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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "                  " }
{TEXT 256 24 "Complex Line Integrals I" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 53 "Part 1:  The definition of \+
the complex line integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "Load the plots package." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plot
s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 39 "1.  First we define the four functions." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f
1:=z->1/z;\nf2:=z->z^2;\nf3:=z->(conjugate(z))^2;\nf4:=z->exp(z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Our first curve " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"
" }{TEXT -1 35 " will be the piece of the parabola " }{XPPEDIT 18 0 "y
=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "0
" "6#\"\"!" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "1+i" "6#,&\"\"\"F$%\"
iGF$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 55 "Z1:=t->t+t^2*I;\ncomplexplot(Z1(t),t=0..1
, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The complex line integral of the s
quare function,  " }{XPPEDIT 18 0 "f[2]" "6#&%\"fG6#\"\"#" }{TEXT -1 
8 ",  over " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT -1 21 " i
s calculated below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 81 "f:=f2; f(z);\nZ:=Z1; Z(t);\na:=0; b:=1;\n
Int(f(Z(t))*diff(Z(t),t),t=a..b);\nevalf(%);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "We chec
k that the value of the integral is independent of the parametric repr
esentation of  the curve " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }
{TEXT -1 12 ". Enter (as " }{XPPEDIT 18 0 "Gamma1" "6#%'Gamma1G" }
{TEXT -1 51 ") another parametric representation for this curve." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 65 "Gamma1:=t->t^2+t^4*I;\ncomplexplot(Gamma1(t),t=0..1, thickness=2
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 31 "Calculate the line integral of " }{XPPEDIT 18 0 "f
[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "C[1];" "6
#&%\"CG6#\"\"\"" }{TEXT -1 38 " using this parametric representation.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f:=f2; f(z);\nZ:=Gamma1
; Z(t);\na:=0; b:=1;\nInt(f(Z(t))*diff(Z(t),t),t=a..b);\nevalf(%);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 52 "Now enter yet another parametric representation for " }
{XPPEDIT 18 0 "C[1];" "6#&%\"CG6#\"\"\"" }{TEXT -1 75 " , and repeat t
he calculation. (You will need to decide on the formula for " }
{XPPEDIT 18 0 "Gamma[2];" "6#&%&GammaG6#\"\"#" }{TEXT -1 43 " and the \+
bounds on the parametric interval." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "Gamma2:=t->??+??*I;\ncomplexplot(Gamma1(t),t=??..??, \+
thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "You need to enter the bounds on th
e parametric interval before calculating the integral below." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f:=f2; f(z);\nZ:=Gamma2; Z(t
);\na:=??; b:=??;\nInt(f(Z(t))*diff(Z(t),t),t=a..b);\nevalf(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 44 "2. Calculate the values of the integral of  " }{XPPEDIT 
18 0 "f4" "6#%#f4G" }{TEXT -1 55 "  over each of the three parametric \+
representations of " }{XPPEDIT 18 0 "C[1]" "6#&%\"CG6#\"\"\"" }{TEXT 
-1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 62 "3.  Here are the parametric representations of the two curves \+
" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "C[2]" "6#&%\"CG6#\"\"#" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "C[3]" "6#&%\"CG6#\"\"$" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "Z2:=t->t+t*I;\nZ3:=t->cos(5*t+3*Pi/2)-sin(3*t)*I;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "curve2:=complexplot(Z2(t),t=0..1, \+
thickness=2):\ncurve3:=complexplot(Z3(t),t=0..Pi/2, thickness=2, color
=blue):\ndisplay(curve2,curve3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Record your r
esults and observations here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "     Integral of " }{XPPEDIT 18 0 "f[2];
" "6#&%\"fG6#\"\"#" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "C[2];" "6#&%
\"CG6#\"\"#" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "     Integral of " }{XPPEDIT 18 0 "f[2];
" "6#&%\"fG6#\"\"#" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "C[3];" "6#&%
\"CG6#\"\"$" }{TEXT -1 4 "    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "     Integral of " }{XPPEDIT 18 0 "f[4];
" "6#&%\"fG6#\"\"%" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "C[2];" "6#&%
\"CG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 17 "     Integral of " }{XPPEDIT 18 0 "f[4];" "6#&%
\"fG6#\"\"%" }{TEXT -1 6 " over " }{XPPEDIT 18 0 "C[3];" "6#&%\"CG6#\"
\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 13 "Observations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 14 "4.  The curve " }{XPPEDIT 18 0 "C[4]" "6#&%\"CG6#\"\"%" }
{TEXT -1 32 " may be parametrized as follows:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Z4:=t->cos
(t)+sin(t)*I;\ncurve4:=complexplot(Z4(t),t=0..Pi, thickness=2,view=[-1
..1,-1..1]):\ndisplay(curve4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Enter a parametric rep
resentation " }{XPPEDIT 18 0 "Z5" "6#%#Z5G" }{TEXT -1 15 " for the cur
ve " }{XPPEDIT 18 0 "C5" "6#%#C5G" }{TEXT -1 28 " and evaluate the int
egrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "Z5:=t->??+??*I;\ncurve5:=complexplot(Z5(t),t=0..Pi,vi
ew=[-1..1,-1..1]):\ndisplay(curve5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Calculate the
 line integrals of " }{XPPEDIT 18 0 "f[1]" "6#&%\"fG6#\"\"\"" }{TEXT 
-1 9 " through " }{XPPEDIT 18 0 "f[4]" "6#&%\"fG6#\"\"%" }{TEXT -1 94 
" and record the results. The code here calculates the integrals numer
ically -- to accommodate " }{XPPEDIT 18 0 "f[3]" "6#&%\"fG6#\"\"$" }
{TEXT -1 43 ". Pay attention to the size of the results." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "f:=f1; f(z);\nZ:=Z4; Z(t);\n
a:=0; b:=Pi;\nInt(f(Z(t))*diff(Z(t),t),t=a..b);\nevalf(%);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f:=f1; f(z);\nZ:=Z5; Z(t);\n
a:=0; b:=Pi;\nInt(f(Z(t))*diff(Z(t),t),t=a..b);\nevalf(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "Record your results here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "     Integrals for " }{XPPEDIT 18 0 "f[1]
;" "6#&%\"fG6#\"\"\"" }{TEXT -1 11 ":     over " }{XPPEDIT 18 0 "C[4];
" "6#&%\"CG6#\"\"%" }{TEXT -1 26 "                     over " }
{XPPEDIT 18 0 "C[5];" "6#&%\"CG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "     Integrals for \+
" }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 11 ":     over " 
}{XPPEDIT 18 0 "C[4];" "6#&%\"CG6#\"\"%" }{TEXT -1 26 "               \+
      over " }{XPPEDIT 18 0 "C[5];" "6#&%\"CG6#\"\"&" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "     In
tegrals for " }{XPPEDIT 18 0 "f[3];" "6#&%\"fG6#\"\"$" }{TEXT -1 11 ":
     over " }{XPPEDIT 18 0 "C[4];" "6#&%\"CG6#\"\"%" }{TEXT -1 26 "   \+
                  over " }{XPPEDIT 18 0 "C[5];" "6#&%\"CG6#\"\"&" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "     Integrals for " }{XPPEDIT 18 0 "f[4];" "6#&%\"fG6#\"
\"%" }{TEXT -1 11 ":     over " }{XPPEDIT 18 0 "C[4];" "6#&%\"CG6#\"\"
%" }{TEXT -1 26 "                     over " }{XPPEDIT 18 0 "C[5];" "6
#&%\"CG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 23 "Part 2: Experimentation
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Recor
d the results of your applet experiments here." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 257 19 "Part 3: Calculation" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "1.  Enter your comparisons here
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 88 "Modify the instructions below to carry out the calculatio
ns requested in Steps 2 and 3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Z6:=t->cos(t)+sin(t)*I;\ncom
plexplot(Z6(t),t=0..2*Pi);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "f:=f1; f(z);\nZ:=Z6; Z(t);\na:=0; b:=1;\nInt(f(Z(t))*diff(Z(t),t
),t=a..b);\nevalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "2. Record the results of your co
mputations and the comparisons here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "3. Record the results of
 your computations and the comparisons here." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 31 "4. Enter your explanation here." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 7 "Summary
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 27 "1. Write your summary here." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "2. Make \+
your conjecture here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "3. Define your functions and carry
 out your test here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "4.  State y
our final conjecture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }