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My Maple Assignment on Web

1. Hierarchy of arithmetic operations: Use your examples (at least three examples and execute them in maple) to prove which arithmetic operations are carried out first in Maple and if they are equal priority, in which directions they are carried?

> 8/4/2;

Maple always starts from the division on the left to the right.In this expression , the calculation is done by (8/4)/2 meaning that (8/4) is done first and then the result is divided by 2.Thus (8/4)/2=2/2=1.

> 8/(4/2);

Maple gives priority to the parentheses first than other arithmetical operators.In this expression,the order of evaluation is 8/(4/2),meaning that (4/2) is done first then 8 is divided by it.Thus 8/(4/2)=8/2=4.

> 8+4/2;

Maple gives priority to the multiplication or division first followed by addition or subtraction.In this expression,the calculation is done by 8+(/4/2) .This meaning that 4 is divided by 2 first and then added to 8.Thus,8+4/2=8+ 2=10.

2.Using your own examples ( one examples each and execute them in Maple)clearly bring out all the differences between the following Maple symbols and commands:

a); and :

> 3*x^2-5*x+1=0;

> 3*x^2-5*x+1=0:

b)= and :=

> x=3;

> x:=3;

c)? and ???

> ?spacecurve;

Description

The spacecurve function defines one or more curves in three space. The first argument can be a list of points or a spacecurve. A spacecurve is a list of size three or more. The first three components are considered to be the parametric representations of the x , y , and z coordinates. Additional arguments include specifying a range for the particular curve via t=a..b , or the number of points that are to be used in drawing the curve via numpoints=n (the default is 50). The first argument can also be a set of such lists, in which case several three-dimensional curves are drawn.
Remaining arguments are interpreted as options which are specified as equations of the form option = value . These can be used to specify global default values such as t=a..b or numpoints=n .
Remaining options are the same as those found in plot3d with the exception of specifying a grid size. For example, the option axes = BOXED specifies that the space curves include a boxed axis bounding the plot. See also ?plot3d[options] .
The result of a call to spacecurve is a PLOT3D structure which can be rendered by the plotting device. The user may assign a PLOT3D value to a variable, save it in a file, then read it back in for redisplay. See ?plot3d[structure] .
spacecurve may be defined by with(plots) or with(plots,spacecurve) . It can also be used by the name plots[spacecurve] .

Examples

>    with(plots):
spacecurve([cos(t),sin(t),t],t=0..4*Pi);
spacecurve({[sin(t),0,cos(t),t=0..2*Pi],[cos(t)+1,sin(t),0,numpoints=10]},
                t=-Pi..Pi,axes=FRAME);
spacecurve({[t*sin(t),t,t*cos(t)],[4*cos(t),4*sin(t),0]},t=-Pi..2*Pi);
knot:= [ -10*cos(t) - 2*cos(5*t) + 15*sin(2*t),
         -15*cos(2*t) + 10*sin(t) - 2*sin(5*t), 10*cos(3*t), t= 0..2*Pi]:
spacecurve(knot);
helix_points := [seq([10*cos(r/30),10*sin(r/30),r/3],r=0..240)]:
spacecurve(helix_points);
spacecurve({helix_points,knot});

See Also

read , save , plots[tubeplot] , plot3d[option] , plot3d[structure]

> ???spacecurve;

Description

Examples

See Also

d)expression and function.

> f:=(x+1)/x;

> f:=x->(x+1)/x;

Sum and add

> restart;

> sum(cube(k),k=1.100);

> add(cube(k),k=1.100);

3.Use of the Help facilities in Maple :Use of the help commands in Maple and explain the use of the three Maple commands (not discussed in the class)with your own examples.

1) vector commands

> v1:=vector([a,b,c]);

A vector in Maple is represented as a one dimensional array whose indices start at 1.
Vectors can be input either directly, as a one dimensional array, or using the vector command in the linear algebra package. For example, array([1,x,x^2]) creates a vector of length 3 .
The ith entry of a vector U is accessed, and assigned to, using the subscript notation U[i]. For example, U[i] := 1/V[i] assigns the ith entry of the vector U to be the inverse of the ith entry of the vector V.
The map function can be used to apply a function to each entry of a vector. For example, map(simplify,V) simplifies each entry of the vector V and map(diff,V,x) differentiates each entry of the vector V with respect to x.

2) logplots

> restart;

> with(plots);

> logplot({exp(x),exp(x^2),exp(2*x)},
x=0.1..10,1..10^6);

logplots is semi-logarithmic plot of functions where the vertical axis is in logarithmic scale

Calling Sequence

logplot( f , h , v ,...)

Parameters

f - function(s) to be plotted

h - horizontal range

v - (optional) vertical range

Description

The routine logplot creates a two dimensional plot with the vertical axis given as a logarithmic scale. A typical call to the logplot function is logplot(f(x), x=a..b) , where f is a real function in x and a..b specifies the horizontal real range on which f is plotted.
The logplot function provides support for two-dimensional log plots of one or more functions specified as expressions, procedures, parametric curves, or lists of points. (When plotting a procedure, operator notation must be used.) See ?plot[function] for more information on plotting functions.
A call to logplot produces a PLOT data structure, which is then printed.
The horizontal and vertical range arguments h and v define the axis labels and the range over which the function(s) are displayed. They take one of the following forms: string , low..hi , or string=low..hi , where low and hi are real constants.
Remaining arguments are interpreted as options which are specified as equations of the form option = value . These options are the same as for those found with the plot command.
The command with(plots,logplot) allows the use of the abbreviated form of this command.

rootlocus

> with(plots,rootlocus):
rootlocus( (s^3-1)/s, s, -5..5 );

Description

The rootlocus command plots the complex roots of the equation

1 + k f(s) = 0

as k runs over the range r = a..b . Since f(s) = n(s)/d(s) is a rational function in s , this is equivalent to tracing the paths of the complex roots of the polynomial
d(s) + k n(s) for k in a..b
Maple's fsolve command is used to compute the roots of the polynomials. Remaining arguments are interpreted as options which are specified as equations of the form option = value . Standard plot options are supported.

The polynomial is initially solved for n equally spaced points k in the range r = a..b . The optional argument numpoints = n specifies the number of points. The default is 49.
Next, the code tries to pair up the roots for adjacent values of k by choosing those closest to each other. If two sets of roots for k=k1 and k=k2 are not sufficiently distinguishable from one another, the algorithm will compute a new set of roots for at k = (k1+k2)/2 . The optional argument adaptive = false will turn this off.
Finally, the code tries to trace out the different curves. It tries to join up adjacent points with line segments. Sometimes this code can be fooled and the result is a messy plot. The optional argument style=point will turn this off by just displaying the roots as points.
The command with(plots,rootlocus) allows the use of the abbreviated form of this command.

4. Use of the units and Scientific Constants :Select any three problems involving different types of units from your textbooks and carry out the complete calculations including the units using Maple.State the problems fully and comments whether Maple gives the corrects final anser along with approriate units.(Refer the eaxmples discussed in the class.)

>

> restart;with(Units):with(Units[Natural]);

>

Warning, the name Unit has been rebound

Warning, the assigned name polar now has a global binding

Warning, these protected names have been redefined and unprotected: *, +, -, /, <, <=, <>, =, Im, Re, ^, abs, add, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, argument, ceil, collect, combine, conjugate, convert, cos, cosh, cot, coth, csc, csch, csgn, diff, eval, evalc, evalr, exp, expand, factor, floor, frac, int, ln, log, log10, max, min, mul, normal, root, round, sec, sech, seq, shake, signum, simplify, sin, sinh, sqrt, surd, tan, tanh, trunc, type, verify

>

5.Find out the dimensions of three different physiacal properties using Maple.

> restart;with(Units);

>

> GetDimensions();

> GetDimension(moment_of_inertia);

> GetDimension(magnetic_flux_density);

> GetDimension(thermodynamic_temperature);

6.Calculate the volume occupied by 1 kg of mercury at 25 C.

> restart;

> with(ScientificConstants);

> GetElement(Hg,density);

> mass:=1000*g;T=25*degC;

> density:=13.5336*g/cm^3;

> volume:=mass/density;

7.Use Maple to calculate the number of water molecules in 1 g of water.

> mass:=1*g;

> GetElement(H,atomicweight);

> GetElement(O,atomicweight);

> mH2O:=(2*1.00794+15.9994)*g;

> mol:=mass/mH2O;

> GetConstant(Avogadro_constant);

> N[A]:=0.602214199e24;

> molecules:=mol*N[A];

8.What is the computer algebra systems and what are its advantages?

Computer algebra system is Symbolic Computation .Symbolic computation is computation with symbols representing mathematical objects,including
-integers,real, and complex
-polynomials

-derivatives and integrals

-systems of equations,series expansions of functions.

-objectve is to obtain closed form,exact numbers.

The advantages of this system is

-Thousand of built in functions

-many options for simplifying expressions.

9.Determine whether y+8/x-2=x+6 is linear nor not?(Hint:solve for y and comments) .

> restart;

> solve((y+8)/(x-2)=x+6,y);

This equation is not linear as it has highest power which is 2.

10.Show that z=1,y=2,z=3 are the solutions for x+2y-3z=4.(Hint:use subs)

> restart;

> subs(z=1,y=2,x=3,x+2*y-3*z=4);

Yes,the value of x,y z are the solutions for x+2y-3z=4 as both sides are same.

11.A)Solve the equations :x-y=-3 and x+2y=3.Plot these equations in the same graph .Do these graphs cross each other?What is the meaning of the point of intersection?

> restart;

> with(plots);

Warning, the name changecoords has been redefined

> e1:=x-y=-3;

> solve(%,y);

> y1:=%;

> e2:=x+2*y=3;

> solve(%,y);

> y2:=%;

> solve({e1,e2},{x,y});

> plot({y1,y2},x=-infinity..infinity);

Yes ,this graph cross each other and the meaning of the point of intersection is where the graph is cross each other.The point of intersection is (-1,2).

B)Plot the equations :y=-x-3 and y=-x+2.From the graphs what can you say about the existence of solutions for this set of equations?

> restart;

> y1:=-x-3;y2:=-x+2;