# Math 2270
# MAPLE TUTORIAL
#  September  2001
#      The purpose of this tutorial is to get you going with MAPLE,
# probably in our Math Lab.  You can follow this tutorial to log onto
# the Math Lab machines, familiarize yourself with how they operate,
# introduce yourself to the software MAPLE, and use it to do some basic
# matrix computations. related to the linear algebra we've been talking
# about. Depending on your previous eperience you may want to skip
# various sections.  
#      Your actual Maple assignment can be found by going to the Maple
# page from our course homepage, at
# http://www.math.utah.edu/~korevaar/2270fall01.html.  (This tutorial
# lives there too.)
#       Our class projects can be done in versions five and above of
# Maple, with minor changes.  The Math Lab uses version 7, whereas the
# EMCB and Marriott Labs have version 6.  A student version of Maple for
# home computers is available from the bookstore for approximately $100.
# 
# 1)  Using the Math Computer Lab:
# The Math Department Computer Lab is located in the South Physics 205, 
# i.e. on the East (uphill) end of the second floor of then brick
# building immediately north of the Math building JWB.  Room 205 is just
# inside the East entrance.  Until construction of the new Math Center
# is complete this winter, Physics and Math share this lab.  Useful
# information about most aspects of the lab can be found by following
# links from the Math Department web pages, e.g. starting at
# http://www.math.utah.edu/ugrad/lab.  You can find out about X-windows,
# Unix commands, Maple, Netscape, file editing, mail, etc.
# 
# 1a) Logging on:
# Almost all students who are registered in Math 2270, or in any other
# math class,  will already have accounts set up in our lab.  Since
# these accounts are created from University class lists, 
# late-registering students may not be accounted for.  If you fit in
# this case the lab assistant can help you apply for an account -
# provided you have brought your student I.D. to show that you are a
# University student.
#      The lab machines are left on - move the mouse or hit a key to
# wake yours up.   There are two possibilities for the dialog box which
# appears.  
# 
# (A) If you get a dialog window asking you to "select a server to log
# on to", use the mouse to select one which has ".math" in its name. 
# 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.) 
# 
# (B)  If  instead you get a dialog window which says "Welcome to
# Orion", highlight "options" with your mouse, and from its submenu
# choose "remote login" and then "choose host from list".  This will get
# you the window in (A), from which you proceed as above.  If you are
# confused, ask the lab assistant.
# 
# Your login name is made out of your initials.  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 with the same initials, then rather than  c-gskf,
# they are assigned login names with a numerical suffix,  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 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.  One 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.  You can make
# multiple local windows by pressing the right mouse button and
# highlighting "local".  
#   
# 1b)  Changing password:  Some time 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.  It takes about 15 minutes for a new password
# to take effect.  If a password is not changed within the first several
# weeks of class, then your computer account will be disabled for
# security reasons.  Make sure to write down your password in a secure
# place, in case your memory is not perfect.
# 
# 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:    
# Xwindows are not so different from P.C. windows or Mac windows; on the
# other hand they have their own idiosyncracies, so you should
# experiment to see how to open and close windows, resize them,  move
# them about, and find them if they happen to get hidden. 
# 
# For your Math 2270 purposes you need to be able to open Netscape (to
# download files) and Maple (to work on them).  Open  Netscape by
# finding it on one of your mouse buttons, under the choice "browsers",
# or by moving your cursor into a local window and then typing netscape
# &, followed by <ENTER>.  The & tells X-windows to create a separate
# process for netscape, keeping your local window free for more UNIX
# commands.  Once netscape opens, go to the Math undergraduate pages at
# http://www.math.utah.edu/ugrad, and follow the links to explore 
# computer lab information.  
# 
# Finally, open a Maple window one of two ways:  from the middle mouse
# button you may choose xmapleV7, or you can type the command xmaple &,
# followed by <enter>, into a local window.  The Math Department's site
# license for version 7 of Maple only allows a certain number of people
# to use it at once.  If this number is exceeded you won't be able to
# run version 7, but you can use the Mouse to open xmapleV5.1, for which
# our site license allows unlimited use.  
# 
# 
# 3) Maple:   
#  If you are starting the tutorial at this point (because you're doing
# it on your own at another location or already knew the Math Lab
# system), you should have opened a mapleV6 or mapleV7 (worst case V5)
# window as well as a web browser.
# 
# 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 menu
# items.  This document you are reading is a Maple document even though
# it is largely text.  You will learn by doing - and by asking your
# neighbor or the lab assistant.
# 
# New User's Tour:  
# Under the Help button at the top right of your Maple window you will
# find the choice "New User's Tour".  This is a good way to learn
# generally about what Maple can do.  After making this menu choice
# Maple may hide the beginning of the tour behind other active windows; 
#  use the "Window" menu item to bring the tour window to the front.  
# 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.
# 
# Entering Maple commands: 
# To give you a flavor of what Maple can do, we will try a few commands.
#  Move your cursor into the "Untitled" (new) Maple window which you
# created in step (2).  Commands begin on  lines 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 incorrectly align 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 with the cursor
# and cryptic diagnostic phrases.  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.
>                  
> (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.
> (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;
> diff(f(x),x);         #the same problem done in two steps.  
>           #The first line shows the format for defining
>           #functions in Maple. (Did you use ``shift-ret''?)  

                                              3
                       f := x -> exp(sin(x)) x


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

> int(t^2*exp(t),t);   #``integrate (t^2)*exp(t) with respect 
>           #to t'' (Maple doesn't put in the integration constant.)

                   2
                  t  exp(t) - 2 t exp(t) + 2 exp(t)

> int(t^3*exp(sin(t)),t);  #this shows that Maple is not God:
>           #If it can't find an elementary-function
>           #antiderivative it just echos what you put in.
>           #you should get:  

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

> 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

                             .5112814089

> Pi;exp(1);evalf(Pi);evalf(exp(1));infinity;  
>           #some important numbers
# 
# 
# Entering Text:
# Scroll to somewhere in your worksheet and add some text with the ``T''
# menu item.  (If you move your cursor to a command field and the press
# the [> button with your mouse, Maple makes a new command field
# directly below it.  If you move to the new command field and press the
# T button you will have a text field into which you may write, using
# the rudimentary word processing tools on the Maple menu bar.
# 
# Now scroll to the top of your file and insert 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. 
# 
# Saving Files:
# 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.  Unfortunately our UNIX
# is not set up so that you can visually see your directory/file tree
# like in PC windows or Macs.  You use various UNIX commands in your
# local window to organize your directories.  For more information about
# this,  go to http://www.math.utah.edu/ugrad/helpforunix.html.    
# 
# It will probably happen some time that you will crash Maple long after
# your last save.  This will not make you feel happy but will teach you
# an important lesson.  
#     
# Printing:
# 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.
# 
# Downloading Maple documents:
#  This document is available online in several formats, if you follow
# the links from the 2270-2 homepage
# athttp://www.math.utah.edu/~korevaar/2270fall01.html.  Files with
# suffix ".mws" or ".txt" can be downloaded from your browser and then
# opened from Maple.  The ".mws" suffix means the file is a Maple
# Worksheet, and should open from Maple to look just like this file. 
# The ".txt" suffix is Maple text, and you should only try it if the
# ".mws" file doesn't open.  It's a cruder form of this file, but
# potentially more universal.  The ".pdf" versions of the file are for
# reading from your browser (with Acrobat reader), but you can't execute
# these from Maple.
# 
# Go to the course homepage address using your browser, follow the link
# to the Maple page, and find and save the 2270proj1a.mws file to your
# home directory.
# 
# Now return to your Maple window and use the ``file'' menu item to open
# 2270proj1a.mws.  It should appear in the central box after you choose
# "open".  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. 
# Notice you can use the "Window" menu item at the top of your Maple
# window to change between various open files.  (Sometimes when you open
# a new file it goes to the back of your pile.  Then use the Window
# option to bring it back to the front.)
# 
# If you know the exact URL of a Maple file, and you are in version 6 or
# 7 of Maple, you can use the "open URL" option under "File" to have
# Maple do the downloading and opening for you.  This saves steps if it
# works correctly.
# 
# If you couldn't open the Maple worksheet file, go back to your browser
# and save the ".txt" version of the tutorial.  Then go back to Maple,
# and follow these directions:  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.
#  
# Execution Groups:
# 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 already
# learned how to insert new prompts or new text wherever your cursor is,
# by using the [> and T buttons on your toolbar.
# 
# 
# 4a)  Linear Algebra,  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
# linear algebra?  It is  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.
#   
#       Can Maple do matrix operations, or even define matrices????  Of
# course!!!  Here's how you would try to find the right commands if you
# didn't know them already, or if I wasn't going to tell you below:
# 
#      Click on the Help option at the upper right corner of your Maple
# window.  A little window opens with further choices.  Pick Topic
# Search, by highlighting it, and a help window should appear.  (If it
# doesn't, it's hidden behind your worksheet; use the ``window'' option
# in your menu to bring it to the front.)  Try searching for  linear
# algebra.  At this point you should get a help window about the linear
# algebra library.  There seem to be a lot of commands. You could try
# finding the one you wanted from this page, or you could do a more
# specific search.  For example, either look up matrix, or click on it
# from the linear algebra page.  Notice if you go to the bottom of a
# help window, you can often figure out how the command works without
# reading all the fine print above.  
#      We can use matrices if we load the linear algebra library of
# procedures:
# 
> with(linalg);   #to hide the list of commands in this
>      #package use a colon instead of a semicolon
# 
# And now we copy the commands from the matrix help window:  (For long
# ones we would use our mouse!)
> matrix(2,2,[5,4,6,3]);  #a 2 by 2 matrix with
>      #successive entries as indicated

                               [5    4]
                               [      ]
                               [6    3]

> matrix([[5,4],[6,3]]);  #same matrix

                               [5    4]
                               [      ]
                               [6    3]

#  So that's how to make a matrix.  By the way,  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 work station.  Or you can keep them
# around and return to your worksheet with the ``window'' menu option.
# 
# 4b)   Some Linear Algebra computations:  Can you figure out the syntax
# of the commands and their meanings?  Some of these commands will be
# useful later,  use the help windows for more details about them.  This
# simple example could easily be done by hand; larger ones could not.
> A:=matrix([[1,1,1],[2,3,2],[3,8,2]]);
>       #coefficient matrix for a linear system,
>       #``:='' is used to define the object on
>       #its left by the construction on its right
> A[2,3];  #one of the entries of A
> b:=vector([1,1,-3]);  #the right-hand side for
>        #a linear system Ax=b
> augAb:=augment(A,b); #the augmented matrix
> C:=rref(augAb); #compute the reduced row echelon form
>       #of the augmented matrix
> x:=col(C,4);  #read off the solution vector to Ax=b 
> evalm(A&*x)=evalm(b); #check your answer
>       #NOTE A good way to do matrix operations use the evalm
>       #command.  Addition is +, but matrix multiplication
>       #is &*.  (Scalar times matrix is *.)
> x:=linsolve(A,b);  #another way to solve linear systems.
> Ainv:=inverse(A); #the inverse matrix (it it exists)!
> x:=evalm(Ainv &* b);  #yet another way, for nonsingular matrices,
>       #to solve Ax=b
> evalm(A&*Ainv); evalm(Ainv&*A);  #just checking!
> evalm(A^3); evalm(A&*A&*A); evalm(A+3*A); 
>       transpose(A);
>          #matrix powers, multiplications, addition,
>          #transpose (what's that?) 
# 
#