (x, "primer 2", y, factor(x-5*y));
y^2, primer 2, y, y*(y-5)
At the pointing of the butts, the command screams print display expressions through coma in natural mathematical form, and the command lprint display information in the style of a series of displays and are reinforced with one type of space.
The Maple package can be used to analyze and graphically interpret numerical information contained in a text file captured by the package, as well as other software add-ons. As a rule, in a text file, numbers are written in rows. To read numeric information from a text file, you can use the following command:
readdata (“file name”, change type( integer / float - The remaining type is installed for the purpose), number healer);
Before using this command, you must activate the additional commands:
readlib(readdata):
[> restart;
[> readlib(readdata):
[> ff:=readdata("aa.txt",integer,8);
[ > x:=ff;
[ > y:=x;
[ > y1:=ff;
[ > f:=readline("aa.txt");
Changed indexation ff is connected with the fact that the numbers are supplied in what appears to be a two-dimensional array, in which the number of rows in the array corresponds to the number of rows, and the number of rows is indicated by the remaining command parameter readdata . How the command screams from the pointed butt readline display numerical data in variable form string .
12. VIKORISTANNA MATHEMATICAL PACKAGEMAPLEFOR SCIENTIFIC DOSLIDZHEN
This section will look at how Maple solves applied engineering tasks. We aim to show the capabilities of the Maple package with advanced engineering tasks related to the advanced research modes of robot ownership, it is important to consider the design and technological parameters, complexes and illustrate the capabilities of the software and command modes of the robot and koristuvacha in the middle of Maple. Below are fragments of the investigation, which are accompanied by short explanations.
12.1. Investigation of the flow of variable parameters of the flat grinding chamber and the flow effect on energy supply
12 .1.1. Statement of the problem
Streamlins have a variety of impact components and consist of a split-type apparatus (one or several units), in which the gas-energy stream imparts fluidity to the particles of the material that are formed, and the chambers which involves the interaction of material flows with each other and/or with special liquids surfaces. As energy carriers in stream valleys, wind is most often stagnated, and sometimes inert gas, water vapor, and combustion products.
Strumenevy broom allows the broom to be combined with mixing, drying and other technological processes. And the robot in a closed cycle will ensure minimal sawing of the saw in the middle.
Any jet machine includes a projector, which is a chamber in which the energy of two flows (main and projected) is mixed and exchanged, and a grinding chamber in which the mixed flow interacts and. With the acceleration of energy transfer at the ejector discharge tubes, the particles are transported to the grinding chamber, and then to the jet zone (Fig. 12.1).
The stream that comes out of the heating tube does not immediately fill the entire cross-section of the marked chamber; the stream at the entrance to it breaks through the walls and then collapses, looking like a strong stream, from the middle to the surface of the section. The surface of the section is unstable, vortices arise on it, as a result of which the liquid mixes with the excess medium.
When the jet from the flame tube is completed, the fluid flow in its outlet section 1-1 at all points the intersection is equal to each other. By stretching the cob, the axial fluidity is constant according to the magnitude and the previous fluidity in the section of the heating tube V 0 . In the area of the tricutaneous ABC (Fig. 12.1.) at all points of the fluid flow stream the energy flow is equal to each other and also equal V 0 - This area creates the so-called core of the jet. Further, the axial liquidity changes step by step and continues at the main level l basic axial fluidity V OS V 0 .
Small 12.1. Diagram of the jet at the grinding chamber
It is clear that the fluidity of energy supply from the section of the heating tube to the surface of the jets changes according to the law
,
(12.1)
de V z – fluidity of energy supply with a grinding chamber on the riser z view through the discharge tube, m/s;
V 0 - Energy density at the cut of the heating tube, m/s;
z 0 - Stand in front of the outlet tube to the plane of the nostrils, m.
When there is a significant change in the kinetic energy of the end circuit of the body, it is necessary to know the work of the forces of intercomponent interaction of particles of the material and energy. This robot lies in the force vector of the dynamic influx of energy onto the particle, which is calculated as such
,
(12.2)
de R - Vector of the force of the dynamic inflow onto the particle, N;
F m - Area of cutting of a particle, m2;
,
(12,3)
Significantly
,
(12.8)
de m - Mass of a piece of trimmed material, kg.
,
(12.9)
de 
- Hardness of particles of trimmed material, kg/m.
Viraz (12.7) I can see
.
(12.10)
The roving can be removed to determine the fluidity of the particles added to the material in the grinding chamber at the section between the pipes distributing to the area of mutual interaction of the jet streams.
A system of differential equations that describes the process of changing the fluidity of particles and energy transfer in the grinding chamber through the fermentation tube until the constriction flows are closed
.
(12.11)
Vidstan l Store – between the cut of the heating tube and the middle surface of the grinding chamber is chosen wisely
,
(12.12)
de d tr = 18 diameter of the heating tube, mm.
Department: Information technologies
Laboratory robot
On the topic: " SYNTAX, BASIC OBJECTS AND COMMAND SYSTEM MAPLE "
Moscow, 2008 r_k
Robot goals :
· know the main objects and changes of the Maple system;
· know and note the commands that are used when working with objects and changes of the Maple system;
· Know the syntax of the basic mathematical functions of the Maple system.
Enter
Maple analytical calculation system is an interactive system. This option means that the user must enter the command or the Maple operator in the worksheet input area and press the key
Kozhen operator chi team obov'yazkovo will end Let's separate with a sign. The Maple system has two such signs – a dot with a clod (;) and a dot (:). Whenever a proposition ends with a point, it is calculated and the result is displayed in the display area. If you select a double key as a separator, the command is canceled, and the results of its work are not displayed in the display area of the working box. This is true, for example, when programmed in Maple, if there is no need for the output of any intermediate results contained in the loop operators, since the output of these results can take up a lot of space in the workbench, and you can achieve a certain amount of time on your image.
Here and below the Maple commands, the entry in the form of the syntax of the Maple language is reviewed. However, when editing applications, it becomes necessary to display commands in mathematical notation, following the command Options ® Input Display ® Standard Math Notation Set the display mode to normal.
Maple has implemented its own language, in addition to which it is necessary to merge the client with the system. The basic concepts are objects and changes, including a definition using admissible mathematical operations.
In the simplest way objects, with which you can practice Maple , є numbers, constants and series.
Numbers
Numbers in the Maple system can be of various types: goals, simple fractions, radicals, floating point numbers and complex numbers. The first three types of numbers allow you to calculate exact numbers (without rounding) of different mathematical expressions, implementing exact arithmetic. Numbers with a dot that float are nearby, in which the number of significant digits is surrounded. These numbers serve as approximations (or approximations) of exact Maple numbers. Complex numbers can be either exact, since the active and explicit parts are represented by exact numbers, or approximate, since given the active and explicit parts of a complex number, floating-point numbers are formed.
Whole numbers are specified by looking at the sequence of digits from 0 to 9. Negative numbers are specified with a minus sign (–) in front of the number, zeros before the first non-zero digit are not significant and add to the value of the integer. The Maple system can work with whole numbers of considerable magnitude, many digits are practically enclosed by the number 228. Calculation with whole numbers involves several arithmetic operations (add +, add –, multiply *, sub /) and number factorial (!).
Maple represents a large integer number that does not appear on the row of the display area and the backslash symbol (\) as the continuation symbol on the display row. The remaining command calculates the number of digits in the result of the previous calculation. It has an operation% as a parameter, which is a more manual form of sending to the result of the previous operation. Maple has two more similar operations that identify the results of the previous and previous commands. Their syntax looks, obviously, like this:
Maple has a wide range of commands that allow you to calculate operations that are specific to the processing of integer numbers: decomposition into simple multipliers (ifactor), calculation of private (iquo) and excess (irem) during the calculation of the subdivision operation, finding the largest literal dilator of two whole numbers ( igcd), vykonannaya reversal, which is the whole number, we will forgive (isprime) and much more.
To verify the calculation of the part and the surplus of two integer numbers, the operation is based on the result of the first (calculation of the part) and the forward (calculation of the surplus) commands. The result of the isprime() command is a Boolean constant true (true) or false (false).
By typing a command in the worksheet input area? integer, you can retrieve a list of all commands for working with integers
Primary fractions ask for additional operations under two whole numbers. Please note that Maple automatically performs the shot reduction operation. All basic arithmetic operations can be performed on simple fractions.
If, when a fraction is given, the sign shortens (div. remains calculated in the application), then such a “fraction” is treated by the Maple system as an integer.
It is often not easy to see the result as a single fraction, and it is due to the deliberate transformation of them in tens of fractions. For this purpose, the evalf() command is used, which approximates the leading fraction by numbers with a point that floats, vikoryst, and ten significant digits in the mantis of their data. If the accuracy is not sufficient by default, it can be set by another parameter of the assigned function.
These dozens of manifestations are not identical Maple objects. Less decimal expression approximation exact value, represented by a fraction.
Radicals are specified as the result of adding whole or fractional numbers to the fractional stage, or calculating the square root of them using the sqrt() function, or calculating the root n th stage for the additional function surd (number, n). The operation of raising a step is indicated by the symbol ^ or a sequence of two stars (**). When putting the shot into the world, place it in the round arms, as well as display the shot. When radicals are given, possible simplifications are also generated, associated with the addition of the sign of the radical of the maximum possible value.
Calculation with wholes, fractions and radicals absolutely accurate, While working with these types of data, the Maple program does not routinely round up floating point numbers.
Numbers with floating point are specified as a whole and shot parts, separated by a tenth dot, with a leading number sign, for example, 3.4567 -3.415. Numbers with a floating point can be specified in the so-called exponential form, in which case after the spoken number with a floating point or the primary integer, called the mantissa, a symbol e or e is placed, after which the whole is specified number зі sign (indicator of step). This form of notation means that the mantissa of the trace is multiplied by ten exponents of the number corresponding to the exponent of the exponent in order to subtract the value of the number written in such exponential form. For example, 2.345e4 corresponds to the number 23450.0. In this way, it is possible to represent numbers that are very large in absolute terms (the index of a step is a positive number) or very small (the index of a step is a negative number).
Mathematical calculations are made up of numbers using additional arithmetic operations. Symbols of arithmetic operations on Maple are re-arranged in the table. 1.
Table 1. Arithmetic operations
The sequence of arithmetic operations corresponds to the standard rules of precedence of operations in mathematics: first, reduction is carried out at the level, then multiplication and subdivision, and finally, addition and subdivision. Everything ends up to the right. The factorial calculation operation has the highest priority. To change the sequence of arithmetic operations, turn the round arms.
If all the numbers in the expression are whole numbers, fractions or radicals, then the result is also presented with these types of data, and if the expression has a number with a floating point, then the result of calculating such a “mixed” virus will also be a number with a floating point point, as only in Virase has no radical. And here the radical is calculated exactly, and the coefficient is calculated either exactly, or the appearance of the numbers with a floating point depends on the type of factors.
The Maple analytical calculation system will always be able to produce calculations with absolute accuracy. If you don’t leave it out, arithmetic with speech numbers is included.
Maple can be used from complex numbers . For an explicit unit
in Maple the constant is vikoriated I. The problem of a complex number does not differ from the basic problem in mathematics.