<\body> with CAS> <\expand|make-title>

> <\abstract> Some famous computer algebra systems (CAS) will be integrated into the Linux-based platform as CAI tools for the higher Mathematical courses. The mini memo will cover some \ topics as follows: <\expand|enumerate-numeric> Maxima: This software is famous for its "Symbolic calculation", including finding limit of functions, differentiation, integration of functions, series expansion etc; Octave: This software has full features about "Numerical calculation" and \ simple syntax to be programmed.\ Gnuplot: The powerful plotting package, it almost has no problem to deal with the scientific or data plotting especially for 2D plots. : Certainly, you have known if you can see this article. Word processing and "GUI" of CAS. It is the simplest way for you to make mathematical document from input to output, I think. All the packages will be integrated into bootable Linux CDROM. In other words, you can continue your paper work in any place with this platform ( and disk for storing data) even in net-coffee store. Now, let us explore the journey and hope you enjoy it! In this chapter, the individual basic introductions about , core of this book, and the tools about computer algebra systems, CAS, and graph will be given. > This is not only introduction for the people who want to learn how to use , but a note for newbie with some experiences. We try to follow the rules to describe how to making documents. From now on, [] means the option, , in the menu bar. While you execute the from X-terminal, you will see the window. If you are first use this application, the document is also opened in the worksheet windows. Press the button in the menu bar, [File] > [New], to open a new buffer for writing document directly or just as doing something "presume document setting": <\expand|enumerate-numeric> [Document] > [style] > [format that you want to write], almost styles generally used in , for example: article, letter etc., can be found in this menu.\ [Text] > [Title] \ > [Make title] \ > [Abstract], author, date and address can also be added.\ [Text] > [Section] and so on. The content part of article can be manipulated just as any work processing programs do. \ Generally, part 1 and 2 \ can be omitted if your do not hear about or simply type a document. For ers, the document uses "letter" style. If you do not care about how to make content with any mathematical symbols, now stop reading. If text input can not satisfy you, just go continuously. The philosophy for to manipulate the contents is similar to does ( but not equal to). Content inputs are divided into three types: modes.\ <\expand|enumerate-numeric> Type text as any WYSIWYG editors do. Math content, i.e. mathematical symbols together with text content in one line. Then:\ <\expand|enumerate-alpha> Press the menu bar []>, or [Text] > [Mathematics] > [Formula] or press , to change the mode from text mode into math mode. The third line of menu bar changes for suitable mathematical choice icons. But the usages are still functional mostly.\ Press [>] or back to the text mode after inputting mathematical content. The symbols are just like these of usages in . For example: backslash> produces >. Display Math content, i.e. independent line or lines for Mathematical symbols.\ <\expand|enumerate-alpha> Choose []> > [Equation], for one-line mathematical input, corresponding command is double US dollar sign; choose []> > [Equations], multi-line mathematical input, corresponding command is environment. Use [>] to change next cell and [Enter] to change new line. The exact cell position is shown at the right-down corner.\ How do I know at which is in "text" mode or in "math" mode? The same environment is enclosed by a> rectangle. Yor can move by the mouse or the "right arrow", [>>], and "left arrow", [>>], keys to change the mode. Now, let us show some examples: <\example> (Riemann Integral) \ \ \ \ [Text][Environment] >> [Example], just like , it is auto-numbered. Press [Enter] to input the following: \; Let > be defined on the closed interval >, and let >> be a partition of > given that <\expand|equation*> a=x\x\x\\\x\x=b > where x>> is the width of the >th subinterval. If >> is any point in the >th subinterval, then the sum <\expand|equation*> <\with|paragraph mode|left> <\with|right margin|> f(c)\x, \ \ \ \ > \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \c\x> is called the >> of > for the partition of >>. \; The parts in blue color are in math mode (press and then [>>] back to the text mode again) and the parts in red are in display Math mode ( press []> > [Equation] and move cursor by mouse into next line back to the text mode). The messages shown at the bottom line of may help you at which mode you are and what you have done. <\example> As above, if the limit <\equation> \\<\|\|\> \0>f(c)\x> exists where \\<\|\|\>=i\n>\|x-x\|>>> then > is integrable on > and the limit is denoted by <\equation> \\<\|\|\> \0>f(c)\x=f(x) d x.> The limit is call the . The number of equation is produced by [Mathematics] >[Number equation] in display Math mode. \; In these two examples, just type as you do in environment, for example >>> means upper script input and >> means lower script input, but not all same. <\example> (Gauss Law) In Math or display Math mode, we can adjust the script by: <\expand|itemize-dot> Press [Mathematics] > [Script] >[Script below] or [Script above], to add new line below or above the math symbol. This is as like the effect of limit>> command after normal mathematical command.\ Input symbols under or above other symbols in display math mode, as above adding script space ahead, <\expand|equation*> <\with|mode|text> <\equation> |S>\\\ \ \ \ d S =4\q After working, choose [File] >[Save] and give a filename, for example \ . This will create a file named as in your home directory. If you work on Knoppix Linux, everything will be disappeared after turning off the power since it works on RAM. You had better to use the following command to backup worksheet: <\code> \ mcopy filename.tm a: Now you can turn off the system. Then you can fire up your and enjoy! Not only the powerful ability of publishing text and math documents, CAS applications can be executed within session of , solving problems and making graphs. The computation ability of CAS will be introduced later and how-to making graphs will be introduced first. Some \ CAS sessions allow to produce in-line postscript graph in worksheets, i.e. Gnuplot, Octave, but some are not, for example Maxima. For the latter case, users have to embed the result graphs into the worksheet. \ <\example> How can do with CAS? The answer is very simple: , including , for example Maxima and Yacas,, for example Octave, and . <\note> Remember, Before quit out from the CAS session, remember to close session by pressing the third icon in the third line of menu bars. Here, we will give a brief introduction for CAS working within . <\definition> is called a function of , subset of real numbers, if there is only unique \> for each point, in . <\example> Maxima session: [Text] >[Session] > [Maxima] <\session|maxima|default> <\output> GCL (GNU Common Lisp) \ Version(2.5.0) 週五 \ 2月 \ 7 19:50:19 CST 2003 Licensed under GNU Library General Public License Contains Enhancements by W. Schelter Maxima 5.9.0rc3 http://maxima.sourceforge.net Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. This is a development version of Maxima. The function bug_report() provides bug reporting information. \; <\input|<\with|color|red> (C1) > f(x):=x^3+sin(x) <\output> <\with|mode|math> >fx:=x+sin x> \; <\input|<\with|color|red> (C2) > f(%pi) <\output> <\with|mode|math> >\> \; <\input|<\with|color|red> (C3) > h(x):=block([u:x,w], if (u\0) then w:sin(x) else w:-sin(x), u+w) <\output> <\with|mode|math> >hx:=u:x,w,u\0w:sin xw:-sin x,u+w> \; <\input|<\with|color|red> (C4) > h(1) <\output> <\with|mode|math> >sin 1+1> \; <\input|<\with|color|red> (C5) > h(-2.0) <\output> <\with|mode|math> >-1.090702573174318> \; <\input|<\with|color|red> (C6) > integrate(x*((x-1)/(x+1))**(1/2),x); <\output> <\with|mode|math> >4*

>+1|8>-

>-1|8>+

>->|-x-1|x+1>+x-1|x+1>+4>> \; <\input|<\with|color|red> (C7) > diff(d6,x) <\output> <\with|mode|math> >4*--+x-1|x+1>-x-1|x+1>*3*

>->|-x-1|x+1>+x-1|x+1>+4>+-x+1>|16*>*

>+1>--x+1>|16*>*

>-1>+>*-x+1>|2>--x+1>|2*>>|-x-1|x+1>+x-1|x+1>+4>> \; <\input|<\with|color|red> (C8) > plot2d(x+sin(x),[x,0,%pi]); <\output> <\with|mode|math> >0> \; <\input|<\with|color|red> (C9) > \; Here, you can directly define functions or use > structure to define complicate function as (C3). Here, is defined as if 0> and as if 0>. Plotting graph is the best way to understand function. Maxima also applies plot functions, . The graph is produced in another X-terminal but not in the worksheet. Use another KDE applications to produce the graph and embed the graph into the worksheet by:\ <\expand|enumerate-alpha> Press [KDE] > [graph] >[ksnapshot] to catch the graph and save it, Press [Insert] > [Image] > [Insert Image] to embed the picture file and scale the sizes of height and width if necessary. <\example> (3D Brownian Motion Particle's Trajectory) Although the graphs of produced by Octave can be embedded into worksheet in general, but it does not always work. example, the following Octave code is is used to simulated Brownian motion within 1 second with time interval, sec. Then output the 3D trajectory of particle in encapsulated postscript format: <\expand|enumerate-roman> Prepare the image file in the one of following formats: PS, EPS, pdf, fig, png, tif, xpm, fig etc. Or, Make graph from CAS and save the screen-shot or store it directly into storage. For\ <\expand|enumerate-alpha> Open [Text] >[Session] >[shell], execute the following octave code to make graph: <\session|octave|default> <\output> GNU Octave, version 2.1.45 (i686-pc-linux-gnu). Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003 John W. Eaton. This is free software; see the source code for copying conditions. There is ABSOLUTELY NO WARRANTY; not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE. \; Report bugs to \bug-octave@bevo.che.wisc.edu\. \; <\input|octave\ > t_total=1; <\input|octave\ > t_step=500; <\input|octave\ > dim=3; <\input|octave\ > init_po=[0,0,0]; <\input|octave\ > increment=sqrt(t_total/t_step)*randn(t_step,dim); <\input|octave\ > po=[init_po;increment]; b(1,:)=zeros(1,dim); <\input|octave\ > gset parametric;\ <\input|octave\ > gset xrange [-2:2];\ <\input|octave\ > gset yrange [-2:2]; <\input|octave\ > gset zrange [-2:2]; <\input|octave\ > for i=2:t_step b(i,:)=po(i,:)+b(i-1,:); endfor <\input|octave\ > gset set term postscript enhanced color eps; <\input|octave\ > gset xtics 1;gset ytics 1; gset ztics 1; <\input|octave\ > gset output '/home/cchuang/texmacs/brownian.eps'; <\input|octave\ > gsplot b title "3D Brownian Motion"; <\input|octave\ > \; After executing the code, brownian.eps is created. Terminate the \ session and back to the environment. Now choose [Insert] >[Image] > [Insert image] to embed the result graph as follows:\ |eps>||||||> Octave can use the gnuplot syntax to generate graph, but cannot interpreter the resulted PostScript code. Theirfore output the graph and embed it back.\ If the effect is not good-looking enough, you can consider another application session, Gnuplot. Gnuplot is more flexible since it applies many utility functions to manipulate graph, including the off-line modification. For example, press [Text] >[Sessions] >[Gnuplot] enters the session. Some simple examples as follows: <\session|gnuplot|default> <\output> \; <\input|GNUplot] > plot [-pi/2:pi/2][-2:2] x+sin(x) <\output> |ps>||||||> \; <\input|GNUplot] > set noclip ~set yrange[-30:10] ~plot x*(abs((x-4)/(x+4)))**(1./2.)\ <\output> |ps>||||||> \; <\input|GNUplot] > \; \; <\input|GNUplot] > set key left ~set xlabel 'x:[cm]' ~ plot [-pi/2:pi/2] x+sin(x) <\output> |ps>||||||> \; <\input|GNUplot] > set xlabel 'x:[cm]' ~ plot sin(x) <\output> |ps>||||||> \; <\input|GNUplot] > set pm3d hidden 100 ~set style line 100 lt 3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~set nosurface ~set size 0.7,1 ~set view 80,180,1,1 \ ~set noborder \ \ \ \ \ \ \ \ \ \ \ \ ~set noxtics ~set noytics ~set noztics ~set parametric ~set samples 36 \ \ \ \ \ \ \ ~set isosamples 20,36 ~set ticslevel 0 \ \ ~set nocolorbox \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~set urange [0:pi] ~set vrange [0:2*pi] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ~splot sin(u)*cos(v),sin(u)*sin(v),cos(u) <\output> |ps>||||||> \; <\input|GNUplot] > set xrange[-5:5] ~set yrange[-2:2] ~f(x)=x\0?-1:1 ~plot f(x) <\output> |ps>||||||> \; <\input|GNUplot] > \; Due to the auto-scaling effect, not all the range of domain of is shown in the first graph. After given the range of variable of , we can observe the graph more explicitly. But how about "" in single line with other plotting command lines? Due to bug interpretation of the interface of gnuplot, tm_gnuplot, gnuplot cannot handle this code with ghostscript. Instead of inputting and in different lines, input ">>" \ to add the set option in normal plotting command. Try directly \ execute the above code to see what about the differences between two statements. Almost CAS's have their plotting philosophy. For example, Octave, the other famous one, not only has its built-in command, >>>>, but also fully supports the Gnuplot functions, only by adding a pre "" before the relative plotting commands.\ <\session|octave|default> <\output> GNU Octave, version 2.1.45 (i686-pc-linux-gnu). Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003 John W. Eaton. This is free software; see the source code for copying conditions. There is ABSOLUTELY NO WARRANTY; not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE. \; Report bugs to \bug-octave@bevo.che.wisc.edu\. \; <\input|octave\ > x=linspace(-10,10,1000); <\input|octave\ > y=x+sin(x); <\input|octave\ > plot(x,y,";Function y=x+sin(x);"); <\output> \; \; |ps>||||||> \; <\input|octave\ > gset term postscript enhanced eps color; <\input|octave\