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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 
"" 0 "" {TEXT 256 42 "Maple for Line Integrals and Vector Fields" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Here's how to use   Maple    to h
elp visualize and also evaluate the line integral  of the vector field
                        " }}{PARA 0 "" 0 "" {TEXT -1 75 "             \+
                                          (P, Q) = (x-y, x+y) " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "over a c
urve---say the upper half of the unit circle..      (This vector field
 is illustrated in  the book in several places. )" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "First, let's plot the vec
tor field, using the   fieldplot command: " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x := 'x'; y := 'y'; P := x-
y;  Q := x+y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fieldplot(
 [P,Q], x=-3..3, y=-3..3, grid=[10,10] );" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 71 "Now let's integrate this vector field along a curve; say \+
the upper half" }}{PARA 0 "" 0 "" {TEXT -1 66 "of the unit circle.    \+
The first step is to parametrize the curve:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "x := cos(t);  y := sin(t);" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Now we find the  \+
dx  and   dy   parts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dx
 := diff(x,t);   dy := diff(y,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
68 "Now we put it all together, integrating   Pdx + Qdy  over the curv
e:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int( P*dx + Q*dy, t =
 0 .. Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 73 "Note that the integral is taken from t=0 \+
to t=Pi because that corresponds" }}{PARA 0 "" 0 "" {TEXT -1 40 "to th
e UPPER HALF of the unit circle.   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 66 "Note also that the answer is positive --
- could you have predicted" }}{PARA 0 "" 0 "" {TEXT -1 36 "this from t
he field  plot   picture?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 68 "One can now change any part of  what's above to ca
lculate other line" }}{PARA 0 "" 0 "" {TEXT -1 141 "integrals.       B
e sure to start at the top (where P and Q are defined)... otherwise Ma
ple might get confused about conflicting definitions." }}}}{MARK "5 0 \+
0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }