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{SECT 0 {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 22 "restart; with(pl
ots); " }{MPLTEXT 1 217 0 "" }}}{SECT 1 {PARA 207 "" 0 "" {TEXT 218 
12 "Integration " }{TEXT 218 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 
219 42 "Maple can integrat e functions numerically" }{TEXT 219 0 "" }}
}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 19 "int(x^2, x = 0..1);" }
{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 220 15 "or symbo
lically" }{TEXT 220 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 
11 "int(x^2,x);" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 208 "" 0 "" 
{TEXT 219 115 "Maple can integrate complex functions to give a complex
 number.  It simply integrates the real and imaginary parts." }{TEXT 
219 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 26 "int(x^2 + I*
 x, x = 0..1);" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 208 "" 0 "" 
{TEXT 219 46 "You can define piecewise continuous functions." }{TEXT 
219 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 52 "f := piecewi
se(t<0 , 0 , t<1 and t>0, t^2, t>1 , 1);" }{MPLTEXT 1 217 0 "" }}}
{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 19 "plot(f, t = -2..2);" }
{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 221 16 "You can \+
use the " }{TEXT 222 9 "piecewise" }{TEXT 221 78 " function to define \+
square waves.   Alos useful is a built in function called " }{TEXT 
221 10 "Heaviside." }{TEXT 221 0 "" }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 223 31 "plot(Heaviside(t), t = -2 ..2);" }{MPLTEXT 1 223 0 
"" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 223 52 "plot(Heaviside(t-1
/2) - Heaviside(t-1), t = -2 ..2);" }{MPLTEXT 1 223 0 "" }}}}{SECT 1 
{PARA 207 "" 0 "" {TEXT 218 14 "Fourier Series" }{TEXT 218 0 "" }}
{EXCHG {PARA 208 "" 0 "" {TEXT 219 137 "Maple can compute a few coeffi
cients for the Fourier series of x^2 in complex form.  We approximate \+
it just in the interval from 0 to 1. " }{TEXT 219 0 "" }}}{EXCHG 
{PARA 206 "> " 0 "" {MPLTEXT 1 217 10 "f :=  x^2;" }{MPLTEXT 1 217 0 "
" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 221 11 "By setting " }{TEXT 224 7 
"N := 5 " }{TEXT 221 44 "we compute 10 terms of the Fourier series.  "
 }{TEXT 221 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 6 "N:= 5
:" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 
75 "for k from -N to N do c[k] := evalf(int(f*exp(-I*k*x*(2*Pi)), x=0 \+
..1)) od:" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 219 
56 "Now compute the fourier series using the coefficients.  " }{TEXT 
219 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 217 55 "FourierSerie
s := sum(c[j]*exp(I*j*x*2*Pi), j = -N ..N);" }{MPLTEXT 1 217 0 "" }}}
{EXCHG {PARA 208 "" 0 "" {TEXT 219 126 "We now plot the real part of t
he fourier series.  Actually, since the function is real, the imaginar
y part of the series is 0." }{TEXT 219 0 "" }}}{EXCHG {PARA 206 "> " 0
 "" {MPLTEXT 1 217 77 "RealPartOfFourierSeries := evalc(FourierSeries \+
+ conjugate(FourierSeries))/2:" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 
206 "> " 0 "" {MPLTEXT 1 217 44 "plot(\{f,RealPartOfFourierSeries\}, x
 = 0..1);" }{MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 219 
25 "Now try problems 1 and 2." }{TEXT 219 0 "" }}}{EXCHG {PARA 206 "> \+
" 0 "" {MPLTEXT 1 217 0 "" }}}{EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 
217 0 "" }}}}{SECT 1 {PARA 212 "" 0 "" {TEXT -1 24 "Multiple variable \+
series" }}{EXCHG {PARA 213 "" 0 "" {TEXT 201 150 "Construct multiple v
ariable series in a similar way.  Use nested sums and loops to do this
.  Be sure to set you indexing variables back to variables. " }}}
{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 0 60 "j := 'j': k := 'k': sum(su
m(a[j,k],j = -2 ..2), k = -2 ..2);" }}}{EXCHG {PARA 214 "> " 0 "" 
{MPLTEXT 1 -1 85 "j := 'j': k := 'k': for j from -2 to 2 do for k from
 -2 to 2 do print(a[j,k]) od; od;" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
213 "" 0 "" {TEXT 201 26 "plot thes functions using " }{TEXT 225 12 "c
ountourplot" }{TEXT 201 24 "  for two variables and " }{TEXT 225 8 "im
plicit" }{TEXT 225 6 "plot3d" }{TEXT 201 20 " for three variables" }}}
{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 0 43 "contourplot(x^2 + y^2, x =
 0..1, y = 0..1);" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 0 66 "impl
icitplot3d(x^2 + y^2 + z^2 =1 , x = 0..1, y = 0..1, z = 0..1);" }}}}
{PARA 215 "" 0 "" {TEXT 226 0 "" }}{PARA 216 "" 0 "" {TEXT -1 0 "" }}}
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