maole 8.mws - [Server 1]

KOS 1110 Computers in Science

Assignment 4 - Questions in Introduction to Maple

Due date Thursday, 29-9-2005, 1:00pm

Instructions: All the answers (except the last question) should be done in Maple work sheet. Use the text mode to type the questions and any comments. After you have answered all the questions, export your file as 4yourname.rtf. Open this file in MS Word and save this file as 4yourname.doc file. Print out this file. Send me the printed form (.doc file) and the electronic form (softcopy) through the email. Remove the output from your Maple work sheet file (.mws), save the file as .htm file and publish in your web page. For the last question include a print out from your website after your have uploaded the web page. Include the web page address in your answer.

1. Computer Algebra System: What is computer algebra system (symbolic computing)?

Symbolic computation is computation with symbols representing mathematical objects, including integers, real and complex, polynomials, derivatives, integrals, systems of equations and series expansions of functions. The objective is to obtain closed form and exact solutions. Thousand of built in functions can be done through this system. It also has many options for simplifying expressions.

2. What are the differences between symbolic computing and numerical computing?

Symbolic computing is computation with symbols representing mathematical objects, including integers, real and complex, polynomials, derivatives, integrals, systems of equations and series expansions of functions. The objective is to obtain closed form and exact solution. Thousand of built in functions can be done through this system. It also has many options for simplifying expressions. Internal representation for rational numbers is different from floating point representations. For rational numbers, numbers of significant digits and maximum size number far exceed typical floating-point representation in numerical computation.

Numerical computing handle operations such as (+,-,*,/) on numbers, plus computation of numeric value for roots of equations, mathematical functions, integrals and derivatives. the computing is carried out on numbers alone. The results most often not exact since floating point approximations is used.

3. What are the advantages of symbolic computing compared to the numerical computing?

Symbolic computing can obtain closed form and exact solutions as it use internal representation for rational numbers that is different from floating point representations.

Numerical computing results most often not exact since floating point approximations is used. Floating point representations often lead to representation or truncation error. It will try to represent irrational numbers and also lead to propagation errors because it accumulated errors arising from a sequence of calculations. This system having word length problem as the allowable values for exponent depend on computer, computing language or calculator.

4. Hierarchy of arithmetic operations: Use your own 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 of equal priority, in which directions they are carried?

Example 1:

addition (+) and subtraction (-)

> x:=12+5-8;

If there are (+) and (-) in one expression (the same priority operator), the calculation will start from left to right.

Example 2:

multiplication (*) and division(/)

> x:=6*7/2;

If there are multiplication(*) and division(/) in one expression,(the same priority operator), the calculation will start from left to right.

Example 3:

multiplication(*) and addition (+)

> x:=12*3+4;

if there are multiplication(*) and addition (+) in one expression,(the different priority operator), the multiplication is calculated first and followed by addition.

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

a) ; and :

b) = and :=

c) ? and ???

d) expression and function

e) sum and add

a); and ;

> GetConstant :=Faraday_constant;

> GetConstant:=Faraday_costant:

From the example above, when we use (;),it will display the execution result.

However if we use(:), it will not display the execution result.

b) = and :=

> restart;

> x=4+3*60+5^2;

> x:=6+2*60+5^2;

from the exampe above,when we use equation(=)sign, it means that the variable which is in example x is having a value of the expression.

if we use assign operator (:=),the result will assign to the x. so that the x will have the same value of the expression.

c) ? and ???

?= using single question mark syntax (?) causes the topic to open with the example section expanded.

??? = using triple question mark syntax (???) causes the topic to open with the example section expanded, and all other sections contracted.

d) expression and function

Expression

> restart;

> f:=7*x^3+12*x^2-9*x-15;

> f(2);

> x:=2;

> f;

Function

> f:=x->7*x^3+12*x^2-9*x-15;

> f(2);

the first example is expression which there is no variable on the left side.

the second example is a function which these variable on the left side and assigning a value to variable does not alter the definition.

e) sum and add

> restart;

> add(7*x+5,x=3);

> sum(7*x+5,x=3);

> add(7*x+5,x=1..s);

Error, unable to execute add

> sum(7*x+5,x=1..s);

sum is the symbolic summation. it is used to compute formula for definite or indefinite sum. if Maple cannot compute a closed form, Maple returns the sum 'unevaluated'.

a typical example is sum (k,k=0...n=1), which returns the formula n^2/2-n/2. to add a finite sequence of values, rather than compute a formula, use the add command.

the add fuction is used to add up an explicit sequence of values. for example, add (k,k...0.9) returns 45. although the sum command can be used to compute explicit sum, it is recommended that the add command be used in programmes if an explicit sum is needed.

6. Use of Help facilties in Maple: Use the help commands in Maple and explain the use of three Maple commands (not discussed in the class) with your own example.

Command 1= algebra

> combine(Int(x,x=a..b)-Int(x^2,x=a..b));

Command 2= energy conversion

Convert the speed of light into units of action (1 planck = 1 joule second).

> convert(m/s, dimensions, energy=true);

> convert(299792458., units, m/s, planck, energy=true);

Command 3= conjugate equation

> with(VariationalCalculus);

> ConjugateEquation(A*sin(t)+B*cos(t), [A,B],[1,0],t,0);

The conjugate points to 0 are the zeros of this.

7. Use of 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 problem fully and comment whether Maple gives the correct final answer along with appropriate units. (Refer, but don't reuse the examples discussed in the class or in the help menu)

Problem 1 : find the momentum of a person in your car on the highway

> Units[UseSystem](FPS);

> with(Units[Natural]);

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

> momentum:= 150*lbs * 60*mph;

> evalf(convert(momentum,system,SI),4);

Problem 2: How much a room (10x10x2 m3) completely full of pure gold actually weighs?

> restart:

> with(Units[Natural]):

Warning, the assigned name polar now has a global binding

> with(ScientificConstants):

> rho[Au]:= evalf(Element(gold,density,units));

> Volume:=(10*m)*(10*m)*(2*m);

> g_SI := evalf(Constant(g,units));

> Weight := evalf (rho[Au]*Volume*g_SI,3);

Problem 3

Magnitude of the rate constant

> restart;

> with(ScientificConstants);

> K:=rate/[A]^2;

> rate:=3.0*11^5*(m*(-1)/(s*(-1)));

> A:=0.1;

> evalf(K);

8. Find out the dimensions of three different physical properties using Maple.

> restart;

> with(Units);

Warning, the name GetUnit has been rebound

> GetDimensions();

> GetDimension (absorbed_dose);

> GetDimension (moment_of_inertia);

> GetDimension (illuminance);

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

> restart;

> with(ScientificConstants);

> GetElement(Hg,density);

> mass:=1*kg;T:=25*degC;

> Volume:=mass/density;

> mass:=1000*g;

> density:=13.6*g/cm^3;

> Volume;

> restart;with(ScientificConstants);

> GetElement(mercury,symbol,meltingpoint,atomicweight);

> PV:=n*R*T;

> V:=n*R*T/P;

> R:=8.314*J/(mol*(K)); T:=(273+25)*K; P:=(1*atm);

> n:=(1000/80)*mol;

> evalf(V);

> V;

The result show the value is by atm. the unit atm indicate value of 1.the result is

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

> restart;

> with(Units);

> with(ScientificConstants);

Warning, the name GetUnit has been rebound

> GetElement(H,atomicweight); GetElement(O,atomicweight);

> waterMole:=2*Element(H,atomicweight)+(Element(O,atomicweight));

> evalf(waterMole);

the measurement is in kilogram. Convert to amu.

> convert(%,units,kg,amu);

> 1/%;

The value is mass per mole. Divide by atomic weight

> %*evalf(Constant(N['A']));

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

> restart;

> with(plots):

> f:=(y+8)/(x-2)=x+6;

> solve(f,{y});

> plot(-20+x^2+4*x,x=-infinity..infinity);

From the graph,we know that the equation is not linear,but quadratic function.

12. Show that z=1, y=2, x=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);

13. 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;

> solve(x - y = -3 ,{y});

> solve(x + 2*y = 3,{y});

> plot({x+3,(-x/2+3/2)},x=-infinity..infinity);

the graph crossing each other,the point where the graph crossing each other is the solution for x.

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 equation?

> plot({-x-3,-x+2}, x=-infinity..infinity);

there are 2 intersections, meaning that there are 2 solutions for this graph.

c) Plot the equations: x + y = 1 and 2x + 2y = 2. From the graphs what can you say about the existence of solutions for this set of equation?

> solve (x + y = 1,{x}) ;

> solve (2*x + 2*y = 2,{x});

> plot({-y+1.-y+1},y=-infinity..infinity);

Both of the equations are the same.so, every points on the line are the solutions for y.

14. Solve the equations: 2x + y - 2z = 8, 3x + 2y - 4z = 15 and 5x + 4y - z = 1..

> restart;

> solve ({2*x+y - 2*z =8, 3*x+2*y -4*z =15, 5*x+4*y-z = 1},{x,y,z});

15. Write any equation of your own and expand it using Maple. Factorize and simply the result and show that it gives back the starting equation.

> restart;

> f:=(x+y+13)*(x*2+5*y+9);

> expand(%);

> factor(f);

> simplify(f);

16. Write any equation with three variables (x, y and z) and solve it using Maple.

> restart;

> solve(2*x+4*y-8*z=9,x);

> solve(2*x+4*y-8*z=9,y);

> solve(2*x+4*y-8*z=9,z);

17. Download any VRML picture from the internet and build a web page using this picture along with a paragraph describing this object and acknowledging the website from where you downloaded this picture. Upload this web page in your web site. Include the printout of this webpage with your assignment.