# Math 2250
# MAPLE TUTORIAL 
# Fall 2000
# 
#    This document is a tutorial for Math 2250 students who may not have
# done previous work with MAPLE or in our lab, or who may just want to
# brush up on their skills.   The work you are to complete as your first
# Maple assignment is  the section 1.4 computing project, described on
# pages 43-44 in the text.  The MAPLE commands you will need in order to
# do that project are reviewed in this tutorial.  A  precise template 
# for your project answers will be available on Friday September 1, and
# will enlarge slightly on the  Investigation a,b,c of page 44.
# 
# 0) A preliminary word of Advice:  Students often approach the task of
# reading mathematical material as if they were reading a novel; they
# sort of skim along quickly.  That approach  is O.K. to get an
# overview, but in order to have a chance at real understanding you must
# be prepared to proceed much more slowly, sentence by sentence and
# thought by thought.  Otherwise you will almost certainly find yourself
# partly lost after several paragraphs and completely lost after several
# more.  (This might happen anyway.)  If you are working properly it can
# easily take half an hour  to read through one page of mathematical
# text.  This takes a certain amount of discipline, patience, and
# practice.  With the Math 2250  computer  projects there  is the added
# temptation of having Maple execute  commands in successive command
# sections by repeatedly  hitting  the enter (return) key, without
# pausing to digest the interlaced text or the meaning of the commands. 
# There is a seductive appeal in having  this capability.  Resist it.
# 
# We will assume you are working in the Math Department computer lab. 
# If you are in another lab (e.g. Engineering lab in EMCB or Library lab
# in Marriot) or on your own computer,  you will get started
# differently.
# 
# 1a)  Logging in to a Math Lab machine:  The Math Department Computer
# Lab is located in the South Physics 205, i.e. inside  the 2-story
# brick building just north of the Math building JWB, and south-east of
# the physics classrooms in the circular part of JFB.  Room 205 is  in
# the back (East) of the building,  on the second floor (from the west),
# which equals the level of the ground on the uphill,  east  entrance.
#       The following information about logging in and your initial
# password is summarized from the handout Introduction to the
# Undergraduate Computer Lab Department of Mathematics, University of
# Utah, SLC, Utah 84112 .  This and other useful handouts are available
# on a table in the front of the lab. 
#       Everyone who is registered in Math 2250 should  automatically
# have an account set up in our lab.  These accounts are created from
# University class lists so it sometimes happens that late-registering
# people don't have accounts yet.  If you turn out to be one of these
# people you will need to consult the lab assistant about getting an
# account.  Make sure to bring your student I.D. because the first thing
# the assistant must do is verify that you are a University student.  
#      If your machine looks asleep jiggle the mouse or hit any key to
# wake it back up.   A window should appear which asks you to select a
# server.  Since the Math Department is sharing this lab with Physics,
# there are physics and math server computers.  You want to select one
# from the list of choices  which has ".math" in its name.  If a
# ".physics" computer is highlighted, use your cursor to choose one of
# the ".math" computers, and then O.K. your choice by clicking in the
# "accept" box.  In a few seconds a login window should appear, asking
# for your login name and password, and have  "Mathematics" written in
# red on the top.   (If the login window says "Physics" you have
# accidently chosen a physics server, and you should type "control C" to
# go back to the previous step.)
#       Your default login name is made out of your student I.D. number
# and your actual name, as follows.  All names from classes begin with
# ``c-''.   If your name is Karl Fred GausS, then your login name is
# c-gskf, following the recipe:  c-(first letter of last name)(last
# letter of last name)(first letter of first name)(middle initial).  If
# there are multiple people registered this term who would have the same
# login name, say c-gskf, then they are instead assigned login names as
# c-gskf1, c-gskf2, c-gskf3, etc.  Mr. Gauss would not know beforehand
# which case he fell into, so would probably try c-gskf first,  followed
# by his password.  In case of failure he would then try c-gskf1, then
# c-gskf2, etc, through c-gskf4. Then he would find a lab assistant. 
# After entering your try at a login name, type the ``return'' key  and
# the cursor should be in the password box.  
#      Your initial password is just the c-gskf part of your login name
# followed by the last four digits of your student I.D.  number.  If Mr.
# Gauss has ID number 000735421 then his initial password is gskf5421,
# regardless whether his login name was c-gskf or c-gskf3.    If the
# login fails try again and then try the different login names suggested
# above.  Another possibility is that your account was created using
# your social security number (which used to be used for student ID
# number).  If failure continues find a lab assistant and he/she will
# help you.  
#      Once you are logged in successfully a ``local'' window should
# appear.  Notice that it has various parts: borders on the top (title
# bar), borders on the side (scroll bar), etc.  If you move your mouse
# on its pad your pointer (called cursor) moves around the screen. If
# you want to work in a window, the cursor should be in it.  
#   
# 1b)  Changing password:  Sometime within the first several weeks of
# classes you must change your default password into a personal one. 
# You do this as follows: Get your cursor into a local window.  Type the
# unix command passwd, followed by return, and follow the directions. 
# Your new password should be exactly 8 characters long.  Don't choose a
# word in the dictionary or a proper name.  Composites of dictionary
# words, like strawdog, are good.  Even better is to use one or two
# upper case letters, e.g. strAwdog.  For still more security, use some
# digits, e.g.  strAw4o9.  Note that it takes about 30 minutes for a new
# password to take effect.  Also, you should be aware that if a password
# is not changed within the first two weeks of class, then your computer
# account will be disabled for security reasons.
# 
# 1c)  Logging out: Move the cursor out of all windows (into the
# background), press the left mouse button and choose the last menu
# item: Exit X-Windows.  (You probably don't want to do this now, but at
# least locate the menu item for later.)
#      At this point you are ready to get used to the X-windows:
# 
# 2)  X-windows, opening netscape, maple, mail, more:  Go through the
# document Introduction to Xwindows in the Lab, which you should have a
# copy of.  There should also be copies of this document in the front of
# the room.   Xwindows are like most windows in most ways; your aim here
# is to experiment to see how to open and close windows, resize them, 
# move them about, and find them if they happen to get hidden. When you
# get to the end of the document you should also have opened a NETSCAPE
# window and a MAPLE  window.  Note:  The command for the current 
# (default) version of Maple is xmaple &, which you can type into a
# local window, followed by <enter>.   This is version 6, and you can
# also find it as an option on one of your mouse buttons, it's your
# choice.   If you are using an earlier version of Maple (V5 or V4, for
# example), there are slight
# differences in commands and syntax, and they may confuse you once or
# twice. 
# 
# Further information:  If you want more in-depth information about the
# computing facilities in this lab, you might pick up a copy of the
# handout A Crash Course on CSC Facilities, from the front table.
# 
# 
# 
# 
#      If you are starting the tutorial at this point (because you're
# doing it on your own at another location), you should have opened a
# mapleV6 window and a web browser window. 
# 3a)  Math Department resources:  There is  introductory material about
# Maple on our web pages.  If you wish to see what's available use the
# browser window you made in step (2) above, and go to the Math
# Department  home page http://www.math.utah.edu/.  By following links
# from here you can find current and future course offerings, faculty
# information, and much more.  Since you are students,  interested in
# Maple information, click on "students" (at the upper left of the
# page), and you are directed to http://www.math.utah.edu/ugrad/, the
# undergraduate home page.  If you choose "Undergraduate Computer Lab",
# followed by "Selected software and tools", you will find some links to
# Maple information. 
# 3b)  More Information.  Maple has its own introduction, as well as a
# new user's tour, see item 5 below.
# 
# 4) Maple:   Move your cursor into the "Untitled" (new) Maple window
# which you created in step (2).  Maple is partly just a very fancy
# calculator; it can do practically any undergraduate mathematics
# computation or symbolic manipulation.  You can write programs in Maple
# and draw pictures as well.  If you are doing a homework assignment you
# can intersperse text with computations using the toolbar:  to get a
# computation prompt click on the ``[>'' box near the top.  To insert
# text click on the ``T'' box.  Or you can change command fields
# (starting with "[>") into text fields by putting the cursor into them
# and then choosing "T".  You can use the mouse to cut, paste, and edit
# a document.  You can change fonts, formats, and use other standard
# text editing tools by choosing appropriate menue items.  This document
# you are reading is a Maple document even though it is largely text.
#       To give you a flavor of what Maple can do, we will try a few
# commands.  They should begin on a line having  a command prompt ``>'',
# and should be ended with either a semicolon ; or a colon :  If you end
# with a semicolon you will see visible output, if you end with a colon
# the output will be suppressed even though the command is executed. 
# Maple will not execute a command until you type the ``return'' or
# ``enter'' key.  If you have a multiline command use ``shift-return''
# to change lines without executing. If you mess up your parentheses or
# brackets or do something else which makes your command unexecutable
# you will get a ``syntax error'' message and Maple will try to point
# out your mistake.  After a while you will become good at fixing these
# mistakes but they can be annoying at first.  Spaces are ignored in
# Maple, so you may use them to make input easier to read.  You can
# enter explanatory comments in a command line by inserting a ``#'' to
# the left of the comments; Maple ignores any text after the #. 
# Sometimes this is more informative then entering nearby explanatory
# text, especially if you are explaining various steps in a subroutine.
#      Now, let's try some commands. (You try just the math commands,
# the editorial comments were only added to explain what the particular
# commands are illustrating ! )  Check that you understand what each
# command is doing.
# 
> 3+4; 4+5: 6 * 7;     #one of these computations will not be shown
>           #even though all three will be done, illustrating the 
>           #difference between a semicolon and a colon
>                  
> (3+4)7;       #if you want to multiply you must use *, so after
>           #trying the command as given, insert a * to fix the
>           #resulting syntax error.  You can execute a line or
>           #execution group (bracketed on the left) if
>           #your cursor is anywhere in it. You can move the
>           #cursor with the mouse or the arrow keys. Maple will
>           #try to put it in a good place if it detects an error.
Error, unexpected number

> (3+4)^2/7; 3+4^2/7; evalf(3+4^2/7);  #the evalf  command gives a 
>           #decimal approximation instead of an algebraic 
>           #expression. Notice that if given a choice, Maple 
>           #computes powers first, then multiplies and divides, 
>           #and finally adds or subtracts.
> diff(x^2,x);  #``differentiate x^2 with respect to x''
> diff(exp(sin(x))*x^3,x);  #a harder differentiation problem
>          #you should get output:

                                   3                  2
               cos(x) exp(sin(x)) x  + 3 exp(sin(x)) x

> f:= x-> exp(sin(x))*x^3;  #this is the syntax for defining a 
>          #function, in this case the function we just 
>          #differentiated

> diff(f(x),x);         #should get the same answer as before.  
> int(t^2*exp(t),t);   #``integrate (t^2)*exp(t) with respect
>           #to t'' (Maple doesn't put in the integration constant.)
> int(t^3*exp(sin(t)),t);  #this shows that Maple is not God, you 
>           #will get

                          /
                         |   3
                         |  t  exp(sin(t)) dt
                         |
                        /


>           # since if Maple can't find an elementary function 
>           #antiderivative it just echos what you put in.  
> evalf(int(t^3*exp(sin(t)),t=0..1));  #But you could do
>           #a definite integral (numerically) even if Maple
>           #can't compute an elementary antiderivative
> sum(3^(-n),n=1..100);  #add a geometric series part way,
>           #this is the series 1/3 +1/9 +1/27 + ...

> evalf(%);  #get its decimal value.  The symbol % refers to the 
>           #last thing which Maple has computed, the command evalf 
>           #gets its numerical value
> 
> Sum(3^(-n),n=1..infinity); evalf(%);  #add the series
>          # all the way to infinity.  Sum with captial
>          #S writes the sum but doesn't evaluate it,
>          #but then evalf(%) does.          


> Sum((.001)*(n/1000)^2, n=1..1000); evalf(%);
>          #This is a Riemann sum for the integral of x^2
>          #from 0 to 1, with 1000 equal subdivisions.
>          #Sum with capital S writes the summation, but
>          #doesn't evaluate it.  evalf(%) gives its value.         

> int(x^2,x=0..1);   #this is the exact value of the same integral
> 
> Pi;exp(1);evalf(Pi);evalf(exp(1));infinity;
>           #some important numbers
# 
# It is always a good idea to save your maple file periodically.  Do
# this now using the tool bar, using the "save" option under the "File"
# menu item. The first time you save a new file, and any time you use
# the "save as" option, you will be asked to name your file and say
# where you want to keep it.  You name it in the left part of the box,
# being careful to keep the suffix ".mws" so that Maple knows this file
# is a Maple Work Sheet.  If your directory is new you probably haven't
# made any subdirectories yet (unix command mkdir, in a local window),
# but as you create more files you may wish to organize where you save
# them using the tree structure of  Unix directories, which you can
# follow in the right side of your saving box.  If  you need more help
# saving your file see the instructions in the Introduction to Maple V.4
# in the Undergraduate Computer Lab handout at the front of the lab, or
# ask an assistant.  It will probably happen some time that you will
# crash Maple long after your last save.  This will not make you feel
# happy.  
#      Now we will see how to print a hard copy of our file.   First,
# scroll to somewhere in your worksheet and add some text with the ``T''
# menu item.  Maybe scroll to the top and put the title ``My first Maple
# worksheet'' (center it with the menu option on the right side of the
# toolbar), as well as your name and today's date. When you are doing
# your Maple projects you will be expected to hand in more than a page
# of computations: You will be expected to add text explanations of what
# you've been doing. 
#      Now, go to the file menu option and choose the print option.  You
# get a little printer setup box.  If you then click on the print
# command diamond, followed by ``enter'' or by a click on the print box
# at the bottom of the window, a paper copy will come out of one of the
# printers at the side of the lab.  Do this now.  Alternately, if you
# want to use a different printer, you can use the output to file
# diamond to create a postscript file which you can then print anywhere,
# using the appropriate unix commands.  For example, to print a
# postscript file to the lab printers from a local window, the command
# would be "lpr -P b129lab1",  or "lpr -P b129lab2", followed by the
# return key.  You do not put in the quote marks, but you are careful to
# leave spaces exactly as indicated.  The lpr stands for line printer,
# the -P stands for print, and the b129lab1 or b129lab2 are the names of
# the two printers.  If you have trouble printing ask a lab assistant
# for help.
# 
# 5)  Differential Equations, and using Maple's help windows:  So, it
# looks like Maple might be interesting to use in Calculus, but how do
# we find out what it can do for us in that subject, or in another
# subject, say differential equations?  An answer for DE's can be
# obtained by perusing your Computing Projects textbook  (which you can
# pick up for free at the main Math office JWB 233) , but it is also
# instructive to use the Help directory located at the upper right-hand
# corner of the maple window.  That's what you're going to do now.
#   
#  5a)  In mapleV6 (and V5) there's an online tutorial:  Click on the
# ``Help'' box, and then on the  choice ``New User's Tour''.  Probably
# this tour will superimpose onto your current Maple session.  Or maybe
# you can't see the new tour because its hidden behind your current
# window.  In the latter case use the ``window''menu option to change
# windows.  The tutorial gives examples from  many areas of mathematics,
# including differential equations, which you can peruse at your
# leisure. In this tour you will be able to put your cursor onto any
# command line, type return, and see what the command does.  If you wish
# you can explore now, or you can continue with the Math 2250 notes
# below and come back to the tour later.  To close the new tour (or any
# other top window), use the ``close'' option inside the ``file'' menu
# item.  To keep the tour open but bring another window to the front,
# use ``window'' menu item.
# 
#  5b) Getting an on-line copy of this Math 2250 tutorial:    (This is
# also the process you will go through to get your  project  solution
# templates.)  This xeroxed tutorial is available as either
# http://www.math.utah.edu/~korevaar/2250tutorial.txt   or
# http://www.math.utah.edu/~korevaar/2250tutorial.mws  The first file is
# in "Maple text" format, and will be easier to read from your browser. 
# The second file is in "Maple work sheet" format, and will look ugly in
# your browser, but if you're lucky your own Maple will be able to open
# it directly once you save it from the internet.  The advantage of the
# second format is that plots, formatting and output are reproduced
# exactly as in your xerox copy.  The disadvantage is that some
# computers (not the ones in our lab) may not be able to understand it. 
# The advantage of the first format is that it is more universal.  The
# disadvantage is that Maple output and formatting may be lost.   For
# now, go to the first web address using your browser window.   If
# you're using netscape as  your browser, pick the  ``file'' menu
# option, choosing ``save as''.  Unless you give another name for it you
# will now save this document with the  same name it has on the web,
# namely ``2250tutorial.txt''.  It creates a copy of this file in your
# home directory.  Do it.  
#      Now return to your Maple window and use the ``file'' menu item to
# open``2250tutorial.txt''.  In order to open a ``Maple text'' document,
# which this is,  you must chose open from the file menu option.  In the
# resulting open file dialog box go to the  filetype box at the bottom,
# click on the triangle to see the list of choices, and use your mouse
# to choose ``Maple text.''  At this point ``2250tutorial.txt should
# appear as a choice in the central box.  Click on it with the mouse to
# highlight it and then click ``OK'' or type ``return''.  A copy of this
# tutorial should then appear in your Maple window, as a Maple document
# that you can work in.  The copy is not as pretty as your xerox (the
# execution groups are all single lines, and the text formatting is not
# as neat, and some output may be lost), but it is O.K.  It has text and
# it has Maple input.  (If you had saved the ".mws" version you could
# open it as a maple worksheet.)
#      You can modify the text and input using the toolbar and menu
# options.  You will notice many brackets on the left of the document. 
# These are execution groups.  Maple will execute everything in one
# execution group at once, and then move the cursor to the next
# execution group.  You can create large execution groups by
# highlighting sections of a document, going to the Edit option and
# picking join execution groups.  You can remove brackets by
# highlighting them with the mouse and deleting them with the delete key
# or the menu option.  And you can insert new prompts or new text
# wherever your cursor is, by using the > or T buttons on your toolbar.
# 
# 5c)  An example of help, related to your first project :  You may
# proceed now whether or not you created a copy of tutorial.txt in your
# own directory and opened from Maple. 
#       Reading the material on pages 43-44 you  notice that you will
# need Maple to draw a slope field for you.  Recall this concept:  First
# order differential equations have the standard form      

                             dy
                            ---- = f(x, y)
                             dx

# and their  solutions are  functions y(x)  which make this equation
# true.  Geometrically that means that if y(x) is a solution, and if you
# plot its graph in the x-y plane, then the slope of the graph at each
# point (x,y(x)) is given by the formula f(x,y(x)).  One can therefore
# work backwards, and this is what we did in section 1.3 of the text, 
# to see what graphs of solutions look like: Construct slope
# (directions) fields: at a representative number of points (x,y)  in
# the plane plot short segments having slopes f(x,y).  If you plot
# enough slopes you will be able to sketch in solution curves to the DE.
# The resulting picture is called a slope field or a direction field.  
# Let's use Maple to illustrate these ideas.
#      Does Maple have commands to draw slope fields, to solve general
# differential equations, to graph  the solutions?????? OF COURSE!!!!! 
# Let's try to find them.
#      Click on the Help option at the upper right corner of your Maple
# window.  A little window opens with further choices.  Pick the ``using
# help'' option.  Thus so far you have done help/using help.  You may
# need to change windows (menu item) to see the help window.  Then in
# the grey box at the top left, click on ``Mathematics''.  Then make
# successive choices so that you create the chain
# Mathematics/Differential Equations/DEtools/plotting/dfieldplot.  This
# final word is actually a Maple command to make pictures of direction
# fields.  For many queries, this table of contents approach to getting
# help ususally works best, but you can also use the ``topics search''
# or ``full text search'' options at the top level of help.
#      If you now read about the command dfieldplot, you will see that
# it seems to plot direction fields for first order DE's, among other
# things.  Often you can get an idea of how a command works by skipping
# to the end of the help file and copying one of the sample commands
# into your own worksheet.  For example, use your mouse and menu options
# now to copy the block of  commands from the end of the dfieldplot help
# file and paste them into your worksheet. Then execute the entire block
# by getting your cursor anywhere into the block and typing ``return''.:
#  The block you are getting should look something like (you don't want
# to type this in by hand!): 
> 
> with(DEtools):
> dfieldplot([diff(x(t),t)=x(t)*(1-y(t)),diff(y(t),t)=.3*y(t)*(x(t)-1)],
> ### WARNING: incomplete quoted name;  use ` to end the name
> [x(t),y(t)],t=-2..2,x=-1..2,y=-1..2,arrows=LARGE,title=`Lotka-Volterra
> model`, color=[.3*y(t)*(x(t)-1),x(t)*(1-y(t)),.1]);
> dfieldplot(diff(y(x),x)=1/2*(-x-(x^2+4*y(x))^(1/2)),y(x),x=-3..3,y=-3.
> .2,
> title=`Restricted domain`,color=1/2*(-x-(x^2+4*y)));


# You should execute this block.  You will find that there were three
# commands in it: the first one, ``with(DEtools):'' loaded a library of
# tools for solving differential equations.  If you had ended this
# command with a semicolon instead of a colon Maple would have listed
# all the commands it was loading from this library.  You can find out
# more about all of these tools by using the help files if you wish.
# Then there are two plotting commands.  The first one creates a
# ``velocity'' vector field (with fat arrows) in the plane for a system
# of two first order differential equations, apparently called the
# Lotka-Volterra model.  This is also sometimes called a predator -prey
# system, see chapter 6.3, page 360 for a similar picture.  The second
# picture is closer to what we want.  It is a direction field for the
# differential equation
> diff(y(x),x)=1/2*(-x-(x^2+4*y(x))^(1/2));

              d                             2
              -- y(x) = - 1/2 x - 1/2 sqrt(x  + 4 y(x))
              dx

# The reason Maple only draws slopes above a certain parabola, is that
# the slopes are not defined as real numbers below it, as you can see
# from the right side of the above formula.  Looking at the commands
# gives you an idea of the syntax which is used.  More specific
# information can be found in the complete help file.  For example, the
# command which draws the last plot clearly has a place to put the DE, a
# place to put the x and y range of the picture, and there seem to be
# various other options available as well.
#      To close the help files after you've used them, use
# the``file/close '' sequence in the toolbar, or the equivalent key
# stroke given next to it, which is simultaneous ``control-F4'' on my
# workstation.  Or you can keep them around and return to your worksheet
# with the ``window'' menu option.
#      Having played with these commands, could you figure out how to
# make the slope field part of the picture on page 44?  Of course, the
# other resource you have is your computing projects book.   
# 
#  6) Commands for  your actual first project:  Let's try to make the
# slope field on page 44.  Using my resources I come up with the
# following commands (see also the Computing Projects manual, pages
# 3-5.)  
> with(DEtools):
> deqtn:=diff(x(t),t)=.01*x(t) - .0001*x(t)^2;
> dfieldplot(deqtn,x(t), t=0..1000, x=0..200);

# If you want to include some solution graphs, you can use the command
# DEplot, and get a picture like the one on the top of 44:
# 
> DEplot(deqtn,x(t),t=0..1000,{[x(0)=.1],[x(0)=1],[x(0)=10],
> [x(0)=100],[x(300)=150],[x(600)=150]},x=0..200,
>         colour=`black`,linecolour=`black`,
>         arrows=`line`, dirgrid=[30,30]);

# 
# Now,  you might recognize the DE in this problem as a separable one,
# so you can actually solve it in closed form using the methods of
# section 1.4, and this means Maple can solve it symbolically as well. 
# You can use the steps outlined on page 43, or you can directly use
# dsolve:  The following commands may prove useful for the actual
# project:
> dsolve(deqtn,x(t)); #general solution
> dsolve({deqtn,x(0)=50},x(t)); #initial value problem
> int(1/(x*(100-x)),x); #compute integrals as part of solving
>      #separable equations
> convert(1/(x*(100-x)),parfrac,x); #do partial fractions.