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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 21 "Introduction to Maple" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "We introd
uce Maple in three sections:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 21 "1) Basic calculations" }}{PARA 0 "" 0 "" {TEXT 
-1 11 "2) Calculus" }}{PARA 0 "" 0 "" {TEXT -1 18 "3) Linear algebra.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "It i
s meant as a crash course, and not to replace the usual texts. There i
s also an intro.mws file in the tutorial section to Maple which one ca
lls by issuing " }{TEXT 19 6 "?intro" }{TEXT -1 105 " at the command l
ine. We expect the reader to have taken introductory courses in mathem
atics and physics." }}{PARA 0 "" 0 "" {TEXT -1 89 "Standard references
 are the two books by Michael B. Monagan et al. published by Springer:
" }}{PARA 0 "" 0 "" {TEXT -1 3 "1) " }{TEXT 257 22 "Maple V Learning G
uide" }}{PARA 0 "" 0 "" {TEXT -1 3 "2) " }{TEXT 258 25 "Maple V Progra
mming Guide" }}{PARA 0 "" 0 "" {TEXT -1 151 "Of course, some of the to
pics covered in those texts are dealt with in our worksheets in the ph
ysics context, which makes for more interesting reading." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
259 29 "Basic calculations and graphs" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "Maple as a Calculator:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 114 "Maple is case sensitive, distinguish car
efully upper-case from lower-case typing.We begin by manipulating numb
ers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "1+17;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "answer:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG\"#=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "answer;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Note that \+
" }{TEXT 19 2 ":=" }{TEXT -1 192 " acts as an assignment operator. The
 equal sign has a different meaning. A common beginner's mistake is to
 use the equal sign instead of the assignment operator and to get thor
oughly confused." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "answ=10;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%answG\"#5" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 5 "answ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%ans
wG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The statement " }{TEXT 19 
7 "answ=10" }{TEXT -1 74 " is an equation, which was entered above, bu
t is not usable. The variable " }{TEXT 19 4 "answ" }{TEXT -1 20 " rema
ins unassigned." }}{PARA 0 "" 0 "" {TEXT -1 15 "We could assign" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ans:=3/4;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$ansG#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "evalf(ans);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++
++v!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "The Digits variable co
ntrols how many digits are carried in floating-point calculations. It \+
can be changed to any integer number." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "Digits;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=20;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 36 "We can evaluate Pi now to 20 digits:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 10 "evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5
&QKz*e`EfTJ!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The symbols for
 Greek letters are obtained by typing their English names." }}{PARA 0 
"" 0 "" {TEXT -1 143 "An awkward matter is that lower-case pi is just \+
a symbol, and not the mathematical pi (that is given as Pi). But as sy
mbols they look the same:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"Pi,pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%#PiG%#piG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"5&QKz*e`EfTJ!#>%#piG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "Exponentiation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "ans:=3^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG\"#\")" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "What happens when we combine symbo
ls into a formula?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:=Pi;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG%#PiG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "b:=r^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
bG*$)%\"rG\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A:=
a*b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*&%#PiG\"\"\")%\"rG\"\"#
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(A,5);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"rG\"\"#\"\"\"$\"&;9$!\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "What if we need the answer for a specific
 value of " }{TEXT 260 1 "r" }{TEXT -1 34 "? Several approaches are po
ssible:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(A,r=5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#D" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "evalf(eval(A,r=5),4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"%by!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "evalf(subs(r=5,A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5j4$[uRj
\")R&y!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "But the following c
annot work, as the second argument in evalf is reserved for the precis
ion (number of digits):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
evalf(A,r=5);" }}{PARA 8 "" 1 "" {TEXT -1 61 "Error, wrong number (or \+
type) of parameters in function evalf" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 90 "If we don't match brackets, we get some help, but it take
s some practice to understand it." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "evalf(subs(r=5,A,4);" }}{PARA 8 "" 1 "" {TEXT -1 14 "
`;` unexpected" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "We are physici
sts, and often we wish to use units. One way to use them in Maple expl
icitly is to enter them as variables, and to set them apart from the a
ctual variables by a distinctive naming convention. For example:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "area:=evalf(subs(r=5*_cm,A),
4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%areaG,$*$)%$_cmG\"\"#\"\"\"$
\"%by!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "It is possible to as
sign variables:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "r:=5*_cm;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,$%$_cmG\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(A,3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$)%$_cmG\"\"#\"\"\"$\"$&y!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#Pi
G\"\"\")%$_cmG\"\"#F&\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "and \+
then to unassign them:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r:
='r':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&%#PiG\"\"\")%\"rG\"\"#F%" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 103 "The best practice for work is to keep variables una
ssigned (or general) until required, and to use the " }{TEXT 19 4 "sub
s" }{TEXT -1 4 " or " }{TEXT 19 4 "eval" }{TEXT -1 78 " command to eva
luate for specific numbers (which is needed for plotting). The " }
{TEXT 19 4 "subs" }{TEXT -1 212 " command is used more often when seve
ral variables are to be substituted; note that they will be substitute
d in a certain order. This ordering becomes important when one substit
uted variable depends on the other." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "subs(_cm=1,Pi=3.14,r=4*_cm,A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$)%$_cmG\"\"#\"\"\"$\"%C]!\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "subs(Pi=3.14,r=4*_cm,_cm=1,A);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"%C]!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
148 "As a final step in this initial session we point out the dangers \+
in assigning/unassigning variables before the final steps of forming a
n expression:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 71 "We look at the volume of a cone, and begin with the area \+
of the circle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\")%\"rG\"\"#F%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 48 "Suppose we played around with particular \+
values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=sqrt(7);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*$-%%sqrtG6#\"\"(\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%#PiG\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "No
w we define the volume:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "
V:=1/3*A*h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*&%#PiG\"\"\"%
\"hGF(#\"\"(\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Now we unass
ign " }{TEXT 261 1 "r" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "r:='r':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\")%\"rG\"\"#F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "This looks as before. However:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "V;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&%#PiG\"\"\"%\"hGF&#\"\"(\"\"$" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 19 "The expression for " }{TEXT 265 1 "V" }{TEXT -1 
17 " was formed when " }{TEXT 272 1 "r" }{TEXT -1 73 " was assigned a \+
specific value. This value was used in the definition of " }{TEXT 267 
1 "V" }{TEXT -1 47 ", or more precisely, the actual expression for " }
{TEXT 266 1 "A" }{TEXT -1 22 " was substituted when " }{TEXT 264 1 "V
" }{TEXT -1 24 " was defined, and thus, " }{TEXT 263 1 "V" }{TEXT -1 
24 " has no connection with " }{TEXT 271 1 "r" }{TEXT -1 22 " anymore.
 Unassigning " }{TEXT 269 1 "r" }{TEXT -1 8 " return " }{TEXT 268 1 "A
" }{TEXT -1 48 " to the general expression (it was defined when " }
{TEXT 270 1 "r" }{TEXT -1 52 " was unassigned, but cannot turn back th
e wheel for " }{TEXT 262 1 "V" }{TEXT -1 1 "!" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "We need to assign the volume again, and since " }{TEXT 
273 1 "A" }{TEXT -1 71 " has been turned back to the original general \+
expression, it works now:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "V:=1/3*A*h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG,$*(%#PiG\"\"
\")%\"rG\"\"#F(%\"hGF(#F(\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
140 "Now we know how to use Maple as a smart calculator. We can perfor
m graphs of expressions, i.e., we have a graphing calculator at this p
oint:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(A,r=0..4);" }
}{PARA 13 "" 1 "" {GLPLOT2D 686 211 211 {PLOTDATA 2 "6%-%'CURVESG6$7S7
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{SECT 0 {PARA 3 "" 0 "" {TEXT 275 20 "Maple knows Calculus" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 158 "We begin by resetting Maple. This comes \+
close to quitting the program and restarting it (we lose all variables
, but we do not recover all the memory used up)." }}{PARA 0 "" 0 "" 
{TEXT -1 189 "Many worksheets have this command at their beginning, as
 you may either want to clear memory from a previous calculation withi
n a different worksheet, or re-run the worksheet independently." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "y:=exp(3*x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"yG-%$expG6#,$%\"xG\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
10 "Note that " }{TEXT 276 1 "y" }{TEXT -1 85 " is an expression in Ma
ple. We will learn how to use mappings in Maple further below." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yp:=diff(y,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#ypG,$-%$expG6#,$%\"xG\"\"$F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 58 "The anti-derivative (indefinite integral)
 is calculated by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Y:=int
(y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG,$-%$expG6#,$%\"xG\"\"
$#\"\"\"F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "A definite integral
 between constant boundaries results in a number:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "int(y,x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&-%$expG6#\"\"'#\"\"\"\"\"$#!\"\"F*F)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "evalf(%,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"%U8!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Euler's number is p
rinted as a non-italicized letter e." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "exp(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "This is not to be confuse
d with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "e;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%\"eG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "whic
h is the unassigned variable " }{TEXT 277 1 "e" }{TEXT -1 85 ". This s
ounds a bit like Pi and pi, except that one enters Euler's number via \+
exp(1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "E;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%\"EG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "No
te that the " }{TEXT 19 4 "subs" }{TEXT -1 54 " command introduced ear
lier does not force evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "subs(x=0,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "Maple knows something about limits." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "limit(y,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "limit(y,x
=-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Let us see why we need " }{TEXT 19 8 "sim
plify" }{TEXT -1 13 " quite often." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "s1:=sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G
-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s2:=cos(x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s2G-%$cosG6#%\"xG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s3:=s1^2+s2^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#s3G,&*$)-%$sinG6#%\"xG\"\"#\"\"\"F-*$)-%$cosGF*F,F-F
-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "s3;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*$)-%$sinG6#%\"xG\"\"#\"\"\"F+*$)-%$cosGF(F*F+F+" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(s3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 
"Keep in mind that the latter command did not change the content of th
e variable s3. For that purpose an assignment is required." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "s3;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&*$)-%$sinG6#%\"xG\"\"#\"\"\"F+*$)-%$cosGF(F*F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s3:=simplify(s3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#s3G\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 
"The above simplification can actually be obtained in another way." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s3:=s1^2+s2^2;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#s3G,&*$)-%$sinG6#%\"xG\"\"#\"\"\"F-*$)-%$cos
GF*F,F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "combine(s3);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "or more specifically" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "combine(s3,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 19 7 "combi
ne" }{TEXT -1 89 " command is useful in other contexts where we wish t
o control the display of expressions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "y:=exp(s)*exp(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"yG*&-%$expG6#%\"sG\"\"\"-F'6#%\"tGF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ex
pG6#,&%\"sG\"\"\"%\"tGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"combine(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,&%\"sG\"\"\"
%\"tGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#%\"sG\"\"\"-F%6#%\"tGF(" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 19 8 "combine " }
{TEXT 2 3 "and" }{TEXT 19 7 " expand" }{TEXT 2 45 " commands allow us \+
to switch hence and forth." }}{PARA 0 "" 0 "" {TEXT -1 133 "Often it i
s necessary to pull out parts of expressions. While this is tedious at
 times, and one tries to avoid it, here is the trick." }}{PARA 0 "" 0 
"" {TEXT -1 47 "An expression has a certain number of operands:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "One can p
ull out the operands one-by-one:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "op(1,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%
\"sG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "op(2,y);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$expG6#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 71 "Suppose we want the argument of the exp function in the s
econd operand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nops(op(2
,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "op(1,op(2,y));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "There is no harm \+
in practicing this a bit on complicated expressions. It shows how Mapl
e assembles expressions in a tree-like hierarchical fashion." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 285 "Now we introduce mappings. They a
re not strictly required for simple function assignments, as we could \+
do everything using expressions. Nevertheless they allow more flexibil
ity in substituting arguments, and provide an easy first step towards \+
subroutines (called procedures in Maple). " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "g:=t->exp(-t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"gGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$expG6#,$9$!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "g;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"gG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$%\"xG!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "In the latter step we used the mapping " 
}{TEXT 279 1 "g" }{TEXT -1 30 " to produce the expression in " }{TEXT 
278 1 "x" }{TEXT -1 151 ". We can plot expressions (as before), and ma
ppings as well, but the latter work differently. Be careful to observe
 what works, and what doesn't below:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "plot(g(x),x=-1..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Many beginners get very confused
 by the above. There is always a logical reason when an empty plot is \+
produced, or when a plot fails!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 176 "It is safer to work with expressions in \+
Maple. Often we manipulate them, and then want to use them as mappings
 in the end. There is a command to turn expressions into mappings:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y:=exp(-2*x)*sin(x)+exp(x)*
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 22 "Note how the function " }{TEXT 280 1 "y" }
{TEXT -1 47 " (blue) has an extremum point (maximum) around " }{TEXT 
281 1 "x" }{TEXT -1 112 "=0.75, and the derivative (green) vanishes th
ere. Make other observations about the function and its derivative." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Note th
at the display of differentiated mapping may appear confusing, as mapp
ings are referred to by just the function name:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "D(sin);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$co
sG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Mappings can be combined in
 Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g1:=t->exp(-t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1GR6#%\"tG6\"6$%)operatorG%&ar
rowGF(-%$expG6#,$9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "g2:=s->sin(2*s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2GR6#%
\"sG6\"6$%)operatorG%&arrowGF(-%$sinG6#,$9$\"\"#F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "g3:=g1@g2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#g3G-%\"@G6$%#g1G%#g2G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "g3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#,$-
%$sinG6#,$%\"xG\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 206 "Of \+
course, the order is important as we are nesting functions (mappings) \+
here. Also note that we need brackets if we want to evaluate directly \+
the combined mapping at some value of the independent variable." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "(g2@g1)(x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$sinG6#,$-%$expG6#,$%\"xG!\"\"\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 62 "We can test out our knowledge of function
s and their inverses:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(l
n@exp)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$expG6#%\"xG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 
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0 "" {MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%#lnG6#-%$expG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ex
pand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$expG6#%\"xG" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "Now, Maple can be very fussy, wh
ich can drive physicists up some trees... In those cases use carefully
 the following (which simply assumes nice things about the variables, \+
which might be true - or sometimes not!)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "simplify(%,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We will have to le
arn about the " }{TEXT 19 6 "assume" }{TEXT -1 77 " system, which allo
ws to make assumptions about variable in a controlled way." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(x,real);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 11 "ln(exp(x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%#x|irG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ln(exp(s))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$expG6#%\"sG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "If we wish to drop the assumption \+
on " }{TEXT 282 1 "x" }{TEXT -1 17 ", we unassign it." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:='x':" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "exp(ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"x
G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ln(exp(x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#-%$expG6#%\"xG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 30 "Mappings can also be iterated." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "y:=sin@@3;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"yG-%#@@G6$%$sinG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%#@@G6$%$sinG\"\"$6#%
\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#--%#@@G6$%$sinG\"\"$6#\"\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+uZI%y'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(sin
(sin(sin(1))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+uZI%y'!#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Note that Maple reserves the notat
ion sin^" }{TEXT 283 1 "n" }{TEXT -1 105 " for the case of the repeate
d application of the sine function. It will never display sine raised \+
to the " }{TEXT 284 1 "n" }{TEXT -1 84 "th power as is usually done in
 Math texts, as this leaves room for ambiguity. sin^2(" }{TEXT 287 1 "
x" }{TEXT -1 25 ") in Maple means sin(sin(" }{TEXT 286 1 "x" }{TEXT 
-1 13 ")), not (sin(" }{TEXT 285 1 "x" }{TEXT -1 5 "))^2." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 90 "The system of assumptions is important in
 integral evaluations. Suppose we wish to obtain:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "int(exp(-alpha*r),r=0..infinity);" }}{PARA 
6 "" 1 "" {TEXT -1 68 "Definite integration: Can't determine if the in
tegral is convergent." }}{PARA 6 "" 1 "" {TEXT -1 34 "Need to know the
 sign of --> alpha" }}{PARA 6 "" 1 "" {TEXT -1 57 "Will now try indefi
nite integration and then take limits." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%&limitG6$,$*&,&-%$expG6#,$*&%&alphaG\"\"\"%\"rGF/!\"\"F/F1F/F/
F.F1F1/F0%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ass
ume(alpha>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(exp(-a
lpha*r),r=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$%
'alpha|irG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 "Maple has sma
rtened up compared to previous versions, and tells us that assumptions
 are needed to obtain the result. It normally displays variables with \+
assumptions on them by a tilde attached to the variable, but this feat
ure can be turned off [easiest through the menu, or via " }{TEXT 19 
24 "interface(showassumed)=0" }{TEXT -1 53 "]. However, we can always \+
find out about assumptions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "about(alpha);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Originally alpha, re
named alpha~:" }}{PARA 6 "" 1 "" {TEXT -1 47 "  is assumed to be: Real
Range(Open(0),infinity)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 149 "We finish this section by a real-life ph
ysics example. We solve the Newton equation for free fall at the surfa
ce of a planet (constant acceleration)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"NE:=m*diff(v(t),t)=m*g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NEG/*&%
\"mG\"\"\"-%%diffG6$-%\"vG6#%\"tGF/F(*&F'F(%\"gGF(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 2 77 "We begin by writing the integrals in unevaluated f
orm (upper-case version of " }{TEXT 19 3 "int" }{TEXT 2 156 "). We hav
e to introduce a new dummy variable s for time, in order not to confus
e the integration boundary with the integration variable. We make use \+
of the " }{TEXT 19 3 "lhs" }{TEXT 2 5 " and " }{TEXT 19 3 "rhs" }
{TEXT 2 48 " functions to pull out the sides of the equation" }{TEXT 
-1 11 ". The mass " }{TEXT 288 1 "m" }{TEXT -1 32 " is divided out of \+
the equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "NE1:=Int(l
hs(NE/m),t=0..s)=Int(rhs(NE/m),t=0..s);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$NE1G/-%$IntG6$-%%diffG6$-%\"vG6#%\"tGF//F/;\"\"!%\"sG-F'6$%\"
gGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We can carry out the inte
gration using the " }{TEXT 19 5 "value" }{TEXT -1 9 " command." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "NE1:=value(NE1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$NE1G/,&-%\"vG6#%\"sG\"\"\"-F(6#\"\"!!\"\"
*&%\"gGF+F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Now we introduce
 the relationship between velocity and position (keeping " }{TEXT 289 
1 "s" }{TEXT -1 22 " as the time variable)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "NE2:=diff(x(s),s)=solve(NE1,v(s));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$NE2G/-%%diffG6$-%\"xG6#%\"sGF,,&-%\"vG6#\"\"!\"
\"\"*&%\"gGF2F,F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "NE2:
=Int(lhs(NE2),s=0..t)=Int(rhs(NE2),s=0..t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$NE2G/-%$IntG6$-%%diffG6$-%\"xG6#%\"sGF//F/;\"\"!%\"t
G-F'6$,&-%\"vG6#F2\"\"\"*&%\"gGF:F/F:F:F0" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "sol:=value(NE2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$solG/,&-%\"xG6#%\"tG\"\"\"-F(6#\"\"!!\"\",&*&-%\"vGF-F+F*F+F+*&%\"
gGF+)F*\"\"#F+#F+F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solv
e(sol,x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"xG6#\"\"!\"\"\"*
&-%\"vGF&F(%\"tGF(F(*&%\"gGF()F,\"\"#F(#F(F0" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 12 "We used the " }{TEXT 19 5 "solve" }{TEXT -1 20 " comman
d to isolate " }{TEXT 291 1 "x" }{TEXT -1 1 "(" }{TEXT 290 1 "t" }
{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 128 "Maple has a built-in d
ifferential equation solver, which can crack a large percentage of sol
vable linear differential equations." }}{PARA 0 "" 0 "" {TEXT -1 86 "F
irst we re-define NE as the differential equation for position as a fu
nction of time:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "NE:=m*di
ff(x(t),t$2)=m*g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NEG/*&%\"mG\"
\"\"-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F/\"\"#F(*&F'F(%\"gGF(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(NE,x(t));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$solG/-%\"xG6#%\"tG,(*&%\"gG\"\"\")F)\"\"#F-#F
-F/*&%$_C1GF-F)F-F-%$_C2GF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "N
o boundary (initial) conditions were specified, and therefore we have \+
the general solution with two integration constants." }}{PARA 0 "" 0 "
" {TEXT 293 1 "x" }{TEXT -1 1 "(" }{TEXT 292 1 "t" }{TEXT -1 149 ") ha
s not been assigned. There are two options. The more convenient one is
 actually to not assign x(t), but to extract the solution as an expres
sion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rhs(sol);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,(*&%\"gG\"\"\")%\"tG\"\"#F&#F&F)*&%$_C1GF&F
(F&F&%$_C2GF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "assign(sol);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"gG\"\"
\")%\"tG\"\"#F&#F&F)*&%$_C1GF&F(F&F&%$_C2GF&" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 160 "If one wants to use x(t) again (even for re-defining a
 differential equation), one has to unassign (this is why never making
 the assignment is more convenient):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "x(t):='x(t)':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
189 "One can also find the special solution by specifying initial cond
itions (either as numbers or as symbols). One needs the D differential
 operator to specify the condition for the derivative." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "IC:=x(0)=x0,D(x)(0)=v0;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"xG6#\"\"!%#x0G/--%\"DG6#F(F)%#v
0G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "One solves the set of diffe
rential equation(s) and initial conditions:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "sol:=dsolve(\{NE,IC\},x(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$solG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Thi
s used to work in previous versions, and does again in Maple 6. In Map
le 5 one needs to give Maple a hint:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "sol:=dsolve(\{NE,IC\},x(t),method=laplace);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "It is useful to assign dsolve to \+
a variable. In that case when the empty set is returned (which means: \+
\"I couldn't come up with a solution\", not \"There is no solution\") \+
one sees it right away (otherwise an empty line is produced)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 294 28 "and some \+
linear algebra too!" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "In this pa
rt we deal with Maple's handling of linear algebra problems. Vectors p
lay an important role in physics. We learn that Maple has (at least) t
wo ways of dealing with them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 174 "The first way is through objects called \+
lists. Lists can be ordered (tables) or unordered. They allow to appen
d objects, similarly as sets represent accumulations of objects." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 224 "Vectors \+
are something more, there are mathematical operations defined on them \+
which are performed by arithmetic on the components. Maple has an arra
y construct to deal with vectors and matrices, and a special package c
alled " }{TEXT 19 6 "linalg" }{TEXT -1 184 " that contains many useful
 procedures to compute operations such as determinant, matrix inverse,
 eigenvalues, etc. In Maple 6 a new linear algebra package has been ad
ded; it is called " }{TEXT 19 13 "LinearAlgebra" }{TEXT -1 162 ", and \+
allows superior handling of numerical linear algebra problems. For com
patibility with Maple 5 we will refrain from using the new package for
 the time being." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 49 "We first show the pitfalls of using simple lists:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "xv:=[1,4,9];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#xvG7%\"\"\"\"\"%\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "xv[1]; xv[2]; xv[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "pri
nt(xv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"%\"\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(xv);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7%\"\"\"\"\"%\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "xv;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"\"\"%\"
\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "This looks easy: we define
d a quantity " }{TEXT 19 2 "xv" }{TEXT -1 105 ", and were able to acce
ss its components using square brackets. Maple refers to this construc
t as a list." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "whattype(xv);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%listG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 282 "There was no need to declare the object. However, it is \+
clear from the first line that Maple had enough information to deal wi
th the construct, namely an ordered list of 3 objects. We cannot trivi
ally add another, e.g., xv[4]:an error message is produced by referrin
g to the object:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xv[4];" }}{PARA 
8 "" 1 "" {TEXT -1 33 "Error, invalid subscript selector" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 115 "The reference to an element of the list \+
using the subscript in square brackets is equivalent to the op() const
ruct:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(xv);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "o
p(2,xv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 123 "Suppose we wish to expand the list by adding anot
her element. The following doesn't work, as it generates a list in a l
ist." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xv1:=[xv,16];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$xv1G7$7%\"\"\"\"\"%\"\"*\"#;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 140 "We need to first extract the elements fr
om the previous list. This can be performed the long way or with a sho
rtcut (see ?list for details):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "o
p(1..nops(xv),xv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"\"\"%\"\"*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "op(xv);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"\"\"\"\"%\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 92 "We are ready to expand our list and re-use its name (be careful
 with recursive definitions)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "xv
:=[op(xv),16];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xvG7&\"\"\"\"\"%
\"\"*\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xv[4];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 195 "For many purposes the work with lists in Maple is the most eff
icient way to deal with multi-component objects. It is possible for Ma
ple to run out of steam when lists contain 100 entries or more." }}
{PARA 0 "" 0 "" {TEXT -1 61 "We note that the following practice does \+
something different:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "yv[1]:=1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#yvG6#\"\"\"F'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "yv[2]:=8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%#yvG6#\"\"#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "yv[
3]:=27;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#yvG6#\"\"$\"#F" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(yv);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%&TABLEG6#7%/\"\"\"F(/\"\"#\"\")/\"\"$\"#F" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "whattype(yv);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%'symbolG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "yv[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 108 "Tables indexed by integers are ordered, \+
but the way they are stored and displayed is not always predictable." 
}}{PARA 0 "" 0 "" {TEXT -1 147 "There are functions to convert between
 tables and lists; in older Maple versions one had to watch what happe
ned when the entries were out of order." }}{PARA 0 "" 0 "" {TEXT -1 
208 "Tables are extremely flexible: they provide a very general pointe
r structure, whereby the indexing does not have to be continuous. In f
act the index can almost be anything. The table simply provides the li
nk." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "yv[5]:=101;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#yvG6#\"\"&\"$,\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "yv[4]:=87;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%#yvG6#\"\"%\"#()" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "yv[i]:=2*i;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#yvG6#%\"iG,$F'
\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "yv[j]:=2*j-1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#yvG6#%\"jG,&F'\"\"#!\"\"\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(yv);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%&TABLEG6#7)/%\"jG,&F(\"\"#!\"\"\"\"\"/F,F,/F*\"
\")/\"\"$\"#F/\"\"%\"#()/\"\"&\"$,\"/%\"iG,$F:F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "convert(yv,list);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7),&%\"jG\"\"#!\"\"\"\"\"F(\"\")\"#F\"#()\"$,\",$%\"iGF
&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "It appears that the numeric
ally indexed entries are coming out in proper order, and that the orde
r of generally defined entries is not quite predictable." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 16 "Arrays in Maple.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "To d
eal with vectors and matrices where the indexing is in terms of intege
r numbers Maple has the array construct. In the " }{TEXT 19 6 "linalg
" }{TEXT -1 141 " package there are special procedures called vector a
nd matrix which define the required arrays. It is possible to define a
rrays outside the " }{TEXT 19 6 "linalg" }{TEXT -1 153 " package. Sinc
e we are interested in the functions contained in the linalg package, \+
we use the vector and matrix constructs from it to define the arrays.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "
" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, new definition for trace" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdeco
mpG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(
augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)ch
arpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspanG%*companionG%'co
ncatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG%(delcolsG%(delro
wsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenv
ectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponentialG%'extendG%,ffg
ausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG
%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG
%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%
'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG
%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'
mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'orthogG%*permanentG
%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%
'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smith
G%,stackmatrixG%*submatrixG%*subvectorG%)sumbasisG%(swapcolG%(swaprowG
%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%*vecpotentG%
(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 
"We displayed the commands that become available with this package." }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xv:=vector([1,4,9]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#xvG-%'vectorG6#7%\"\"\"\"\"%\"\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "yv:=vector([1,8,27]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#yvG-%'vectorG6#7%\"\"\"\"\")\"#F" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 58 "These are column vectors displayed as row
s to save space. " }}{PARA 0 "" 0 "" {TEXT -1 144 "One can refer to ve
ctor elements using an index in square brackets, which is analogous to
 list elements, but the op construct works differently:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "xv[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(xv);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "N
ote that this is not the dimension of the vector! There is a special c
ommand for that:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "vectdim(xv);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "op(xv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6
#7%\"\"\"\"\"%\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "op(o
p(xv));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$;\"\"\"\"\"$7%/F$F$/\"\"#\"
\"%/F%\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "whattype(xv)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'symbolG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 107 "If one had to access the elements via the op constr
uct, it would be a real nuisance (as compared to lists)." }}{PARA 0 "
" 0 "" {TEXT -1 47 "The inner product can be calculated as follows:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dotprod(xv,xv);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"#)*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We can \+
write a small loop to calculate this:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 6 "dp:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to 3 do
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dp:=dp+xv[i]*xv[i]; od:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "dp;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dot
prod(xv,yv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$w#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "Edit the above loop and verify the calcul
ation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
228 "Note that we have referred to the elements of the vector in the s
ame way as list elements were addressed. In fact, we could write our o
wn set of procedures, and always work with lists. We turn the do-loop \+
above into a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 111 "We pass two arguments (the vectors to be multiplied
) and check with an if-then-fi construct whether they match." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 
0 40 "dprod:=proc(arg1,arg2) local i,n1,n2,dp;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "n1:=vectdim(arg1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "n2:=vectdim(arg2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "if n1 <> \+
n2 then RETURN(`Mismatch of sizes in dprod procedure`); fi;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 6 "dp:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "for i from 1 to n1 do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dp:=d
p+arg1[i]*arg2[i]: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "dp;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "dprod(xv,xv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#)
*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Note how a procedure returns
 the last executed statement." }}{PARA 0 "" 0 "" {TEXT -1 114 "It also
 has its error checking (which could be improved to check the type of \+
arguments passed into the procedure)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "dprod(xv,[1,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(xv);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"\"\"\"%\"\"*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 102 "This worked even though [1,1,1] is a list and not
 a vector. vectdim must be working properly on lists:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "vectdim([1,2,3,4]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dpr
od(xv,[1,2,3,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%EMismatch~of~siz
es~in~dprod~procedureG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d
otprod(xv,[1,2,3,4]);" }}{PARA 8 "" 1 "" {TEXT -1 44 "Error, (in dotpr
od) arguments not compatible" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "M
aple's dotprod procedure works also with lists:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "dotprod([1,2,3],[1,2,3]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "We now def
ine a matrix:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix([[1,2,3]
,[4,5,6],[7,8,9]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrix
G6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "We can check some properties:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "trace(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We obtained the sum of t
he diagonal elements. Next we calculate the determinant:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "det(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Obviously the rows/columns
 are not independent of each other. Thus, no inverse exists." }}{PARA 
0 "" 0 "" {TEXT -1 28 "We try a small modification." }}{PARA 0 "" 0 "
" {TEXT -1 135 "Observe how elements of a matrix are addressed, and ho
w vectors and matrices are not displayed by default, but evalm has to \+
be invoked." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=A;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"BG%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "B[3,3]:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6$\"\"$F
'\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "B;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 
"A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "evalm(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m
atrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(B);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "A; e
valm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"
(\"\")\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(A-B);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"AG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%\"AG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Something
 apparently absurd happened:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 12 "We assigned " }{TEXT 326 1 "B" }{TEXT -1 27 " t
o pick up the content of " }{TEXT 327 1 "A" }{TEXT -1 33 ". Then we ch
anged one element in " }{TEXT 328 1 "B" }{TEXT -1 32 ". That should no
t have affected " }{TEXT 329 1 "A" }{TEXT -1 23 ", but obviously it di
d." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Binv:=inverse(B);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%BinvG-%'matrixG6#7%7%#!\"#\"\"$#!\"%F,\"
\"\"7%F*#\"#6F,F+7%F/F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Note
 that since the inverse exists, the system of equations" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 325 1 "B" }{TEXT -1 1 " " }
{TEXT 324 1 "x" }{TEXT -1 3 " = " }{TEXT 323 1 "b" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "has a unique solution. It
 can be obtained as follows: we pick some right-hand-side " }{TEXT 
322 1 "b" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "bv:=vec
tor([1,3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bvG-%'vectorG6#7%
\"\"\"\"\"$\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "xsol:=e
valm(Binv &* bv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xsolG-%'vector
G6#7%#\"\"\"\"\"$F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Obser
ve that the " }{TEXT 19 2 "&*" }{TEXT -1 27 " operator works inside th
e " }{TEXT 19 5 "evalm" }{TEXT -1 6 " call." }}{PARA 0 "" 0 "" {TEXT 
-1 84 "We verify the solution (a mathematician accepts a solution only
 after verification):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(B &*
 xsol - bv);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F
'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Thus we have found a vector \+
" }{TEXT 19 4 "xsol" }{TEXT -1 11 " such that " }{TEXT 19 12 "B&*xsol \+
= bv" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 116 "Of course, in pr
actice one would not perform a matrix inverse to solve a single system
 of equations, as it requires " }{TEXT 298 1 "n" }{TEXT -1 48 " times \+
more operations to find an inverse of an " }{TEXT 297 1 "n" }{TEXT -1 
4 "-by-" }{TEXT 296 1 "n" }{TEXT -1 40 " matrix than to solve the line
ar system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 194 "We demonstrate now that a homogeneous system of linear equatio
ns has interesting solutions if the coefficient matrix is singular (ot
herwise the solution is unique, and only the trivial solution " }
{TEXT 321 1 "x" }{TEXT -1 23 " = [0,0,0] is allowed)." }}{PARA 0 "" 0 
"" {TEXT -1 23 "We redefine our matrix " }{TEXT 320 1 "A" }{TEXT -1 1 
":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A:=matrix([[1,2,3],[4,5,6],[7
,8,9]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"
\"\"\"#\"\"$7%\"\"%\"\"&\"\"'7%\"\"(\"\")\"\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "zerov:=vector([0,0,0]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&zerovG-%'vectorG6#7%\"\"!F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "xsol:=vector([x1,x2,x3]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%xsolG-%'vectorG6#7%%#x1G%#x2G%#x3G" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "evalm(A&*xsol-zerov);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'vectorG6#7%,(%#x1G\"\"\"%#x2G\"\"#%#x3G\"\"$,(F(\"
\"%F*\"\"&F,\"\"',(F(\"\"(F*\"\")F,\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "solve(%,xsol);" }}{PARA 8 "" 1 "" {TEXT -1 35 "Error,
 (in solve) invalid arguments" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "
This error message means that the solve procedure does not accept vect
ors as arguments." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqs:=evalm(A&*
xsol-zerov);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG-%'vectorG6#7%,
(%#x1G\"\"\"%#x2G\"\"#%#x3G\"\"$,(F*\"\"%F,\"\"&F.\"\"',(F*\"\"(F,\"\"
)F.\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "whattype(eqs);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'symbolG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "eqs[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%#x1
G\"\"\"%#x2G\"\"#%#x3G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "sol:=solve(eqs,xsol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG6
\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "This didn't work. " }{TEXT 
19 5 "solve" }{TEXT -1 104 " does not work on lists. We need to conver
t them to sets (it escapes me why one should need to do this):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "convert(eqs,set);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#<%,(%#x1G\"\"\"%#x2G\"\"#%#x3G\"\"$,(F%\"\"(F'\"\")F
)\"\"*,(F%\"\"%F'\"\"&F)\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "sol:=solve(convert(eqs,set),convert(xsol,set));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$solG<%/%#x2G,$%#x3G!\"#/%#x1GF)/F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Note that x3 is a parameter (pick
ing a number makes the solution unique), while x1 and x2 are determine
d from it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 203 "We now proceed with the linear algebra eigenvalue problem. In \+
classical mechanics it is needed in the calculation of the principal a
xes of a rigid body (diagonalization of the moments-of-inertia tensor)
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Give
n a square (" }{TEXT 300 1 "n" }{TEXT -1 4 "-by-" }{TEXT 299 1 "n" }
{TEXT -1 9 ") matrix " }{TEXT 305 1 "A" }{TEXT -1 41 " we are seeking \+
a solution of the problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 308 1 "A" }{TEXT -1 1 " " }{TEXT 307 1 "x" }{TEXT -1 10 "
 = lambda " }{TEXT 306 1 "x" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "where " }{TEXT 309 1 "x" }{TEXT -1 5 " are " }
{TEXT 301 1 "n" }{TEXT -1 72 "-by-1 column vectors, and lambda is a sc
alar that can be complex-valued." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "One can prove that if the matrix " }
{TEXT 310 1 "A" }{TEXT -1 81 " is real and symmetric, the eigenvalues \+
lambda are real-valued. There are always " }{TEXT 302 1 "n" }{TEXT -1 
80 " of them, although the same answer can appear with a multiplicity \+
higher than 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "The proof that a solution to the problem can be found is \+
by construction. Consider the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 313 1 "B" }{TEXT -1 3 " = " }{TEXT 312 1 "A" }
{TEXT -1 10 " - lambda " }{TEXT 311 1 "I" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "where lambda is an unknown scalar,
 and " }{TEXT 314 1 "I" }{TEXT -1 25 " the unit matrix of size " }
{TEXT 304 1 "n" }{TEXT -1 4 "-by-" }{TEXT 303 1 "n" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 84 "The above problem then is a homogeneous s
ystem of equations with coefficient matrix " }{TEXT 315 1 "B" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 90 "A homogeneous system has non-t
rivial solutions only if its coefficient matrix is singular." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Thus, we can fi
nd solutions to the eigenvalue problem when det(" }{TEXT 316 1 "B" }
{TEXT -1 105 ")=0. This becomes a condition on lambda, which can then \+
be found in a straightforward way as shown below." }}{PARA 0 "" 0 "" 
{TEXT -1 21 "The solution vectors " }{TEXT 317 1 "x" }{TEXT -1 126 " t
o be found separately for each eigenvalue lambda(i) are determined fro
m the solution of the homogeneous system of equations " }{TEXT 319 1 "
B" }{TEXT -1 1 " " }{TEXT 318 1 "x" }{TEXT -1 86 " = 0. They will not \+
be unique in the sense that the length of the vector is arbitrary." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The geome
tric interpretation of the problem is as follows:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "For an arbitrary vector \+
" }{TEXT 333 1 "x" }{TEXT -1 28 " we have the property that  " }{TEXT 
334 1 "y" }{TEXT -1 3 " = " }{TEXT 335 1 "A" }{TEXT -1 1 " " }{TEXT 
336 1 "x" }{TEXT -1 38 " points in a direction different from " }
{TEXT 337 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "For a given matrix " }{TEXT 339 1 "A" }
{TEXT -1 47 " we are looking for solution vectors such that " }{TEXT 
340 1 "A" }{TEXT -1 42 " does not rotate them. The only effect of " }
{TEXT 338 1 "A" }{TEXT -1 86 " on these vectors is a stretch operation
. The amount of stretching is given by lambda." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We first define a matrix
 that is real and symmetric. We can simply add the transpose of a non-
symmetric matrix to it:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Am:=eval
m(transpose(A)+A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AmG-%'matrixG
6#7%7%\"\"#\"\"'\"#57%F+F,\"#97%F,F.\"#=" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "I3:=array(identity,1..3,1..3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#I3G-%&arrayG6&%)identityG;\"\"\"\"\"$F)7\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(I3);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Bm:=evalm(Am-lambda*I3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BmG-%'matrixG6#7%7%,&\"\"#\"\"\"%'l
ambdaG!\"\"\"\"'\"#57%F/,&F0F,F-F.\"#97%F0F3,&\"#=F,F-F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "cpoly:=det(Bm);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&cpolyG,(%'lambdaG\"#'**$)F&\"\"#\"\"\"\"#I*$)F&\"
\"$F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "The characteristic \+
polynomial is of order " }{TEXT 332 1 "n" }{TEXT -1 80 ", i.e., in our
 case of order 3. The fundamental theorem of algebra asserts that " }
{TEXT 330 1 "n" }{TEXT -1 79 " roots can be found over the field of co
mplex numbers to a polynomial of order " }{TEXT 331 1 "n" }{TEXT -1 1 
"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(cpoly,lambda);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!,&\"#:\"\"\"*$-%%sqrtG6#\"$@$F&F&
,&F%F&F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!$\"+(GZ;H$!\")$!*(GZ;HF&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The eigenvalues are real, as we co
nstructed " }{TEXT 19 2 "Am" }{TEXT -1 84 " to be symmetric. The fact \+
that 0 is an eigenvalue implies that the original matrix " }{TEXT 19 
2 "Am" }{TEXT -1 13 " is singular:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "det(Am);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 49 "The solution of roots of polynomials with highe
r " }{TEXT 342 1 "n" }{TEXT -1 56 " is difficult (impossible in closed
 form in general for " }{TEXT 341 1 "n" }{TEXT -1 50 ">5) and one has \+
to resort to numerical techniques." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "For low " }{TEXT 343 1 "n" }{TEXT -1 70 " \+
they can be found as explained above by using the following routines:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "charpoly(Am,mu);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*$)%#muG\"\"$\"\"\"F(*$)F&\"\"#F(!#IF&!#'*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(Am);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%\"\"!,&\"#:\"\"\"*$-%%sqrtG6#\"$@$F&F&,&F%F&F'!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To find the eigenvectors \+
requires more work, but for small " }{TEXT 344 1 "n" }{TEXT -1 47 " th
e following works (if the roots are simple):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "eigenvects(Am);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%
\"\"!\"\"\"<#-%'vectorG6#7%F%!\"#F%7%,&\"#:F%*$-%%sqrtG6#\"$@$F%F%F%<#
-F(6#7%,&#!#5\"#<F%F/#F%F;,&#\"\"(\"#MF%F/#F%F@F%7%,&F.F%F/!\"\"F%<#-F
(6#7%,&F9F%F/#FDF;,&F>F%F/#FDF@F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
110 "The structure of the answer is: eigenvalue, multiplicity, set of \+
eigenvectors (even if the multiplicity is 1);" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "The answer for lambda=0 is easy to read." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "To obtain numerically \+
calculated eigenvalues (and eigenvectors) consult the following help p
age:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?Eigenvals" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }